CHAPTER 7 𝔏 AND ^{#}𝔏 ON THE CONLEY INDICES OF NORMALLY

HYPERBOLIC INVARIANT SUBMANIFOLDS

Throughout this chapter assume M to be of class C^{r+1} and ξ to be a vectorfield on M of class C^{r} where r ≥ 1. As before dim(M) = m. For simplicity, all homology is taken with coefficients in R. Also, for each integer k ≥ 0, let D_{c}^{k} denote the closed k-disk of radius c centered at 0 ∈ R^{k} and let S_{c}^{k−1} denote its bounding k − 1-sphere. In particular, S_{c}^{−1} = ∂D_{c}^{0} = ∅. As usual, D^{k} := D_{1}^{k} and S^{k−1} := S_{1}^{k−1}. Also, the standard inner product and its associated norm on a Euclidean space are respectively denoted by ⟨· | ·⟩ and ∥ · ∥.

A. Summary of Results.

7.1 Definition. Let η = (E, p, X) be a vector bundle of rank k (k ≥ 0) over X paracompact, choose a Euclidean metric ∥ · ∥ for E, and let E_{≥1} := {v ∈ E : ∥v∥ ≥ 1}. The Thom space of η, denoted T(η), is the quotient space E/E_{≥1} with basepoint [E_{≥1}]. The definition does not depend on the choice of Euclidean metric: this is obvious for k = 0; for k ≥ 1 see [Hus, Definition 15.1.2]. Note that when k = 0, T(η) ≡ X ⨿ {∗} with basepoint ∗ ∉ X but is not a homeomorph of X ∨ S^{0} where S^{0} is identified with {x_{0}} ∪ {∗} for any choice of basepoint x_{0} ∈ X because as pointed spaces they are not homotopy equivalent even though as unpointed spaces they are homeomorphic. The lack of homeomorphism in the pointed case complicates the statement of Theorem 7.6 below.

7.2 Definition. (A) Let S be a compact C^{1} submanifold without boundary of M invariant under the action of a C^{1} flow ψ in M. Call S a normally hyperbolic invariant submanifold of ψ if, and only if, there is a continuous splitting T M|S = N_{S}^{u} ⊕ T S ⊕ N_{S}^{s} so that for each t ∈ R, T ψ_{t} leaves invariant N_{S}^{u}, N_{S}^{s}, and T S, and for each u ∈ S, there exists τ ≡ τ(u) ∈ R so that ∥T_{u}ψ_{τ} |N_{S}^{s}∥ < min{m(T_{u}ψ_{τ} |T S), 1} and ∥T_{u}ψ_{τ} |T S∥ > max{m(T_{u}ψ_{τ} |N_{S}^{u}), 1} where for a linear map between normed spaces m(L) = inf{∥Lv∥ : ∥v∥ = 1}. Note m(L) = ∥L^{−1}∥^{−1} when L is invertible. This definition is taken from [R] and is equivalent to ψ_{t} being eventually, relatively 1-normally hyperbolic for each t ∈ R as defined in [HPS]. Let us refer to N_{S}^{u} and N_{S}^{s} as the expanding and contracting subbundles of the normal bundle of S.

(B) Let ψ and S be as in (A). The strong stable manifold at u ∈ S, denoted W^{ss}(u), consists of those points v ∈ M with the property that the distance between ψ_{t}(v) and ψ_{t}(u) goes to zero as t → ∞ at an asymptotic rate that is at least as fast as ∥T_{u}ψ_{τ} |N_{S}^{s}∥. The strong unstable manifold at u is defined similarly by reversing the time direction. For u a point in an isolated invariant set that is also normally hyperbolic and which admits a block B, the local strong stable manifold within B at u , denoted W^{ss}(u; B), consists of those points in W^{ss}(u) whose positive semiorbits under the flow lie in B. The local strong unstable manifold within B at u is defined analogously, but using negative semi-orbits. For λ = u, s, W^{λ}(S) is fibered by the family {W^{λλ}(u) : u ∈ S}. When B is a block with corners, transversality at the boundary implies that W^{λλ}(u; B) (λ = u, s) is a neat submanifold of B; also {W^{λλ}(u; B) : u ∈ S} fibers W^{λ}(S; B).

7.3 Definition. Let S be a closed, connected submanifold of M that is orientable over R and normally hyperbolic relative to the flow generated by a vectorfield on an open neighborhood of S, and assume that N_{S}^{u} and N_{S}^{s} are R-orientable. Then associated (i) to a choice of R-orientation classes for the expanding and contracting subbundles of the normal bundle, (ii) to a choice of fundamental class for S, and (iii) to the given orientation O^{M} of M is a sign σ (i.e., either σ = 1 ∈ R or σ = −1 ∈ R) that arises in the computation of the intersection pairing on the Conley indices of S. The sign σ is defined as follows.

Set ℓ := dim(S), let o_{S} ∈ H_{ℓ}(S; R) denote the chosen fundamental class of S, and let ν_{S} ∈ H^{ℓ}(S; R) denote the generator satisfying ⟨ν_{S}, o_{S}⟩ = 1. Next, identify the normal bundle of S with a total tubular neighborhood N_{S} = (U, p, S) of S, and write N_{S} = N_{S}^{u} ⊕ N_{S}^{s}. Under these identifications, for λ = u, s, let E_{S}^{λ} ⊂ U be the total space of N_{S}^{λ}, assume N_{S}^{λ} has rank k_{λ} ≥ 0, and let τ^{λ} ∈ H^{k_{λ}}(E_{S}^{λ}, E_{S}^{λ} \ S; R) be an R-orientation of N_{S}^{λ}. As the Whitney sum N_{S}^{u} ⊕ N_{S}^{s} is the pull-back by the diagonal map on S of the product bundle N_{S}^{u} × N_{S}^{s} and where Δ̄ is the map on U to the total space of that product bundle induced by the pull-back operation, Δ̄^{∗}(τ^{u} ×τ^{s}) is an orientation class of (U, p, S). Thus, p^{∗}ν_{S} ⌣ Δ̄^{∗}(τ^{u} ×τ^{s}) generates H^{m}(U, U \S; R) as a consequence of the Thom Isomorphism Theorem. On the other hand, let o_{S}^{U} be the fundamental class generating H_{m}(U, U \ S; R) that under the excision induced isomorphism has image o_{S}^{M} ∈ H_{m}(M, M \ S; R) where o_{S}^{M} is the fundamental class along S determined by the given orientation O^{M} of M. Define

σ ≡ σ(τ^{u}, τ^{s}, o_{S}, O^{M}) := ⟨p^{∗}ν_{S} ⌣ Δ̄^{∗}(τ^{u} × τ^{s}), o_{S}^{U}⟩ .

As defined, σ ostensibly depends on the choice of tubular neighborhood of S. That it does not is a consequence of the fact (e.g., see [H, Theorem 4.5.3]) that any two tubular neighborhoods of S are isotopic in M; details are left to the reader.

7.4 Theorem. For S as in Definition 7.3, the following hold:

(1) S is an isolated invariant set of the flow generated by ξ;

(2) [C (S)] = [T(N_{S}^{u})] and [C∗(S)] = [T(N_{S}^{s})];

(3) H̃_{k}C (S; R) ≃ R for k = k_{u}, k_{u} +ℓ and is the zero module for k < k_{u} and k > k_{u} + ℓ; in particular, each R-orientation class τ^{u} of N_{S}^{u} determines a unique generator o_{k_{u}}^{u} of H̃_{k_{u}}C (S; R), and τ^{u} and a choice of fundamental class o_{S} ∈ H_{ℓ}(S; R) together determine a unique generator o_{k_{u}+ℓ}^{u} of H̃_{k_{u}+ℓ}C (S; R);

(4) H̃_{k}C∗(S; R) ≃ R for k = k_{s}, k_{s} + ℓ and is the zero module for k < k_{s} and for k > k_{s} + ℓ; in particular, each R-orientation class τ^{s} of N_{S}^{s} determines a unique generator o_{k_{s}}^{s} of H̃_{k_{s}}C∗(S; R), and τ^{s} and a choice of fundamental class o_{S} ∈ H_{ℓ}(S; R) together determine a unique generator o_{k_{s}+ℓ}^{s} of H̃_{k_{s}+ℓ}C∗(S; R); (5) given R-orientations τ^{u} of N_{S}^{u} and τ^{s} of N_{S}^{s} and given a fundamental class

o_{S} of S, then, where σ = σ(τ^{u}, τ^{s}, o_{S}, O^{M}), the corresponding generators o_{k}^{λ} (k = k_{λ}, k_{λ} + ℓ; λ = u, s) satisfy

#(o_{k_{u}+ℓ}^{u} ⌢_{•} o_{k_{s}}^{s}) = σ , ^{#}(o_{k_{u}}^{u} ⌢_{•} o_{k_{s}+ℓ}^{s}) = (−1)^{ℓk_{u}}σ

and

o_{k_{u}+ℓ}^{u} ⌢_{•} o_{k_{s}+ℓ}^{s} = σo_{S} ;

(6) there exists a cofinal family 𝔅(S) of neighborhoods of S so that for each B ∈ 𝔅(S), B is a C^{r} isolating block with corners for S, and in the sense of Definition 6.7, W^{λ}(S; B) geometrically represents o_{k_{λ}+ℓ}^{λ} and, for each u ∈ S, W^{λλ}(u; B) geometrically represents o_{k_{λ}}^{λ} (λ = u, s).

As stated in the Introduction, the proof of Theorem 7.4 will be carried out in full only for the special cases of S a hyperbolic critical point and S a hyperbolic closed orbit. These two cases are stated below as Theorems 7.5–6. A proof of the general case will be given elsewhere and depends on the construction of an isolating block B as described in the Introduction that is the total space of the Whitney sum of two disk bundles.

The version of Theorem 7.4 for hyperbolic critical points will be stated based on the following:

Hypotheses HCP. The point u_{0} is a critical point of ξ whose Hessian dξ at u_{0} has (counting multiplicities) P eigenvalues with positive real part and N with negative real part where P + N = m.

For the definition of the Hessian of a vectorfield at a critical point as just used see [Ab-R].

As is well-known, u_{0} satisfies hypotheses HCP if, and only if, u_{0} is a hyperbolic critical point of ξ in which case the singleton S := {u_{0}} is certainly an R-orientable, normally hyperbolic invariant submanifold where in the notation of Definition 7.3, k_{u} = P, k_{s} = N and ℓ = 0. Also, it is immediate from the definitions that the local stable and strong local stable manifolds coincide in the critical point case and similarly for the local unstable manifolds. Since the Thom space of a vector bundle of rank k over a point is obviously a pointed k-sphere, Theorem 7.4 has as corollary the following result where in its statement ∗ ∈ S^{k} denotes any chosen basepoint.

7.5 Theorem. If u_{0} ∈ M satisfies hypotheses HCP, then the conclusions of Theorem 7.4 hold for S := {u_{0}} where k_{u} = P, k_{s} = N, and ℓ = 0. In particular

(1) [C (S)] = [(S^{P} , ∗)] and [C∗(S)] = [(S^{N}, ∗)];

(2) given R-orientations τ^{u} of E_{S}^{u} and τ^{s} of E_{S}^{s}, the corresponding generators o_{P}^{u} of H̃_{P} C (S; R) and o_{N}^{s} of H̃_{N} C∗(S; R) satisfy

#(o_{P}^{u} ⌢_{•} o_{N}^{s}) = σ(τ^{u}, τ^{s}, [u_{0}], O^{M})

where [u_{0}] is the homology class of u_{0} regarded as a 0-simplex;

(3) if M = R^{m}, o_{P}^{u} # o_{N}^{s} generates H_{m}(R^{m}, R^{m} \ {0}; R).

The manner in which the orientation classes τ^{u} and τ^{s} determine the generators o_{P}^{u} and o_{N}^{s} is described below in Lemma 7.13 and the proof of Theorem 7.15.

The version of Theorem 7.4 for hyperbolic closed orbits will be stated based on the following:

Hypotheses HCO. S is a closed orbit of ξ with positive prime period τ that has (counting multiplicities) P Floquet multipliers with modulus greater than one and N Floquet multipliers with modulus less than one where P + N = m − 1.

As is well-known, S is a hyperbolic closed orbit of ξ if, and only if, hypotheses HCO are satisfied in which case S is an R-orientable, normally hyperbolic invariant submanifold relative to ξ where in the notation of Definition 7.3, k_{u} = P, k_{s} = N, and ℓ = 1. Thus, Theorem 7.4 has with a bit of work (see Lemma 7.24 and Propositions 7.25–26) the following result as corollary where in its statement, for any integer k ≥ 0, S^{k}RP^{2} denotes the k^{th} iterated reduced suspension of the projective plane relative to any choice of basepoint ∗ ∈ RP^{2}.

7.6 Theorem. If S satisfies hypotheses HCO, then the conclusions of Theorem 7.4 hold with k_{u} = P, k_{s} = N, and ℓ = 1. In particular,

(1) either P = 0 and [C (S)] = [S^{1} ⨿ {∗}] or P ≥ 1 and, as N_{S}^{u} is or is not orientable over Z, either [C (S)] = [S^{P +1} ∨ S^{P}] or [C (S)] = [S^{P −1}RP^{2}];

(2) either N = 0 and [C∗(S)] = [S^{1} ⨿ {∗}] or N ≥ 1 and, as N_{S}^{s} is or is not orientable over Z, either [C∗(S)] = [S^{N+1} ∨ S^{N}] or [C∗(S)] = [S^{N−1}RP^{2}];

(3) given R-orientations τ^{u} of N_{S}^{u} and τ^{s} of N_{S}^{s} and given a fundamental class o_{S} of S, then, where σ ≡ σ(τ^{u}, τ^{s}, o_{S}, O^{M}), the corresponding generators o_{j}^{u} of H̃_{j}C (S; R) (j = P, P + 1) and o_{k}^{s} of H̃_{k}C∗(S; R) (k = N, N + 1) satisfy

#(o_{P +1}^{u} ⌢_{•} o_{N}^{s}) = σ , ^{#}(o_{P}^{u} ⌢_{•} o_{N+1}^{s}) = (−1)^{P} σ ,

and

o_{P +1}^{u} ⌢_{•} o_{N+1}^{s} = σ o_{S} .

The manner in which the orientation classes τ^{u} and τ^{s} determine the generators o_{P}^{u} and o_{N}^{s} and the manner in which τ^{u} and τ^{s} together with a fundamental class o_{S} determine o_{P +1}^{u} and o_{N+1}^{s} is described below in Lemmas 7.28 and 7.29 and in the proof of Theorem 7.31.

Note that statement (1) of Theorem 7.5 implies that statement (2) of that theorem completely determines 𝔏 and ^{#}𝔏 for S = {u_{0}} where u_{0} satisfies hypotheses HCP. Similarly, statements (1) and (2) of Theorem 7.6 imply that statement (3) of that theorem completely determines 𝔏 and ^{#}𝔏 when S satisfies hypotheses HCO.

B. Computational Preliminaries. Made here are the estimates needed to construct isolating blocks under the assumption of hypotheses HCP or HCO and the computations underlying statement (5) of Theorem 7.4 and therefore of statement (2) of Theorem 7.5 and statement (3) of Theorem 7.6.

The first result of this subsection will in effect compute the intersection pairing for hyperbolic critical points and hyperbolic periodic orbits. For other R-orientable, normally hyperbolic invariant submanifolds it only computes the intersection pairing on generating elements of H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S) of lowest and highest order; no more can be said without additional knowledge about the homology of S although it will clear from the proof how to use such additional information when available to obtain additional values of the intersection pairing. The result requires hypotheses, stated presently, that motivate the construction of isolating blocks made in Theorems 7.11 and 7.21. In stating them, for reasons that will become clear in the course of stating and proving Theorem 7.21 below, when reference is made to a k-disk bundle, it is not assumed that the structure group is O(k) or a subgroup thereof, but only that the structure group is some subgroup of Homeo(R^{k}) whose elements preserve the Euclidean norm. Also, if η is a k-disk bundle, its associated k − 1-sphere bundle is denoted by η̇.

Hypotheses and Definitions NHIS.

(1) S is a closed, connected, ℓ-dimensional, R-orientable C^{r} submanifold of M;

(2) N_{S} = (U, p, S) is a tubular neighborhood of S and admits a continuous splitting N_{S} = N_{S}^{u} ⊕ N_{S}^{s} where N_{S}^{λ} is a k_{λ}-plane bundle with total space E_{S}^{λ} ⊂ U (λ = u, s);

(3) τ^{λ} ∈ H^{k_{λ}}(E_{S}^{λ}, E_{S}^{λ} \ S; R) is an R-orientation class for N_{S}^{λ} (λ = u, s);

(4) o_{S} ∈ H_{ℓ}(S; R) is a fundamental class of S;

(5) S is an isolated invariant set of some flow in U, and N (S), the family of neighborhoods of S, admits a cofinal family 𝔅𝔚 with each B ∈ 𝔅𝔚 a C^{r} isolating block with corners having the properties:

(a) B is the total space of a subbundle β of N_{S}, and the splitting of N_{S} induces a splitting of β = ω^{u} ⊕ω^{s} with ω^{λ} a k_{λ}-disk bundle (λ = u, s);

(b) B^{+} is the total space of ω̇^{u} ⊕ ω^{s} and B^{−} is the total space of ω^{u} ⊕ ω̇^{s};

(c) W^{λ} := W^{λ}(S; B) is the total space of ω^{λ} (λ = u, s) and W^{u} and W^{s} are topologically transverse at some point u ∈ S.

For λ = u, s, note that W^{λ} is a manifold with boundary and that ∂W^{λ} is the total space of ω̇^{λ}. Use p_{λ}: E_{S}^{λ} → S to denote the restriction of p to E_{S}^{λ}, and use W_{u}^{λ} and ∂W_{u}^{λ} to denote respectively the fiber of ω^{λ} at u and the fiber of ω̇^{λ} at u. Hypotheses NHIS(1)–(5), the Thom Isomorphism Theorem, and the observations that (W^{u}, ∂W^{u}) ⊂ (B, B^{+}) and that (W^{s}, ∂W^{s}) ⊂ (B, B^{−}) allow the definition of generating classes for H̃_{∗}C (S) and H̃_{∗}C∗(S) as follows:

(6) for λ = u, s, let o_{k_{λ}}^{λ} be the generator of H_{k_{λ}}(W_{u}^{λ}, ∂W_{u}^{λ}; R) satisfying

⟨τ^{λ}|(W_{u}^{λ}, ∂W_{u}^{λ}), o_{k_{λ}}^{λ}⟩ = 1 ;

(7) for λ = u, s, let o_{k_{λ}+ℓ}^{λ} be the generator of H_{k_{λ}+ℓ}(W^{λ}, ∂W^{λ}; R) satisfying

p_{λ ∗}(τ^{λ}|(W^{λ}, ∂W^{λ}) ⌢ o_{k_{λ}+ℓ}^{λ}) = o_{S} ;

(8) let o_{k_{u}}^{u} and o_{k_{u}+ℓ}^{u} be the respective canonical images (see Definition 6.7) in H̃_{∗}C (S) of o_{k_{u}}^{u} and o_{k_{u}+ℓ}^{u} and assume these are independent of B ∈ 𝔅𝔚;

(9) let o_{k_{s}}^{s} and o_{k_{s}+ℓ}^{s} be the respective canonical images in H̃_{∗}C∗(S) of o_{k_{s}}^{s} and o_{k_{s}+ℓ}^{s} and assume these are independent of B ∈ 𝔅𝔚.

7.7 Lemma (cf. [D, VIII.11.13 and VIII.13.23]). Under the assumption of hypotheses and definitions NHIS and with σ as in Definition 7.3

#(o_{k_{u}+ℓ}^{u} ⌢_{•} o_{k_{s}}^{s}) = σ , ^{#}(o_{k_{u}}^{u} ⌢_{•} o_{k_{s}+ℓ}^{s}) = (−1)^{ℓk_{u}}σ

and

o_{k_{u}+ℓ}^{u} ⌢_{•} o_{k_{s}+ℓ}^{s} = σo_{S} .

Proof. The computations are done in three steps: first, relative to a suitable chart, local representatives of the various orientation and fundamental classes are determined; second, formal computations are made which if correct give the desired results; third, the formal computations are justified via a lengthy diagram chase. Throughout, the following facts are used without further explicit comment:

(1) the Kronecker pairing is invariant under isomorphisms on homology and cohomology induced by inclusions and homeomorphisms;

(2) intersection pairings on arbitrary pairs of pairs in a manifold are natural with respect to inclusions and with respect to orientation preserving homeomorphisms; see Lemma 2.10 and [D, Proposition VIII.13.21].

To begin, choose a coordinate chart (Q, ψ) at u with the following properties: Q ⊂ U, ψ(Q) = R^{ℓ} × R^{k_{u}} × R^{k_{s}}, ψ(u) = (0, 0, 0), ψ(Q ∩ S) = R^{ℓ} × {0} × {0}, ψ(E_{S}^{u} ∩Q) = R^{ℓ} ×R^{k_{u}} ×{0}, and ψ(E_{S}^{s} ∩Q) = R^{ℓ} ×{0}×R^{k_{s}}, and ψ trivializes the restricted bundles N_{S}|(S ∩ Q) and N_{S}^{λ}|(S ∩ Q) (λ = u, s); in particular p^{−1}(v) ⊂ Q for each v ∈ Q ∩ S. To satisfy the last property, it may be necessary to first shrink the tubular neighborhood U along those fibers in and near Q∩S. Afterward choose B ∈ 𝔅𝔚 satisfying B ⊂ U; below, the classes o_{k_{λ}}^{λ} and o_{ℓ+k_{λ}}^{λ} (λ = u, s) are the ones defined using this B. To facilitate printing set R_{0}^{k} := R^{k} \ {0} for each integer k ≥ 1. Below, i_{∗} and i^{∗} always denote inclusion induced maps arising from the inclusion Q ⊂ U.

First, let π_{λ}: U → E_{S}^{λ} denote the continuous projection arising from the splitting, set π_{Q u} := ψ^{∘} π_{u}|Q^{∘} ψ^{−1}, and consider the ladder of isomorphisms

H^{k_{u}}(U, U \ E_{S}^{s})) ^{i^{∗}}_{≃} H^{k_{u}}(Q, Q \ E_{S}^{s})) ^{ψ^{−1 ∗}}_{≃} H^{k_{u}}(R^{ℓ} × (R^{k_{u}}, R_{0}^{k_{u}}) × R^{k_{s}}) H^{k_{u}}(E_{S}^{u}, E_{S}^{u} \ S) _{i^{∗}}_{≃}_{π_{u}^{∗}}_{≃}  H^{k_{u}}(Q ∩ E_{S}^{u}, Q ∩ E_{S}^{u} \ S)_{ψ^{−1 ∗}}_{≃}_{(π_{u}|Q)^{∗}}_{≃}  H^{k_{u}}(R^{ℓ} × (R^{k_{u}}, R_{0}^{k_{u}}) × {0})_{π_{Q u}^{∗}}_{≃}

where each vertical arrow is invertible because it is the end map of a strong deformation retraction obtained by contracting each fiber of N_{S}^{s} linearly to its zero-point and where i^{∗} in the bottom row is an isomorphism because it restricts an orientation class to an orientation class and the orientation classes here generate as a consequence of the Thom isomorphism theorem because S and Q ∩ S are connected. Under the composite isomorphism of the bottom row, the orientation class τ^{u} corresponds to an element 1 × τ′_{k_{u}} × 1 where τ′_{k_{u}} generates H^{k_{u}}(R^{k_{u}}, R_{0}^{k_{u}}), and since π_{Q u}(x, y, z) = (x, y, 0) for (x, y, z) ∈ R^{ℓ} × R^{k_{u}} × R^{k_{s}}, it follows that under the composite isomorphism of the top row, π_{u}^{∗}τ^{u} corresponds to 1 × τ′_{k_{u}} × 1 ∈ H^{k_{u}}(R^{ℓ} × (R^{k_{u}}, R_{0}^{k_{u}}) × R^{k_{s}}). By interchanging the roles of E_{S}^{u} and E_{S}^{s} and of R^{k_{u}} and R^{k_{s}}, one similarly sees that τ^{s} is represented locally by 1 × 1 × τ′′_{k_{s}} ∈ H^{k_{s}}(R^{ℓ} × {0} × (R^{k_{s}}, R_{0}^{k_{S}})) and that π_{s}^{∗}τ^{s} is represented locally by 1 × 1 × τ′′_{k_{s}} ∈ H^{k_{s}}(R^{ℓ} × R^{k_{u}} × (R^{k_{s}}, R_{0}^{k_{S}})) where τ′′_{k_{s}} generates H^{k_{s}}(R^{k_{s}}, R_{0}^{k_{s}}). Because Δ̄ = (π_{u}, π_{s}) = (π_{u} × π_{s})^{∘} Δ_{U} where Δ_{U}: U → U × U is the diagonal map, it follows from the definition of cup products that the orientation class

Δ̄^{∗}(τ^{u} × τ^{s}) = Δ_{U}^{∗}(π_{u}^{∗}τ^{u} × π_{s}^{∗}τ^{s}) = π_{u}^{∗}τ^{u} ⌣ π_{s}^{∗}τ^{s} .

Then the naturality of cup products implies that under the composite isomorphism

(†) H^{k_{u}+k_{s}}(U, U \ S) ⟶_{≃}^{i^{∗}} H^{k_{u}+k_{s}}(Q, Q \ S) ⟶_{≃}^{ψ^{−1 ∗}} H^{k_{u}+k_{s}}(R^{ℓ} × (R^{k_{u}}, R_{0}^{k_{u}}) × (R^{k_{s}}, R_{0}^{k_{s}}))

Δ̄^{∗}(τ^{u} × τ^{s}) corresponds to 1 × τ′_{k_{u}} × τ′′_{k_{s}} = 1 × τ′_{k_{u}} × 1 ⌣ 1 × 1 × τ′_{k_{s}}. Next, set p_{Q} := ψ|(Q ∩ S)^{∘} p|Q^{∘} ψ^{−1} and consider the ladder of isomorphisms

H^{ℓ}(U)  H^{ℓ}(U, U \ p^{−1}(u)) ^{i^{∗}} H^{ℓ}(Q, Q \ p^{−1}(u)) ^{ψ^{−1 ∗}}_{≃} H^{ℓ}((R^{ℓ}, R_{0}^{ℓ}) × R^{k_{u}} × R^{k_{s}}) H^{ℓ}(S)_{p^{∗}} _{≃}  H^{ℓ}(S, S \ {u}) _{i^{∗}}_{≃}_{p^{∗}} _{≃}_{≃}  H^{ℓ}(Q ∩ S, Q ∩ S \ {u}) _{ψ^{−1 ∗}}_{≃}_{(p|Q)^{∗}} _{≃}  H^{ℓ}((R^{ℓ}, R_{0}^{ℓ}) × {0} × {0})_{p_{Q}^{∗}} _{≃}

where the unlabeled arrows are inclusion induced. Here, each vertical arrow is an isomorphism as the fiberwise linear contraction of the total space of a vector bundle to its zero section is a strong deformation retraction. Also, i^{∗} in the bottom row is an excision isomorphism, and the bottom unlabeled arrow is invertible because the corresponding homology homomorphism maps fundamental class to fundamental class. Under the composite isomorphism of the bottom row, the class ν_{S} ∈ H^{ℓ}(S) Kronecker dual to the given fundamental class o_{S} of S corresponds to a class τ_{ℓ}×1×1 where τ_{ℓ} generates H^{ℓ}(R^{ℓ}, R_{0}^{ℓ}). Then under the composite isomorphism of the top row, p^{∗}ν_{S} corresponds to τ_{ℓ} × 1 × 1 ∈ H^{ℓ}((R^{ℓ}, R_{0}^{ℓ}) × R^{k_{u}} × R^{k_{s}}) because p_{Q}(x, y, z) = (x, 0, 0) for (x, y, z) ∈ R^{ℓ} × R^{k_{u}} × R^{k_{s}}.

We can now compute local representatives for p^{∗}ν_{S} ⌣ Δ̄^{∗}(τ^{u} × τ^{s}) and for the top-dimensional class p_{λ}^{∗}ν_{S} ⌣ τ^{λ} of N_{S}^{λ} (λ = u, s). Append the appropriate identity homomorphism at the left end of sequence (†) and compute the cup products of corresponding successive images of p^{∗}ν_{S} and Δ̄^{∗}(τ^{u}×τ^{s}) along, respectively, the top row of the ladder below sequence (†) and sequence (†) as appended. By naturality of cup products, it follows that under the composite isomorphism

H^{m}(U, U \ S) ≃ H^{m}(U, U \ {u}) ⟶_{≃}^{i^{∗}} H^{m}(Q, Q \ {u})

⟶_{≃}^{ψ^{−1 ∗}} H^{m}((R^{ℓ}, R_{0}^{ℓ}) × (R^{k_{u}}, R_{0}^{k_{u}}) × (R^{k_{s}}, R_{0}^{k_{s}}))

p^{∗}ν_{S} ⌣ Δ̄^{∗}(τ^{u} × τ^{s}) corresponds to τ_{ℓ} × τ′_{k_{u}} × τ′′_{k_{s}} = τ_{ℓ} × 1 × 1 ⌣ 1 × τ′_{k_{u}} × τ′′_{k_{s}}. In similar manner, one shows that the top-dimensional class of N_{S}^{λ}, viz., p_{λ}^{∗}ν_{S} ⌣ τ^{λ}, is represented locally in H^{ℓ+k_{λ}}(ψ(Q ∩ E_{S}^{λ}), ψ(Q ∩ E_{S}^{λ} \ {u})) by

τ_{ℓ} × τ′_{k_{u}} × 1 = τ_{ℓ} × 1 × 1 ⌣ 1 × τ′_{k_{u}} × 1  when λ = u

and by

τ_{ℓ} × 1 × τ′′_{k_{s}} = τ_{ℓ} × 1 × 1 ⌣ 1 × 1 × τ′′_{k_{s}}  when λ = s.

The local representative of p_{λ}^{∗}ν_{S} ⌣ τ^{λ} is used to compute a local representative of o_{ℓ+k_{λ}}^{λ} as follows. Because B ⊂ U, also (W^{λ}, ∂W^{λ}) ⊂ (E_{S}^{λ}, E_{S}^{λ}\S), and as the former pair consists of the total spaces of a disk-sphere subbundle pair of N_{S}^{λ}, the inclusion induced maps on homology and cohomology are isomorphisms. It then follows from the definition of o_{ℓ+k_{λ}}^{λ} that its local representative must be Kronecker dual to the local representative of p_{λ}^{∗}ν_{S} ⌣ τ^{λ}, whence in H_{k_{λ}+ℓ}(ψ(Q ∩ E_{S}^{λ}), ψ(Q ∩ E_{S}^{λ} \ {u})) it is represented by

(−1)^{ℓk_{u}}o_{ℓ} × o′_{k_{u}} × [0] when λ = u

and by

(−1)^{ℓk_{s}}o_{ℓ} × [0] × o′′_{k_{s}} when λ = s.

Next, choose the orientation O^{R^{m}} of R^{m} whose fundamental class around 0, call it o_{m}, locally represents o_{S}^{U} as defined in Definition 7.3. To get an explicit expression for o_{m}, first, note that under the composite isomorphism on homology dual to the composite isomorphism on cohomology given by the bottom row of the second ladder considered, the fundamental class o_{S} corresponds to an element o_{ℓ} × [0] × [0] ∈ H_{ℓ}((R^{ℓ}, R_{0}^{ℓ}) × {0} × {0}) where o_{ℓ} generates H_{ℓ}(R^{ℓ}, R_{0}^{ℓ}), and note that ⟨τ_{ℓ}, o_{ℓ}⟩ = ⟨ν_{S}, o_{S}⟩ = 1. Then, because

⟨τ_{ℓ} × τ′_{k_{u}} × τ′′_{k_{s}}, o_{ℓ} × o′_{k_{u}} × o′′_{k_{s}}⟩ = (−1)^{ℓk_{u}+ℓk_{s}+k_{u}k_{s}}

it follows with σ defined as in Definition 7.3 that the image of o_{S}^{U} under the composite isomorphism (the first unlabeled isomorphism maps fundamental class to fundamental class; the second is an excision isomorphism)

H_{m}(U, U \ S) ≃ H_{m}(U, U \ {u}) ≃ H_{m}(Q, Q \ {u}) ⟶_{≃}^{ψ_{∗}} H_{m}(ψ(Q), ψ(Q \ {u}))

is o_{m} := (−1)^{ℓk_{u}+ℓk_{s}+k_{u}k_{s}}σo_{ℓ} × o′_{k_{u}} × o′′_{k_{s}} which uniquely determines O^{R^{m}}.

We wish to take advantage of formulae (2.5) in computing the intersection pairing on the various local representatives; however, the use of (2.5) restricts us to computing intersection classes of homology classes that can be represented in open pairs. For the moment, let us ignore this requirement and compute formally ignoring the need to meet the hypotheses under which Poincaré duality holds. Also, we use singular cohomology and cup and cap products rather than their Čech counterparts, and to meet the excisiveness requirements for their use the classes used will be regarded as classes in a larger ambient space obtained by replacing each factor {0} with R^{k} where k = ℓ, k = k_{u}, or k = k_{s} as appropriate. The justification for doing so will be given after the formal computations are completed and follows from fact (2) above and naturality of singular cap and cup products.

First, let us compute o_{k_{u}+ℓ}^{u} ⌢_{•} o_{k_{s}+ℓ}^{s}. Regard σ 1 × 1 × τ′′_{k_{s}} as an element of H^{k_{s}}(R^{ℓ} × R^{k_{u}} × (R^{k_{s}}, R_{0}^{k_{s}})). It is straightforward to check that

σ 1 × 1 × τ′′_{k_{s}} ⌢ (−1)^{ℓk_{u}+ℓk_{s}+k_{u}k_{s}}σ o_{ℓ} × o′_{k_{u}} × o′′_{k_{s}} = (−1)^{ℓk_{u}}o_{ℓ} × o′_{k_{u}} × [0]

whence σ 1 × 1 × τ′′_{k_{s}} is formally Poincaré dual to (−1)^{ℓk_{u}}o_{ℓ} × o′_{k_{u}} × [0] when O^{R^{m}} is used to orient R^{m} ≡ R^{ℓ} × R^{k_{u}} × R^{k_{s}}. Formal use of formula (2.5b) then yields the computation (the first equality is only true modulo an isomorphism)

(−1)^{ℓk_{u}}o_{ℓ} × o′_{k_{u}} × [0] ⌢_{•} (−1)^{ℓk_{s}}o_{ℓ} × [0] × o′′_{k_{s}}
= σ 1 × 1 × τ′′_{k_{s}} ⌢ (−1)^{ℓk_{s}}o_{ℓ} × [0] × o′′_{k_{s}}
= σ o_{ℓ} × [0] × [0] ;

from which we conclude that

(7.1)  o_{k_{u}+ℓ}^{u} ⌢_{•} o_{k_{s}+ℓ}^{s} = σ o_{S} .

Similarly, to compute o_{k_{u}+ℓ}^{u} ⌢_{•} o_{k_{s}}^{s} we additionally note that o_{k_{s}}^{s} is by definition Kronecker dual to τ^{s}|(W_{u}^{s}, W_{u}^{s} \ {u}), and the latter is represented locally relative to (Q, ψ) by 1 × 1 × τ′′_{k_{s}} which is easily seen to be Kronecker dual to [0] × [0] × o′′_{k_{s}}. Again, formal use of (2.5b) yields (with the same caveat) the computation

(−1)^{ℓk_{u}}o_{ℓ} × o′_{k_{u}} × [0] ⌢_{•} [0] × [0] × o′′_{k_{s}} = σ 1 × 1 × τ′′_{k_{s}} ⌢ [0] × [0] × o′′_{k_{s}}
= σ [0] × [0] × [0] ;

from which we conclude that

(7.2)  ^{#}(o_{k_{u}+ℓ}^{u} ⌢_{•} o_{k_{s}}^{s}) = σ .

Finally, since τ^{u}|(W_{u}^{u}, W_{u}^{u} \ {u}) is represented locally by 1 × τ′_{k_{u}} × 1 which is Kronecker dual to [0] × o′_{k_{u}} × [0], the latter is the local representative of o_{k_{u}}^{u}. Also, it is easily checked that (−1)^{ℓk_{u}}σ τ_{ℓ} × 1 × τ′′_{k_{s}} ⌢ o_{m} = [0] × o′_{k_{u}} × [0]. Thus again, formal use of (2.5b) yields (with caveat as in the two previous cases)

[0] × o′_{k_{u}} × [0] ⌢_{•} (−1)^{ℓk_{s}}o_{ℓ} × [0] × o′′_{k_{s}}
= (−1)^{ℓk_{u}}σ τ_{ℓ} × 1 × τ′′_{k_{s}} ⌢ (−1)^{ℓk_{s}}o_{ℓ} × [0] × o′′_{k_{s}}
= (−1)^{ℓk_{u}}σ [0] × [0] × [0] .

We conclude that

(7.3)  ^{#}(o_{k_{u}}^{u} ⌢_{•} o_{k_{s}+ℓ}^{s}) = (−1)^{ℓk_{u}}σ .

It remains to justify the formal computations made. Here, only justification for the computation leading to (7.1) will be made; it is left to the reader to carry out similar justifications for those computations leading to (7.2) and (7.3). Note that the computation cannot be justified by invoking invariance of intersection pairings under inclusion directly because the intersection class of an element in H_{ℓ+k_{u}}((R^{ℓ}, R_{0}^{ℓ}) × (R^{k_{u}}, R_{0}^{k_{u}}) × R^{k_{s}}) ⊗ H_{ℓ+k_{s}}((R^{ℓ}, R_{0}^{ℓ}) × R^{k_{u}} × (R^{k_{s}}, R_{0}^{k_{s}})) by definition lies in H_{ℓ}((R^{ℓ}, R_{0}^{ℓ}) × (R^{k_{u}}, R_{0}^{k_{u}}) × (R^{k_{s}}, R_{0}^{k_{s}})) = ⟨0⟩.

First, it is convenient to work within a relatively compact open subset of R^{m} rather than in R^{m} itself, and we first show how to do this via invariance of intersection pairings under inclusions. Toward that end, let D_{ε}^{k} and S_{ε}^{k−1} be defined as at the beginning of the chapter, let B_{ε}^{k} := int(D_{ε}^{k}), and for 0 ≤ ε_{1} < ε_{2}, let A_{ε_{1} ε_{2}}^{k} := B_{ε_{2}}^{k} \D_{ε_{1}}^{k}. Fix t > 0, set X := B_{t}^{ℓ} ×B_{t}^{k_{u}} ×B_{t}^{k_{s}}, and give X the orientation O^{X} induced by the orientation O^{R^{m}} defined earlier. Choose q, r, s ∈ R satisfying 0 < q < r < s < t. Observe that in R^{ℓ} × R^{k_{u}} × R^{k_{s}} there exist inclusions of pairs

\begin{matrix}(R^{ℓ}, R^{ℓ} \ \{0\}) × (R^{k_{u}}, R^{k_{u}} \ \{0\}) × \{0\} \\ i_{1} & ↑ \\ (D_{s}^{ℓ}, S_{s}^{ℓ−1}) × (D_{s}^{k_{u}}, S_{s}^{k_{u}−1}) × \{0\} \\ j_{1} & ↓ \\ (B_{t}^{ℓ}, A_{r t}^{ℓ}) × (B_{t}^{k_{u}}, A_{r t}^{k_{u}}) × B_{r}^{k_{s}} \end{matrix}and\begin{matrix}(R^{ℓ}, R^{ℓ} \ \{0\}) × \{0\} × (R^{k_{s}}, R^{k_{s}} \ \{0\}) \\ i_{2} & ↑ \\ (D_{s}^{ℓ}, S_{s}^{ℓ−1}) × \{0\} × (D_{s}^{k_{s}}, S_{s}^{k_{s}−1}) \\ j_{2} & ↓ \\ (B_{t}^{ℓ}, A_{r t}^{ℓ}) × B_{r}^{k_{u}} × (B_{t}^{k_{s}}, A_{r t}^{k_{s}}) \end{matrix}

each inducing isomorphisms on homology. Also, if we take two singular homology classes, say ξ in the homology of the middle entry in the left column and η from the middle entry in the right column, and form the intersection classes ξ ⌢_{•} η, i_{1 ∗}ξ ⌢_{•} i_{2 ∗}η, and j_{1 ∗}ξ ⌢_{•} j_{2 ∗}η, then the first lies in the Čech homology of the pair (D_{s}^{ℓ}, S_{s}^{ℓ}) × {0} × {0}, the second in that of the pair (R^{ℓ}, R^{ℓ} \ {0}) × {0}), and the third in that of the pair (B_{t}^{ℓ}, A_{r t}^{ℓ}) × B_{r}^{k_{u}} × B_{r}^{k_{s}}. Further, since the inclusions

(R^{ℓ}, R^{ℓ} \ {0}) × {0}) ⟵^{i_{3}} (D_{s}^{ℓ}, S_{s}^{ℓ}) × {0} × {0} ⟶^{j_{3}} (B_{t}^{ℓ}, A_{r t}^{ℓ}) × B_{r}^{k_{u}} × B_{r}^{k_{s}}

induce isomorphisms on homology (note that all three pairs have isomorphic Čech and singular homology: the first and third because they are open pairs; the second because it is a pair of Euclidean neighborhood retracts), it follows from the invariance of intersection pairings under inclusions that to compute i_{1 ∗}ξ ⌢_{•} i_{2 ∗}η it suffices to compute j_{1 ∗}ξ ⌢_{•} j_{2 ∗}η. For the latter we are working with pairs of open pairs in X so that the formulae (2.5) can be used. In particular, we may regard the homology classes (−1)^{ℓk_{u}}o_{ℓ} × o′_{k_{u}} × [0] and (−1)^{ℓk_{s}}o_{ℓ} × [0] × o′′_{k_{s}} as elements of H_{ℓ+k_{u}}((B_{t}^{ℓ}, A_{r t}^{ℓ}) × (B_{t}^{k_{u}}, A_{r t}^{k_{u}}) × B_{r}^{k_{s}}) and H_{ℓ+k_{s}}((B_{t}^{ℓ}, A_{r t}^{ℓ}) × B_{r}^{k_{u}} × (B_{t}^{k_{s}}, A_{r t}^{k_{s}})) respectively.

The justification will now be reduced to diagram chasing, and let us proceed to define various spaces and maps needed to write down the diagrams. Define the pair (K, L), closed relative to X, by the equality

(X \ L, X \ K) := (B_{t}^{ℓ}, A_{r t}^{ℓ}) × (B_{t}^{k_{u}}, A_{r t}^{k_{u}}) × B_{r}^{k_{s}} .

Set C := D_{r}^{ℓ}×D_{r}^{k_{u}} ×D_{r}^{k_{s}}. Then it is easily checked that K\L = D_{r}^{ℓ}×D_{r}^{k_{u}} ×B_{r}^{k_{s}} ⊂ C and that (K ∩ C, L ∩ C) = D_{r}^{ℓ} × D_{r}^{k_{u}} × (D_{r}^{k_{s}}, S_{r}^{k_{s}−1}). Also, define

Y := B_{t}^{ℓ} × B_{t}^{k_{u}} × A_{q t}^{k_{s}} , Z := A_{q t}^{ℓ} × B_{t}^{k_{u}} × B_{t}^{k_{s}} ∪ B_{t}^{ℓ} × A_{q t}^{k_{u}} × B_{t}^{k_{s}}

Next, let us define two homology classes that will be used to define cap product homomorphisms in our diagram; we need an auxiliary diagram. To facilitate printing, set X_{L} := X \ (L ∩ C), X_{K} := X \ (K ∩ C), Y_{L} := Y \ (L ∩ C), and X_{0} := X \ {0}. In the commutative diagram of inclusion induced homomorphisms

H_{m}(X, X_{0})_{≃}  H_{m}(X, X_{K})^{≃}  H_{m}(X, X_{K} ∪ Y)  H_{m}(X_{L}, X_{K} ∪ Y_{L})^{≃}_{i_{0 m}} H_{m}(R^{m}, R_{0}^{m})  H_{m}(X, Y ∪ Z)_{k_{0 m}}  ,

in the top row, the rightmost arrow is an excision isomorphism and the leftmost arrow is an isomorphism because it maps fundamental class to fundamental class. Let o_{0}^{X} ∈ H_{m}(X, X_{0}) be the fundamental class of X whose image in H_{m}(R^{m}, R_{0}^{m}) under the excision isomorphism that is the leftmost vertical arrow is o_{m}. Define ō_{K∩C} ∈ H_{m}(X_{L}, X_{K} ∪ Y_{L}) to be the image of o_{0}^{X} under the composition of the arrows in the top row with the left-pointing arrows replaced by their inverses. Also, define ō_{0}^{X} := i_{0 m}ō_{K∩C} ∈ H_{m}(X, Y ∪ Z). Diagram chasing then shows that also o_{m} = k_{0 m}ō_{0}^{X}. Note too that the underlying inclusion k_{0} is a weak deformation retract. Hence, k_{0 m} is an isomorphism, and ō_{0}^{X} can be characterized as the unique element whose image in H_{m}(X, X_{0}) under the inclusion induced oblique arrow is o_{0}^{X}. Thus, the definition of ō_{0}^{X} does not depend on K or L; we use this later.

We then have the following diagram of R-module homomorphisms

Ȟ^{k_{s}}(K, L)_{e_{1}^{∗}} Ȟ^{k_{s}}(K ∩ C, L ∩ C) H^{k_{s}}(R^{ℓ+k_{u}} × (R^{k_{s}}, R_{0}^{k_{s}}))_{⌢ o_{m}}  H^{k_{s}}(X, Y)_{j^{∗}}  _{e_{2}^{∗}}_{⌢ ō_{0}^{X}}  H^{k_{s}}(X \ (L ∩ C), Y \ (L ∩ C))_{⌢ ō_{K∩C}} H_{ℓ+k_{u}}((R^{ℓ+k_{u}}, R_{0}^{ℓ+k_{u}}) × R^{k_{s}})  H_{ℓ+k_{u}}(X, Z)  H_{ℓ+k_{u}}(X \ (L ∩ C), X \ (K ∩ C))_{e_{3 ∗}} H_{ℓ+k_{u}}(X \ L, X \ K)_{k_{∗}}

where all arrows other than the cap product homomorphisms are inclusion induced with e_{1}^{∗}, e_{2}^{∗}, and e_{3 ∗} excision isomorphisms. The diagram is commutative: the triangles at the bottom are because they consist of inclusion induced morphisms; both rectangles are by naturality of cap products [D, VII.12.6] because ō_{0}^{X} = i_{0 m}ō_{K∩C}, because o_{m} = k_{0 m}ō_{0}^{X}, and because all excisiveness requirements are met because the necessary sets are open.

Every arrow in the diagram is an isomorphism. First, note that the inclusion

of pairs (K ∩ C, L ∩ C) ↪^{j}(X, Y) is a deformation retract, whence as the latter pair is a neighborhood pair of the former pair, the canonical map H^{∗}(X, Y) → Ȟ^{∗}(K ∩ C, L ∩ C) can be identified with j^{∗} and is an isomorphism. Note, therefore, that the Poincaré duality isomorphism ⌢ o^{X}: Ȟ^{k_{s}}(K, L) → H_{ℓ+k_{u}}(X \ L, X \ K) coincides with the composite e_{3 ∗}^{∘} ⌢ ō_{K∩C} ^{∘}e_{2}^{∗}^{∘}(j^{∗})^{−1}^{∘}e_{1}^{∗} (cf. pp. 292–3 and p. 296 of [D]); this implies that ⌢ o_{K∩C} is an isomorphism. Also, ⌢ o_{m} is an isomorphism because our earlier computation of a formal Poincaré dual of (−1)^{ℓk_{u}}o_{ℓ} × o′_{k_{u}} × [0] shows that ⌢ o_{m} maps cyclic generator to cyclic generator. The inclusions inducing the horizontal arrows in the left rectangle are easily seen to be weak deformation retracts so induce isomorphisms. An elementary diagram chase then shows the remaining arrows to be isomorphisms.

It follows, therefore, that if ξ ∈ H_{ℓ+k_{u}}(X \ L, X \ K), then (⌢ o^{X})^{−1}ξ, its Poincaré dual, is uniquely determined by the element x := (⌢ o_{m})^{−1}k_{∗}ξ and conversely. In particular, this is true for ξ := (−1)^{ℓk_{u}}o_{ℓ} × o′_{k_{u}} × [0].

Next, we need (but leave it to the reader to write it down) a second diagram entirely analogous to the diagram just constructed but with (K, L) replaced by (K∗, L∗) where the latter pair is defined by

(X \ L∗, X \ K∗) := (B_{t}^{ℓ}, A_{r t}^{ℓ}) × B_{r}^{k_{u}} × (B_{t}^{k_{s}}, A_{r t}^{k_{s}})

and with Y and Z replaced respectively by Y ∗ and Z∗ defined by

Y ∗ := B_{t}^{ℓ} × A_{q t}^{k_{u}} × B_{t}^{k_{s}} , Z∗ := A_{q t}^{ℓ} × B_{t}^{k_{u}} × B_{t}^{k_{s}} ∪ B_{t}^{ℓ} × B_{t}^{k_{u}} × A_{q t}^{k_{s}} .

Also, the leftmost vertical arrow gets replaced by the arrow

H^{k_{u}}(R^{ℓ} × (R^{k_{u}}, R_{0}^{k_{u}}) × R^{k_{s}}) ⟶^{⌢o_{m}} H_{ℓ+k_{s}}((R^{ℓ}, R_{0}^{ℓ}) × R^{k_{u}} × (R^{k_{s}}, R_{0}^{k_{s}})) .

In direct analogy with the previous case, if η ∈ H_{ℓ+k_{s}}(X \ L∗, X \ K∗), then its Poincaré dual (⌢ o^{X})^{−1}η is uniquely determined by the element y := (⌢ o_{m})^{−1}k′_{∗}η and conversely where

k′_{∗}: H_{ℓ+k_{s}}(X \ L∗, X \ K∗) → H_{ℓ+k_{s}}((R^{ℓ}, R_{0}^{ℓ}) × R^{k_{u}} × (R^{k_{s}}, R_{0}^{k_{s}}))

is the inclusion induced map. In particular, this is true for η := (−1)^{ℓk_{s}}o_{ℓ}×[0]×o′′_{k_{s}}; in which case y = (−1)^{k_{u}k_{s}}σ1 × τ′_{k_{u}} × 1.

To complete the justification, take the tensor product of the two diagrams, lattice point by lattice point, call the result the source diagram, regard it as one face of a three-dimensional diagram whose opposite face, given momentarily, is a diagram, call it the target diagram, of the same type as those just tensored together where the arrows from the source to target are given by taking cup products on the tensor product of cohomology modules (on the Čech modules the cup product is Δ^{∗} ^{∘} ×), noting that each rectangle with two parallel edges cup products and one edge each in the source and target diagrams is commutative by naturality of cup products, and by defining the arrow on the tensor product of homology modules in the unique way possible so that the three-dimensional diagram is commutative. It will turn out that on the bottom-most tensored pair the arrow to the target diagram is (up to an excision isomorphism labeled e_{∗} below) the intersection pairing 𝔏_{V} where

V := ((X \ L, X \ K), (X \ L∗, X \ K∗)) .

To facilitate the printing of the target diagram, set X′′ := X \ ((K ∩ K∗) ∩ C),

X′ := X \(((L∩K∗)∪(K ∩L∗))∩C) and Y′ := Y \(((L∩K∗)∪(K ∩L∗))∩C) .

Also, define W := A_{q t}^{ℓ} ×B_{t}^{k_{u}} ×B_{t}^{k_{s}}. Then the target diagram with an extra excision isomorphism added as a bottom tail to get 𝔏_{V} exactly is the diagram

Ȟ^{k_{u}+k_{s}}(K ∩ K∗, (L ∩ K∗) ∪ (K ∩ L∗))_{e′′_{1 ∗}} Ȟ^{k_{u}+k_{s}}((K ∩ K∗) ∩ C, ((L ∩ K∗) ∪ (K ∩ L∗)) ∩ C) H^{k_{u}+k_{s}}(R^{ℓ} × (R^{k_{u}}, R_{0}^{k_{u}}) × (R^{k_{s}}, R_{0}^{k_{s}}))_{⌢ o_{m}}  H^{k_{u}+k_{s}}(X, Y ∪ Y ∗)_{j^{′′ ∗}}  _{e_{2}^{′′ ∗}}_{⌢ō_{0}^{X}}  H^{k_{u}+k_{s}}(X′, Y′)_{⌢ ō_{K∩K∗∩C}} H_{ℓ}((R^{ℓ}, R_{0}^{ℓ}) × R^{k_{u}} × R^{k_{s}})  H_{ℓ}(X, W)  H_{ℓ}(X′, X′′) H_{ℓ}(X \ ((L ∩ K∗) ∪ (K ∩ L∗)), X \ (K ∩ K∗))_{e′′_{3 ∗}}_{k′′_{∗}} H_{ℓ}(X \ (L ∪ L∗), X \ ((K ∩ K∗) ∪ (L ∪ L∗)))_{e_{∗}}

and is seen to be commutative and to have invertible arrows via a proof entirely analogous to that used to construct and analyze the first diagram. Too, because the cup product on H^{k_{s}}(R^{ℓ+k_{u}} × (R^{k_{s}}, R_{0}^{k_{s}})) ⊗ H^{k_{u}}(R^{ℓ} × (R^{k_{u}}, R_{0}^{k_{u}}) × R^{k_{s}}) to H^{k_{u}+k_{s}}(R^{ℓ} × (R^{k_{u}}, R_{0}^{k_{u}}) × (R^{k_{s}}, R_{0}^{k_{s}})) is an isomorphism—it maps a cyclic generator to a cyclic generator—elementary diagram chasing shows that all arrows in the full three-dimensional diagram are invertible. Let ξ, x, η, and y be as most recently defined. Our last observation yields that ξ ⊗ η because it generates the module at the bottom-most lattice point of the source diagram has image ±o_{ℓ} × [0] × [0] ∈ H_{ℓ}((B_{t}^{ℓ}, A_{q t}^{ℓ}) × B_{r}^{k_{u}} × B_{r}^{k_{s}}) along any path to that module (the bottom-most lattice point of the target diagram, tail added) in the three-dimensional diagram. Then on the one hand, we computed earlier that x ⌢ k′_{∗}η = σo_{ℓ} × [0] × [0], and therefore by the associativity identity relating singular cup and cap products,

σo_{ℓ} × [0] × [0] = x ⌢ k′_{∗}η = x ⌢ (y ⌢ o_{m}) = (x ⌣ y) ⌢ o_{m}
= ((⌢ o_{m})^{−1}k_{∗}ξ ⌣ (⌢ o_{m})^{−1}k′_{∗}η) ⌢ o_{m} .

On the other hand, because traveling down the outer right edge of the source diagram corresponds to (⌢ o^{X}) ⊗ (⌢ o^{X}), the tensor product of Poincaré duality isomorphisms, and because traveling down the outer right edge of the target diagram coincides with the Poincaré duality isomorphism ⌢ o^{X} followed by e_{∗}^{−1}, it follows from a diagram chase in the three-dimensional diagram that

σo_{ℓ} × [0] × [0] = k′′_{∗}(((⌢ o^{X})^{−1}ξ ⌣ (⌢ o^{X})^{−1}η) ⌢ o^{X})

= k′′_{∗}e_{∗}(ξ ⌢_{•}^{V} η)

≡ k′′_{∗}e_{∗}𝔏_{V} (ξ ⊗ η)

where the second equality is an immediate consequence of the rigorous formulation of formula (2.5a). As already noted, the image of ξ ⊗ η in the bottom-most module of the target diagram must generate that module; i.e., necessarily, 𝔏_{V} (ξ ⊗ η) = ±o_{ℓ} × [0] × [0], and this and the just given string of equalities yields the desired result because k′′_{∗} and e_{∗} are inclusion induced, and therefore, leave signs unchanged.

Note that a rigorous formulation of formula (2.5a) is contained in Proposition 9.18 below—specifically see sequence (9.7)—with the caveat that Proposition 9.18 is formulated for a pair V ∈ M_{Δ̸ o}^{(2)} whereas present circumstances dictate that V be a general pair of open pairs that does not satisfy the off-diagonal condition. However, the proof of Proposition 9.18 applies equally well to the general case if suitable modifications are made to the lattice points of the diagrams used in its proof, and the reader, with a little thought, should have no difficulty making the necessary changes. For example, because in the general case (V_{0} ∩V ∗_{1})∪(V_{1} ∩V ∗_{0}) ≠ ∅, the last term in sequences (9.7–9) must be changed from H_{i+j−m}(V_{1} ∩ V ∗_{1}; G ⊗ G′) to H_{i+j−m}(V_{1} ∩ V ∗_{1}, (V_{0} ∩ V ∗_{1}) ∪ (V_{1} ∩ V ∗_{0}); G ⊗ G′); all other necessary changes can be worked out backwards from this one. □

The next lemma gives a condition for obtaining a Lyapunov function and will be used in the construction of isolating blocks for hyperbolic critical points and hyperbolic closed orbits. It is a simple generalization of well-known estimates; e.g., see [H-S, §§9.1–2]. The following notation and hypotheses are needed for its statement.

Let T^{ℓ} denote the product of S^{1} with itself ℓ times when ℓ is a positive integer. Given ω ∈ R_{++}^{ℓ} and an integer k ≥ 1 there is a C^{∞} universal covering map /ω: R^{ℓ} × R^{k} → T^{ℓ} × R^{k} defined for (θ, x) ∈ R^{ℓ} × R^{k} by /ω(θ, x) = (e^{iθ_{1}2π/ω_{1}}, …, e^{iθ_{ℓ}2π/ω_{ℓ}}, x). We will say that a function f on R^{ℓ}×R^{k} to a space Z is multi-periodic in θ ∈ R^{ℓ} with multi-period ω if there exists a function f̃: T^{ℓ} ×R^{k} → Z, necessarily unique, satisfying f̃^{∘} /ω = f.

For u ∈ R^{m}, write u = (θ, x) ∈ R^{ℓ} × R^{m−ℓ} ≡ R^{m}, suppose E is a subspace of R^{m−ℓ} with orthogonal complement E^{⊥} relative to the standard inner product on R^{m−ℓ}, let π_{E}: R^{m} → E and π_{E^{⊥}}: R^{m} → E^{⊥} be the orthogonal projections relative to the standard inner product under the identification R^{m} ≡ R^{ℓ} ⊕ R^{m−ℓ}, and suppose A ∈ L(E, E) with the property that the quadratic form Q on E defined by Q(y) := ⟨Ay | y⟩ is positive definite. Also, let K = {x ∈ R^{m−ℓ} : ∥π_{E^{⊥}}(x)∥ ≤ ∥π_{E}(x)∥ }. Below, L_{χ} denotes Lie differentiation along the trajectories of a vectorfield χ.

7.8 Lemma. If χ is a vectorfield on R^{ℓ} × R^{m−ℓ} whose local representative χ̂ is multi-periodic in θ ∈ R^{ℓ} with multi-period ω ∈ R_{++}^{ℓ} and satisfies

π_{E} ^{∘} χ̂(θ, x) = A π_{E}(θ, x) + R_{E}(θ, x)

where R_{E}(θ, 0) = 0 and D_{2}R_{E}(θ, 0) = 0 for each θ ∈ R^{ℓ}, then

c_{1} := inf{⟨Ax | x⟩ : x ∈ E \ {0}} > 0

and there exists c_{0} > 0 so that

L_{χ}∥π_{E}(θ, x)∥^{2} ≥ c_{1}∥π_{E}(θ, x)∥^{2} if θ ∈ R^{ℓ} and x ∈ K ∩ D_{\sqrt{2}c_{0}}^{m−ℓ}.

Proof. For x ∈ R^{m−ℓ}, write x = x_{1} + x_{2} where x_{1} ∈ E and x_{2} ∈ E^{⊥}, note c_{1} = inf{⟨Ax_{1} | x_{1}⟩ : ∥x_{1}∥ = 1}, and therefore c_{1} > 0 by the positive definiteness of Q and the local compactness of E.

Define n(θ, ρ) := sup{∥D_{2}R_{E}(θ, x)∥ : ∥x∥ ≤ ρ} for θ ∈ R^{ℓ} and ρ > 0. Then n is both lower and upper semi-continuous; hence continuous. The proof of lower semi-continuity follows trivially from the definitions and continuity of ∥D_{2}R_{E}∥; the proof of upper semi-continuity follows easily from the continuity of ∥D_{2}R_{E}∥ and the compactness of any closed ball of finite radius in R^{m−ℓ}. Details are left to the reader.

Continuity of n and the assumption that D_{2}R_{E}(θ, 0) = 0 yield for each θ ∈ R^{ℓ}, a δ(θ) > 0 so that

n(θ′, ρ) < \frac{1}{2\sqrt{2}}c_{1} if ∥θ − θ′∥ < δ(θ) and 0 < ρ < δ(θ).

Writing ω = (ω_{1}, …, ω_{ℓ}), let

C := [0, ω_{1}] × ⋯ × [0, ω_{ℓ}] ⊂ R^{ℓ} .

By the compactness of C and multi-periodicity of χ̂, hence of R_{E}, it follows that there exists c_{0} > 0 so that

0 < ρ < \sqrt{2}c_{0} implies n(θ, ρ) < \frac{1}{2\sqrt{2}}c_{1} for each θ ∈ R^{ℓ}.

Therefore, as R_{E}(θ, 0) = 0 for each θ ∈ R^{ℓ}, the Cauchy-Schwarz inequality, the Mean Value theorem, and the definition of the cone K together imply that (7.4)

|⟨R_{E}(θ, x) | x_{1}⟩| ≤ \frac{1}{2\sqrt{2}}c_{1}∥x_{1}∥ ∥x∥ ≤ \frac{1}{2}c_{1}∥x_{1}∥^{2} for θ ∈ R^{ℓ}, x ∈ K ∩ D_{\sqrt{2}c_{0}}^{m−ℓ}.

Then for (θ, x) ∈ R^{ℓ} × R^{m−ℓ},

L_{χ}∥π_{E}(θ, x)∥^{2} = 2⟨Ax_{1} + R_{E}(θ, x) | x_{1}⟩
≥ 2c_{1}∥x_{1}∥^{2} − 2|⟨R_{E}(θ, x) | x_{1}⟩| .

Hence, (7.4) implies that

L_{χ}∥π_{E}(θ, x)∥^{2} ≥ c_{1}∥x_{1}∥^{2} if θ ∈ R^{ℓ} and x ∈ K ∩ D_{\sqrt{2}c_{0}}^{m−ℓ}. □

7.9 Proposition. If χ̂ = (χ̂_{1}, χ̂_{2}, χ̂_{3}) is a C^{1} map of R^{ℓ} × R^{P} × R^{N} to itself having the form

\begin{aligned}χ̂_{1}(θ, x_{1}, x_{2}) & = a + Θ(θ, x_{1}, x_{2}) & ∈ R^{ℓ} \\ χ̂_{2}(θ, x_{1}, x_{2}) & = Ex_{1} + R_{1}(θ, x_{1}, x_{2}) & ∈ R^{P} \\ χ̂_{3}(θ, x_{1}, x_{2}) & = Cx_{2} + R_{2}(θ, x_{1}, x_{2}) & ∈ R^{N} \end{aligned}  for (θ, x_{1}, x_{2}) ∈ R^{ℓ} × R^{P} × R^{N}

where Θ, R_{i} (i = 1, 2), and the partial derivative D_{j}R_{i} (i = 1, 2, j = 2, 3) vanish on R^{ℓ} × {0} × {0}, where E ∈ L(R^{P} , R^{P}), C ∈ L(R^{N}, R^{N}), and the quadratic forms Q_{1} and Q_{2} defined by Q_{1}(x_{1}) = ⟨Ex_{1} | x_{1}⟩ for x_{1} ∈ R^{P} and Q_{2}(x_{2}) = ⟨−Cx_{2} | x_{2}⟩ for x_{2} ∈ R^{N} are positive definite, and where χ̂ is multi-periodic in θ ∈ R^{ℓ} with multi-period ω ∈ R_{++}^{ℓ}, then

S̃ := T^{ℓ} × {0} × {0}

is an isolated invariant set of the flow generated by the induced vectorfield χ̃ on T^{ℓ} × R^{P} × R^{N} and for some c_{0} > 0,

0 < c < c_{0} implies K_{c} := T^{ℓ} × D_{c}^{P} × D_{c}^{N}

is an isolating block for S̃ with

K_{c}^{+} = T^{ℓ} × S_{c}^{P −1} × D_{c}^{N} and K_{c}^{−} = T^{ℓ} × D_{c}^{P} × S_{c}^{N−1} .

Proof. First, observe that by hypothesis the vectorfield χ̃ is tangent to S̃, whence S̃ is a compact invariant set of the flow generated by χ̃. Thus to finish it must be shown that K_{c} is an isolating block whose maximal invariant set is S̃.

The inner product on M := R^{ℓ} × R^{P} × R^{N} is taken to be the direct sum of the standard inner products on R^{ℓ}, R^{P}, and R^{N}. Let χ be the vectorfield on M whose local representative is χ̂, set X := R^{P} × R^{N}, and define the cones C_{1} and C_{2} in X by

C_{1} := {(x_{1}, x_{2}) ∈ X : ∥x_{2}∥ ≤ ∥x_{1}∥ } and C_{2} := {(x_{1}, x_{2}) ∈ X : ∥x_{1}∥ ≤ ∥x_{2}∥ } .

Also, define f_{1}(θ, x_{1}, x_{2}) = ∥x_{1}∥^{2} and define f_{2}(θ, x_{1}, x_{2}) = ∥x_{2}∥^{2}. Because both are independent of θ, there are corresponding induced functions f̃_{1} and f̃_{2} on the quotient T^{ℓ} × R^{P} × R^{N} given by f̃_{i}(z, x_{1}, x_{2}) = ∥x_{i}∥^{2}.

Lemma 7.8 is applied twice: first with E := R^{P} × {0} and A the operator E, second with E := {0} × R^{N} and A the operator −C. What results upon observing that L_{χ}∥π_{E}(θ, x)∥^{2} is in both cases independent of θ and thereby allowing us to pass to the quotient is that where

c_{1} := inf_{∥x_{1}∥=1}⟨Ex_{1} | x_{1}⟩  and  c_{2} := inf_{∥x_{2}∥=1}⟨−Cx_{2} | x_{2}⟩

c_{1} and c_{2} are positive, and for some c_{0} > 0, where Σ_{i} := T^{ℓ} × C_{i} ∩ D_{\sqrt{2}c_{0}}^{P +N},

(7.5_{i})  (−1)^{i−1}Lχ̃f̃_{i} ≥ c_{i}f̃_{i} on Σ_{i}  (i = 1, 2).

Now observe that

K_{c} = f̃_{1}^{−1}([0, c]) ∩ f̃_{2}^{−1}([0, c]) ,

and as a consequence of (7.5_{i}) (i = 1, 2),

Lχ̃f̃_{1} > 0 on T^{ℓ} × S_{c}^{P −1} × D_{c}^{N}  and  Lχ̃f̃_{2} < 0 on T^{ℓ} × D_{c}^{P} × S_{c}^{N−1} .

Similar to observations made in the proof of Proposition 4.4, for 0 < c < c_{0}, it follows that K_{c} is an isolating block relative to the flow generated by χ̃, that its exit set is

K_{c}^{+} = f̃_{1}^{−1}(c) ∩ K_{c} = T^{ℓ} × S_{c}^{P −1} × D_{c}^{N} ,

and that its entrance set is

K_{c}^{−} = f̃_{2}^{−1}(c) ∩ K_{c} = T^{ℓ} × D_{c}^{P} × S_{c}^{N−1} .

Also, note that ∂K_{c} = K_{c}^{+} ∪ K_{c}^{−}.

It remains to show that K_{c} isolates S̃. We begin by showing that Σ_{1 c} := Σ_{1} ∩ K_{c} is positively invariant relative to K_{c} and that Σ_{2 c} := Σ_{2} ∩ K_{c} is negatively invariant relative to K_{c}. To do so, first, observe that K_{c} = Σ_{1 c} ∪ Σ_{2 c} and that Σ_{1 c} ∩ Σ_{2 c} = K_{c} ∩ h^{−1}(0) where h(z, x_{1}, x_{2}) = ∥x_{1}∥^{2} − ∥x_{2}∥^{2}. Hence, to show the desired invariances it will suffice to show that Lχ̃h > 0 on h^{−1}(0) ∩ K_{c} \ S̃. It is easy to compute that

Lχ̃h(z, x_{1}, x_{2}) = 2⟨Ex_{1} + R_{1}(z, x_{1}, x_{2}) | x_{1}⟩ − 2⟨Cx_{2} + R_{2}(z, x_{1}, x_{2}) | |x_{2}⟩

≥ 2c_{1}∥x_{1}∥^{2} + 2c_{2}∥x_{2}∥^{2} − 2 ∑_{i=1,2} ∥R_{i}(z, x_{1}, x_{2})∥∥x_{i}∥ .

As in the proof of the lemma, shrinking c_{0} if necessary, it follows that

∥R_{i}(z, x_{1}, x_{2})∥ ≤ \frac{1}{2\sqrt{2}} min\{c_{1}, c_{2}\}∥x∥ if ∥x∥ ≤ \sqrt{2}c_{0}.

If h(z, x) = 0, then necessarily ∥x∥ = \sqrt{2}∥x_{i}∥ for i = 1, 2. It follows that

Lχ̃h(z, x_{1}, x_{2}) ≥ min{c_{1}, c_{2}}∥x∥^{2} if x ∈ h^{−1}(0) ∩ K_{c} for 0 < c < c_{0},

whence Lχ̃h > 0 on K_{c} ∩ h^{−1}(0) \ S̃ as desired.

Suppose that u = (z, x_{1}, x_{2}) ∈ K_{c} \ S̃ and that the orbit through u is complete and contained in K_{c}. Now, either u ∈ Σ_{1 c} or u ∈ Σ_{2 c}. Assume the former. Then, by the relative positive invariance of Σ_{1 c}, the positive semi-orbit u · R^{+} remains in Σ_{1 c}. This, however, is impossible, for integrating the inequality of (7.5_{1}) along this semi-orbit over a positive time interval [0, t] yields that f̃_{1}(u·t) > e^{c_{1}t} showing that the positive semi-orbit through u cannot be contained in K_{c}. On the other hand, if u ∈ Σ_{2 c}, then the relative negative invariance of Σ_{2 c} implies that it contains u · R^{−}. This too is seen to be impossible, this time by integrating the inequality of (7.5_{2}) along the negative semi-orbit over an arbitrarily large negative time interval [t, 0]. Thus K_{c} isolates S̃ for 0 < c < c_{0}. □

An immediate corollary of the proof of the proposition obtained by taking ℓ = 0 and /ω the identification {0} × R^{P} × R^{N} ≡ R^{P} × R^{N} is the following:

7.10 Corollary. If χ̂ = (χ̂_{1}, χ̂_{2}) is a C^{1} map of R^{P} × R^{N} to itself having the form

\begin{aligned}χ̂_{1}(x_{1}, x_{2}) & = Ex_{1} + R_{1}(x_{1}, x_{2}) & ∈ R^{P} \\ χ̂_{2}(x_{1}, x_{2}) & = Cx_{2} + R_{2}(x_{1}, x_{2}) & ∈ R^{N} \end{aligned}  for (x_{1}, x_{2}) ∈ R^{P} × R^{N}

where R_{i} and DR_{i} vanish at the origin (0, 0) ∈ R^{P} × R^{N} and where the quadratic forms Q_{1} and Q_{2} defined by Q_{1}(x_{1}) = ⟨Ex_{1} | x_{1}⟩ for x_{1} ∈ R^{P} and Q_{2}(x_{2}) = ⟨−Cx_{2} | x_{2}⟩ for x_{2} ∈ R^{N} are positive definite, then S = {(0, 0)} is an isolated invariant set of the flow generated by the vectorfield χ on R^{P} × R^{N} whose local representative is χ̂ and for some c_{0} > 0, if 0 < c < c_{0}, then B_{c} := D_{c}^{P} × D_{c}^{N} is an isolating block for S with

B_{c}^{+} = S_{c}^{P −1} × D_{c}^{N} and B_{c}^{−} = D_{c}^{P} × S_{c}^{N−1} .

C. Results Leading to the Proof of Theorem 7.5. In the theorems of this subsection and the next we will be working with both the flow generated by a given vectorfield and that generated by a linearization along an isolated invariant set S which is then also an isolated invariant set of the flow generated by the linearization. Further, the flows will have common isolating blocks in a sufficiently small neighborhood of the common invariant set. However, the local invariant manifolds within any such block will differ. Hence, to distinguish between the local invariant manifolds of one flow and those of the other, the generating vectorfield will be appended to the notation as a subscript: for example, W_{ξ}^{u}(S; B) denotes the local unstable manifold of S within B relative to the flow generated by ξ.

Also, the word “isotopy” will not be used in this subsection or the next to mean “ambient isotopy”, but will be used according to the following conventions to simplify the statement and exposition of the proofs of the theorems.

7.11 Definition. (A) Two submanifolds with boundary W_{0}, W_{1} of M will be called isotopic in M if there exists an isotopy of continuous embeddings from W_{0} to W_{1}, i.e., if there exists continuous H: W_{0} × [0, 1] → M so that writing H_{t}(u) := H(u, t), for each u ∈ W_{0} and t ∈ [0, 1], H_{t} is a continuous embedding of W_{0} into M with H_{0} = 1_{W_{0}} and H_{1} a homeomorphism of W_{0} onto W_{1}. Note invariance of domain implies H restricts to an isotopy from ∂W_{0} to ∂W_{1}. Clearly, the notion of isotopy as defined here yields an equivalence relation on submanifolds with boundary of M.

(B) Also of interest to us is the case where for i = 0, 1, (W_{i}, ∂W_{i}) ⊂ (P_{1}, P_{0}) where (P_{1}, P_{0}) is a nested index pair relative to some flow in M. If H is as in part (A) but in fact has image in P_{1} with H_{t}(∂W_{0}) ⊂ P_{0} for t ∈ [0, 1], then call the pairs (W_{0}, ∂W_{0}) and (W_{1}, ∂W_{1}) isotopic in the pair (P_{1}, P_{0}).

7.12 Theorem (cf. [C,I.4.3]). Under the assumption of hypotheses HCP, S = {u_{0}} is an isolated invariant set relative to the flow generated by ξ and also relative to the flow generated by dξ in some chart neighborhood U of u_{0} with chart map ψ satisfying (i) ψ(U) = R^{P} × R^{N}, (ii) ψ(u_{0}) = (0, 0), (iii) T_{u_{0}}ψ(E_{S}^{u}) = R^{P} × {0} and T_{u_{0}}ψ(E_{S}^{s}) = {0} × R^{N}, and there exists a family B_{c}, 0 < c < c_{0}, of C^{r+1}, codimension zero, compact submanifolds with corners of U exhibiting the following properties:

(1) B_{c} is an isolating block with corners relative to the flows generated by ξ and dξ, and with respect to these two flows B_{c} has common exit set B_{c}^{+} and common entrance set B_{c}^{−} that together satisfy ∂B_{c} = B_{c}^{+} ∪ B_{c}^{−};

(2) ψ maps B_{c} onto D_{c}^{P} × D_{c}^{N}, maps B_{c}^{+} onto S_{c}^{P −1} × D_{c}^{N}, maps B_{c}^{−} onto D_{c}^{P} × S_{c}^{N−1}, maps W_{dξ}^{u}(S; B_{c}) onto D_{c}^{P} × {0}, and maps W_{dξ}^{s}(S; B_{c}) onto {0} × D_{c}^{N};

(3) B_{c}/B_{c}^{+} is basepoint homotopic to S^{P} and B_{c}/B_{c}^{−} to S^{N};

(4) for ν = u, s, with W_{ξ c}^{ν} := W_{ξ}^{ν}(S; B_{c}) and W_{dξ c}^{ν} := W_{dξ}^{ν}(S; B_{c}), there exist isotopies of pairs from (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) to (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) in (B_{c}, B_{c}^{+}) and from (W_{dξ c}^{s}, ∂W_{dξ c}^{s}) to (W_{ξ c}^{s}, ∂W_{ξ c}^{s}) in (B_{c}, B_{c}^{−}).

Proof. As is well-known, for all sufficiently small positive ε, T_{u_{0}}M admits an ε-almost proper ordered basis for dξ; i.e., if ε > 0 is sufficiently small there exists a basis of T_{u_{0}}M so that the matrix of dξ relative to this basis has the block form

(7.6)  [ \begin{matrix}E & 0 \\ 0 & C \end{matrix} ]

where E is a P × P expansion matrix in ε-almost proper real Jordan form and C is an N × N contraction matrix in ε-almost proper real Jordan form. In particular, with E or C written in the form D + N where D is block diagonal and N is lower triangular nilpotent, each non-zero entry of N equals ε with ε positive but sufficiently small so that where ⟨ · | · ⟩_{ε} denotes the inner product on R^{P} ⊕ R^{N} making the ε-almost proper basis orthonormal, the quadratic forms Q_{1} on R^{P} and Q_{2} on R^{N} defined by

Q_{1}(x_{1}) = ⟨Ex_{1} | x_{1}⟩_{ε} for x_{1} ∈ R^{P}

and

Q_{2}(x_{2}) = ⟨−Cx_{2} | x_{2}⟩_{ε} for x_{2} ∈ R^{N}

are positive definite.

Let (U, ψ_{0}) be a chart at u_{0} with ψ_{0} mapping U onto R^{m} and sending u_{0} to the origin, and let ξ̂_{0} be the local representative of ξ relative to this chart. Then dξ is represented locally by Dξ̂_{0}(0) and choosing an ε-almost proper basis B_{ε} of T_{u_{0}}M for dξ, the basis is mapped by T_{u_{0}}ψ_{0} to an ε-almost proper basis of R^{m} for Dξ̂_{0}(0). In particular, the matrix of Dξ̂_{0}(0) relative to this basis is the same as the matrix of dξ relative to the basis B_{ε}. Let L be the linear isomorphism from R^{m} to R^{P} × R^{N} obtained by mapping each x ∈ R^{m} to (x_{1}, x_{2}) where x_{1} consists of the first P coordinates of x relative to the ordered basis T_{u_{0}}ψ_{0}(B_{ε}) of R^{m} and where x_{2} consists of the last N coordinates of x relative to this basis.

Define ψ = L^{∘} ψ_{0} and note that the chart (U, ψ) satisfies properties (i)–(iii) in the statement of the theorem. Where ξ̂ denotes the local representative of ξ relative to the chart (U, ψ), the hypotheses of Corollary 7.10 are satisfied when χ̂ := ξ̂ and also with χ̂ := Dξ̂(0, 0), the local representative of dξ relative to the chart (U, ψ), by construction.

Thus, it follows from Corollary 7.10 that for some positive c_{0}, if 0 < c < c_{0}, then B_{c} := ψ^{−1}(D_{c}^{P} × D_{c}^{N}) is an isolating block relative to the flow generated by ξ and also relative to the flow in U generated by dξ, and for both of these flows B_{c} isolates S. Also, the exit and entrance sets of B_{c} as determined by ξ are coincident with those determined by dξ and where B_{c}^{+} (respectively, B_{c}^{−}) denotes the common exit (respectively, entrance) set,

B_{c}^{+} = ψ^{−1}(S_{c}^{P −1} × D_{c}^{N}) and B_{c}^{−} = ψ^{−1}(D_{c}^{P} × S_{c}^{N−1}) .

Because D_{c}^{P} and D_{c}^{N} are contractible it follows when P and N are both positive that B_{c}/B_{c}^{+} is basepoint homotopic to D_{c}^{P} /S_{c}^{P −1}, a homeomorph of S^{P}, and that B_{c}/B_{c}^{−} is basepoint homotopic to D_{c}^{N}/S_{c}^{N−1}, a homeomorph of S^{N}. A separate argument is required to compute B_{c}/B_{c}^{+} if P is zero or to compute B_{c}/B_{c}^{−} if N is zero. For example, if P = 0 and therefore N = m, then D_{c}^{P} = {0} and S_{c}^{P −1} = ∅; hence B_{c}^{+} = ∅ and B_{c}/B_{c}^{+} is homeomorphic to D_{c}^{N} ⨿ {∅} which contracts to {0} ⨿ {∅}, a discrete space with two points and therefore a homeomorph of S^{0}. Similarly, if N = 0, then B_{c}/B_{c}^{−} is homotopic to S^{0}. Thus, the index spaces associated to B_{c} have the required homotopy types.

Note it is a simple consequence of the block diagonal form of the matrix representation of dξ in (7.6) that for c > 0 W_{dξ}^{u}(S; B_{c}) = ψ^{−1}(D_{c}^{P} × {0}) and W_{dξ}^{s}(S; B_{c}) = ψ^{−1}({0} × D_{c}^{N}). Thus, all items in statements (1)–(3) of the present theorem hold.

Finally, let us show that statement (4) holds. With a decrease in c_{0} if necessary, standard proofs of the invariant manifold theorems (e.g., see [Ab-R, Lemma 27.2]) show that there is a C^{r} function G: int(D_{c_{0}}^{P}) → R^{N} so that for 0 < c < c_{0} with G_{c} denoting the restriction of G to D_{c}^{P}, the graph of G_{c} is ψ(W_{ξ}^{u}(S; B_{c})). Note that because W_{ξ}^{u}(S; B_{c}) ⊂ B_{c} and ψ(B_{c}) = D_{c}^{P} × D_{c}^{N}, in fact G_{c} has image in D_{c}^{N}. Let us regard the graph map associated to G_{c} as a section of the trivial N-disk bundle D_{c}^{P} × D_{c}^{N} over D_{c}^{P} so that ψ(W_{dξ}^{u}(S; B_{c})) is the image of the zero section of this bundle. Note that fibers in this bundle over a point in S_{c}^{P −1}, the exit set of D_{c}^{P} relative to the linear flow on R^{P} generated by the matrix E, lie entirely in the exit set of B_{c}. Thus, ψ(W_{ξ}^{u}(S; B_{c})) can be isotoped along fibers of this bundle to the zero section to show that the pair (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) is isotopic to the pair (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) in (B_{c}, B_{c}^{+}). In an analogous manner, one shows that the pair (W_{ξ c}^{s}, ∂W_{ξ c}^{s}) is isotopic in (B_{c}, B_{c}^{−}) to the pair (W_{dξ c}^{s}, ∂W_{dξ c}^{s}). □

7.13 Lemma. Under the assumption of hypotheses HCP and conclusions of Theorem 7.12, if τ^{u} ∈ H^{P} (E^{u}, E^{u} \ {0}; R) is an R-orientation of the vector space E^{u}, then for 0 < c < c_{0}, there exist unique fundamental classes

o_{ξ c}^{u} ∈ H_{P} (W_{ξ c}^{u}, ∂W_{ξ c}^{u}; R)  and  o_{dξ c}^{u} ∈ H_{P} (W_{dξ c}^{u}, ∂W_{dξ c}^{u}; R)

that together satisfy

(1) o_{ξ c}^{u} and o_{dξ c}^{u} have common image in H_{P} (B_{c}, B_{c}^{+});

(2) ⟨i_{c}^{u}^{∗}τ^{u}, o_{dξ c}^{u}⟩ = 1 where i_{c}^{u}: (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) → (E^{u}, E^{u} \ {0}) is the embed-

ding given by the composite

(W_{dξ c}^{u}, ∂W_{dξ c}^{u}) ⟶^{ψ}

(D_{c}^{P} , S_{c}^{P −1}) × {0} ⊂ (R^{P} , R^{P} \ {0}) × {0}

⟶^{(T_{u_{0}} ψ)^{−1}} (E^{u}, E^{u} \ {0}) .

An analogous statement holds for τ^{s} an R-orientation of the vector space E^{s} and the existence of fundamental classes o_{dξ c}^{s} and o_{ξ c}^{s} for W_{dξ c}^{s} and W_{ξ c}^{s} respectively satisfying the analogues of conditions (1) and (2) obtained by replacing B_{c}^{+} with B_{c}^{−} and by replacing E^{u} with E^{s}.

Proof. Because the inclusions D_{c}^{P} ⊂ R^{P} and S_{c}^{P −1} ⊂ R^{P} \ {0} are strong deformation retractions, it follows from the Five Lemma and the cohomology sequences of the pairs (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) and (E^{u}, E^{u} \ {0}) that i_{c}^{u}^{∗} induces isomorphisms on cohomology. Thus i_{c}^{u}^{∗}τ^{u} generates H^{P} (W_{dξ c}^{u}, ∂W_{dξ c}^{u}; R). Hence, for 0 < c < c_{0}, choose o_{dξ c}^{u} to be the unique fundamental class in H_{P} (W_{dξ c}^{u}, ∂W_{dξ c}^{u}; R) dual to i_{c}^{u}^{∗}τ^{u} relative to the Kronecker pairing. Thus, o_{dξ c}^{u} satisfies condition (2). Then condition (i) will be satisfied if, and only if, we choose o_{ξ c}^{u} to be the image of o_{dξ c}^{u} under the isomorphism induced by the end map of the isotopy from (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) to (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) since the two pairs are isotopic in (B_{c}, B_{c}^{+}).

The analogue for the local stable manifolds is proved in the same manner. □

7.14 Definition. Given τ^{u} an R-orientation of E^{u}, for 0 < c < c_{0}, let o_{dξ c}^{u} and o_{ξ c}^{u} be the fundamental classes of W_{dξ c}^{u} and W_{ξ c}^{u} respectively which together satisfy properties (1) and (2) of Lemma 7.13. Call {o_{dξ c}^{u}}_{0<c<c_{0}} the compatible family of fundamental classes determined by the R-orientation τ^{u} of E^{u} for the family of submanifolds W_{dξ}^{u} := {W_{dξ c}^{u}}_{0<c<c_{0}}, and call {o_{ξ c}^{u}}_{0<c<c_{0}} the compatible family of fundamental classes determined by the R-orientation τ^{u} of E^{u} for the family of submanifolds W_{ξ}^{u} := {W_{ξ c}^{u}}_{0<c<c_{0}}.

Similar designations are made for τ^{s} an R-orientation of E^{s} and the families of fundamental classes {o_{dξ c}^{s}}_{0<c<c_{0}} and {o_{ξ c}^{s}}_{0<c<c_{0}} determined by Lemma 7.13 for the families of submanifolds W_{dξ}^{s} := {W_{dξ c}^{s}}_{0<c<c_{0}} and W_{ξ}^{s} := {W_{ξ c}^{s}}_{0<c<c_{0}} respectively.

7.15 Theorem. Under the assumption of hypotheses HCP and conclusions of Theorem 7.12, for S = {u_{0}},

(1) H̃_{P} C (S; R) ≃ R and H̃_{k}C (S; R) ≃ 0 for k ≠ P;

(2) each R-orientation τ^{u} ∈ H^{P} (E^{u}, E^{u} \{0}; R) determines a unique generator o_{ξ}^{u} ∈ H̃_{P} C (S; R) characterized by the property that it is the common canonical image of the members of the compatible family of fundamental classes determined by the R-orientation τ^{u} of E^{u} for the family of submanifolds W_{ξ}^{u};

(3) H̃_{N} C∗(S; R) ≃ R and H̃_{k}C∗(S; R) ≃ 0 for k ≠ N; (4) each R-orientation τ^{s} ∈ H^{N}(E^{s}, E^{s} \ {0}; R) determines a unique genera-

tor o_{ξ}^{s} ∈ H̃_{N} C∗(S; R) characterized by the property that it is the common canonical image of the members of the compatible family of fundamental classes determined by the R-orientation τ^{s} of E^{s} for the family of submanifolds W_{ξ}^{s};

(5) ^{#}(o_{ξ}^{u} ⌢_{•} o_{ξ}^{s}) = σ(τ^{u}, τ^{s}, [u_{0}], O^{M}).

Proof. Statements (1) and (3) are an immediate consequence of statement (3) of Theorem 7.12.

To show that two fundamental classes from a compatible family of fundamental classes determined by an R-orientation τ^{u} of E^{u} for the submanifolds W_{ξ}^{u} have common canonical image in H̃_{∗}C (S; R), it suffices to show that under inclusion induced maps the two fundamental classes have common image in the homology of some index space for S because H̃_{∗}C (S; R) is defined as the direct limit of the homology modules of the index spaces in C (S). Such a procedure will be carried out by (a) showing it true for the compatible family of fundamental classes determined by an R-orientation τ^{u} of E^{u} for the family of submanifolds W_{dξ}^{u}, (b) noting that the index space of the common image is also an index space for the flow generated by ξ, and (c) concluding by using property (1) of Lemma 7.13.

Suppose that τ^{u} is an R-orientation of E^{u} and let {o_{dξ c}^{u}}_{0<c<c_{0}} and {o_{ξ c}^{u}}_{0<c<c_{0}} be the compatible families of fundamental classes determined by τ^{u} according to the prescription of Lemma 7.13 for the families of submanifolds W_{dξ}^{u} and W_{ξ}^{u} respectively. For 0 < c < c_{0}, let k_{c}^{u} be the inclusion (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) ⊂ (B_{c}, B_{c}^{+}) and let k̇_{c}^{u} be the inclusion (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) ⊂ (B_{c}, B_{c}^{+}). Now it is immediate from conclusion (2) of Theorem 7.12 that the pair (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) is a strong deformation retract of the pair (B_{c}, B_{c}^{+}), whence k̇_{c}^{u} is a homotopy equivalence for 0 < c < c_{0}. Also, because (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) is isotopic in (B_{c}, B_{c}^{+}) to (W_{ξ c}^{u}, ∂W_{ξ c}^{u}), it follows that the latter pair is a weak deformation retract of (B_{c}, B_{c}^{+}); i.e., also k_{c}^{u} is a homotopy equivalence for 0 < c < c_{0}. Hence, for 0 < c < c_{0}, the canonical image of o_{ξ c}^{u} in H̃_{P} C (S; R), namely, b̄_{∗}^{+}k_{c ∗}^{u}o_{ξ c}^{u}, generates H̃_{P} C (S; R). To finish proving statement (2) we must show that the canonical image is the same for each c ∈ ]0, c_{0} [. To do so, where 0 < c′ < c′′ < c_{0}, we will use the diagram

(7.7)  (B_{c′}, B_{c′}^{+}) (B_{c′′}, CB_{c′′}^{+})^{[b_{c′}^{+} ]}_{[h]}  B_{c′}/B_{c′}^{+}_{[j̄_{c′}]} B_{c′′}/CB_{c′′}^{+} (B_{c′′}, B_{c′′}^{+})  _{[b_{c′′}^{+} ]}  B_{c′′}/B_{c′′}^{+}_{[j̄_{c′′}]}

of homotopy classes of maps whose construction now begins.

Set CD_{c′′}^{P +} := cl (D_{c′′}^{P} \ D_{c′}^{P}) and set CB_{c′′}^{+} := ψ^{−1}(CD_{c′′}^{P +} × D_{c′′}^{N}). Clearly, S has nested index pair (B_{c′′}, CB_{c′′}^{+}) relative to the flow in U generated by dξ and also relative to the flow generated by ξ. By construction, for λ = c′, c′′, there is an inclusion j_{λ}: (B_{λ}, B_{λ}^{+}) ⊂ (B_{c′′}, CB_{c′′}^{+}) and therefore an inclusion induced map of index pairs j̄_{λ}: B_{λ}/B_{λ}^{+} → B_{c′′}/CB_{c′′}^{+}. The homotopy classes of these inclusions and inclusion induced maps define the slanted arrows and the right vertical arrows of diagram (7.7), and each horizontal arrow there is taken to be the homotopy class of the quotient map from the index pair to its index space.

To define the left vertical arrow in diagram (7.7), we must first for 0 < c, c̄ < c_{0} define an isotopy H^{c c̄} from W_{dξ c}^{u} to W_{dξ c̄}^{u} whose image throughout the isotopy lies in the chart neighborhood U guaranteed by Theorem 7.12 and is mapped by the chart map ψ into R^{P} × {0}. The definition will use the flow in U generated by dξ which will be denoted as the map (u, t) ↦ u∗t. All exit time maps in the definition are taken relative to the flow of dξ. For 0 < c ≤ c̄ < c_{0}, define H^{c c̄} from W_{dξ c}^{u} to W_{dξ c̄}^{u} as follows: for 0 ≤ t ≤ 1,

H_{t}^{c c̄}(u) := \{ \begin{matrix}u & if u = u_{0} \\ u ∗ t τ_{Bc̄}(u ∗ τ_{B_{c}}(u)) & if u ∈ W_{dξ c}^{u} and u ≠ u_{0}. \end{matrix}

Note that if c = c̄, then H_{t}^{c c} = 1_{W_{dξ c}^{u}}. For 0 < c < c̄ < c_{0}, define H^{c̄c} by

H_{t}^{c̄c} := H_{1−t}^{c c̄} ^{∘} (H_{1}^{c c̄})^{−1} for 0 ≤ t ≤ 1.

Geometrically, for 0 < c < c̄ < c_{0}, for 0 ≤ t ≤ 1, H_{t}^{c c̄} leaves the critical point fixed and translates each non-critical orbit segment of W_{dξ c}^{u} forward by a time (scaled by t) determined by the unique point on the orbit segment in the exit set of B_{c} and is the time it takes that point to reach the exit set of Bc̄. Thus, each H_{t}^{c c̄} is one-to-one and maps W_{dξ c}^{u} into W_{dξ c̄}^{u}, and H_{1}^{c c̄} maps W_{dξ c}^{u} onto W_{dξ c̄}^{u}. Continuity of H^{c c̄} is clear except possibly at points (u_{0}, t). To see continuity at such points, let T_{c c̄} = sup{τ_{Bc̄}(u) : u ∈ ∂W_{dξ c}^{u}} and note that T_{c c̄} is finite as a consequence of the inequality (7.5_{1}) as it applies to the proof of Corollary 7.10 and the application of that corollary to the flow in U generated by dξ. Now, if u_{0} ∈ W open, by continuity of the flow, there exists an open neighborhood V of u_{0}, V ⊂ B_{c}, so that V ∗ [0, T_{c c̄}] ⊂ W. It follows that if (u, t) ∈ V × [0, 1] then H^{c c̄}(u, t) ∈ W. Thus H^{c c̄} is an isotopy in W_{dξ c̄}^{u} from W_{dξ c}^{u} to W_{dξ c̄}^{u} for 0 < c ≤ c̄ < c_{0}; hence also for 0 < c̄ < c < c_{0}.

To define the left vertical arrow in diagram (7.7), recall that for 0 < c < c_{0}, ψ(W_{dξ c}^{u}) = D_{c}^{P} × {0}, let π_{1}: R^{P} × R^{N} → R^{P} be the projection to the first factor, let i_{c′}(x) = (x, 0) for x ∈ D_{c′}^{P}, let ℓ_{c′ c′′} be the inclusion D_{c′}^{N} ⊂ D_{c′′}^{N}, define

K_{t} := ψ^{−1} ^{∘} ((π_{1} ^{∘} ψ^{∘} H_{t}^{c′ c′′} ^{∘} ψ^{−1} ^{∘} i_{c′}) × ℓ_{c′ c′′})^{∘} ψ for 0 ≤ t ≤ 1,

and set h := K_{1}. Observe that h maps (B_{c′}, B_{c′}^{+}) to (B_{c′′}, B_{c′′}^{+}) and define the left vertical arrow of diagram (7.7) to be the homotopy class of the map h restricted to (B_{c′}, B_{c′}^{+}).

Diagram (7.7) is commutative. For clearly the upper and lower trapezoids commute being diagrams of the classes of inclusions and quotients, and the left hand triangle commutes because j_{c′′} ^{∘}h is homotopic to the inclusion j_{c′} via the homotopy K_{t} (0 ≤ t ≤ 1).

Claim H_{1 ∗}^{c c̄}o_{dξ c}^{u} = o_{dξ c̄}^{u} for 0 < c, c̄ < c_{0}. For the embedding i_{c}^{u}: (W_{dξ c}^{u}, ∂W_{dξ c}^{u}) → (E^{u}, E^{u}\{0}) of Lemma 7.13 is homotopic to the composite iu ; c̄^{∘}H_{1}^{c c̄} via H^{c c̄}. Thus,

1 = ⟨i_{c}^{u}^{∗}τ^{u}, o_{dξ c}^{u}⟩ = ⟨H_{1}^{c c̄∗} iu ; c̄^{∗}τ^{u}, o_{dξ c}^{u}⟩ = ⟨iu ; c̄^{∗}τ^{u}, H_{1 ∗}^{c c̄}o_{dξ c}^{u}⟩ ,

whence the claim is immediate by definition of o_{dξ c̄}^{u} as the class dual to iu ; c̄^{∗}τ^{u}.

Suppose that 0 < c, c̄ < c_{0}. Let us now show that o_{ξ c}^{u} and o_{ξ c̄}^{u} have common canonical image in H̃_{P} C (S; R). If c < c̄, set c′ := c and c′′ := c̄; otherwise, if c̄ < c, set c′ := c̄ and c′′ := c. Note that k̇_{c′′}^{u} ^{∘} H_{1}^{c′ c′′} = h ^{∘} k̇_{c′}^{u}. Then, by the claim of the previous paragraph H_{1 ∗}^{c c̄}o_{dξ c}^{u} = o_{dξ c̄}^{u}; also H_{1}^{c̄c} = (H_{1}^{c c̄})^{−1}. Hence, if c = c′ < c′′ = c̄, it follows that

(7.8a)  h_{∗}k̇_{c ∗}^{u}o_{dξ c}^{u} = k̇_{c̄∗}^{u}o_{dξ c̄}^{u} ,

but if c̄ = c′ < c′′ = c, it follows that

(7.8b)  h_{∗}k̇_{c̄∗}^{u}o_{dξ c̄}^{u} = k̇_{c ∗}^{u}o_{dξ c}^{u} .

The equations (7.8), the commutativity of diagram (7.7), and the fact—see [K5, Proposition 3.1]—that the homotopy class of an inclusion induced map between index spaces is that of the unique morphism in the Conley index between those index spaces immediately imply that b_{c ∗}^{+}k̇_{c ∗}^{u}o_{dξ c}^{u} and b_{c̄∗}^{+}k̇_{c̄∗}^{u}o_{dξ c̄}^{u} have a common successor in H̃_{P} (B_{c′′}/CB_{c′′}^{+}) where c′′ = c̄ if c < c̄, but c′′ = c if c̄ < c. Because condition (i) of Lemma 7.13 implies that k̇_{c ∗}^{u}o_{dξ c}^{u} = k_{c ∗}^{u}o_{ξ c}^{u} and that k̇_{c̄∗}^{u}o_{dξ c̄}^{u} = k_{c̄∗}^{u}o_{ξ c̄}^{u}, it follows that b_{c ∗}^{+}k̇_{c ∗}^{u}o_{ξ c}^{u} and b_{c̄∗}^{+}k̇_{c̄∗}^{u}o_{ξ c̄}^{u} have a common successor in H̃_{P} (B_{c′′}/CB_{c′′}^{+}) where c′′ = c̄ if c < c̄, but c′′ = c if c̄ < c. It follows that o_{ξ c̄}^{u} and o_{ξ c}^{u} have common canonical image in H̃_{P} C (S; R) because as constructed (B_{c′′}, CB_{c′′}^{+}) is an index pair for S relative to the flow generated by ξ. This completes the proof of statement (2); statement (4) is proved similarly.

Finally, statement (5) follows immediately from Lemma 7.7 using the flow of dξ. □

7.16 The Proof of Theorem 7.5. Under the assumption of hypotheses HCP, statements (1)–(6) of Theorem 7.4 follow from the results proved above: Statement (1) from Theorem 7.12; statement (2) from conclusion (3) of Theorem 7.12, the connected simple system properties of C (S) and C∗(S), and our earlier observation on the Thom space of a vector bundle over a point; and statements (3)–(6) from Theorem 7.15. Note conclusion (3) of Theorem 7.5 follows from conclusion (2) (i.e., statement (5) of Theorem 7.4) and Proposition 2.13. □

D. Results Leading to the Proof of Theorem 7.6. The proof of Theorem 7.6 will be the object of the rest of this section and will be carried out in a lengthy series of steps. Throughout, let us assume that hypotheses HCO hold for the closed orbit S of ξ and let us fix a tubular neighborhood U of S with tubular map p retracting U onto S and with (U, p, S) a vector bundle over S isomorphic to the normal bundle of S. It will also be convenient to use the following notation: if f: V′ → V is a local diffeomorphism onto V and ζ is a vectorfield on V, then the formula ζ′(z) = (T_{z}f)^{−1}^{∘} ζ(f(z)) defines a vectorfield on V′ and we write ζ′ = f^{∗}ζ; i.e., f^{∗}ζ is the only vectorfield χ on V′ that satisfies the relation T f ^{∘} χ = ζ ^{∘} f. If g: V′′ → V′ is a local diffeomorphism onto V′, then (f ^{∘} g)^{∗}ζ = g^{∗}f^{∗}ζ by the chain rule. It is also convenient to use e^{iψ} as an abbreviation for (cos(ψ), sin(ψ)) for ψ ∈ R and is tautological if one identifies the complex numbers with ordered pairs of reals with i = (0, 1) and the usual multiplication and addition.

Underlying the entire proof of Theorem 7.6 is the following lemma that summarizes the treatment of closed orbits and Floquet theory found in [Ab-R], especially the proof of Theorem 28.1 in which the local stable and unstable manifolds of an elementary closed orbit are constructed. Because there are proofs of existence of a tubular neighborhood of a closed submanifold that are valid when the submanifold and ambient manifold are at best C^{2}—for example see [H]—it need not be assumed that M is at least C^{3} as is done in[Ab-R].

7.17 Lemma. There exists a C^{r} universal covering σ: R × R^{P} × R^{N} → U with the properties:

(1) σ(R × {0} × {0}) = S;

(2) σ(θ + 2τ, x_{1}, x_{2}) = σ(θ, x_{1}, x_{2}) for (θ, x_{1}, x_{2}) ∈ R × R^{P} × R^{N}; hence, where /2τ: R × R^{P} × R^{N} → Ũ := S^{1} × R^{P} × R^{N} is defined by

/2τ(θ, x_{1}, x_{2}) := (e^{iθπ/τ} , x_{1}, x_{2})

and σ_{2}: Ũ → U is defined by

σ_{2} := σ^{∘} (/2τ)^{−1} ,

σ_{2}: Ũ → U is a well-defined double cover of U;

(3) where p_{1}: Ũ → S^{1} is projection on the first factor and where c: S^{1} → S is defined by c(e^{iθπ/τ}) = u_{0} · θ for some u_{0} ∈ S,

Ũ ⟶^{σ_{2}} U
p_{1}↓  ↓^{p}
S^{1} ⟶^{c} S

is a vector bundle morphism;

(4) the involutive diffeomorphism j: Ũ → Ũ determined by the double cover σ_{2}: Ũ → U (i.e., j ^{∘} j = 1Ũ and σ_{2} ^{∘} j = σ_{2}) has the form

j(e^{iθπ/τ} , x_{1}, x_{2}) = (e^{i(θ+τ)π/τ} , L_{+}(θ)x_{1}, L_{−}(θ)x_{2})

where for each θ ∈ R, L_{+}(θ) and L_{−}(θ) are linear automorphisms, the former of R^{P}, the latter of R^{N}, with L_{+}(θ) (respectively, L_{−}(θ)) orientation preserving or reversing as N_{S}^{u} (respectively, N_{S}^{s}) is or is not orientable over Z and with L_{±}(θ + 2τ) = L_{±}(θ); (5) there exist matrices E ∈ Gl(P; R) and C ∈ Gl(N; R) in ε-almost proper real

canonical Jordan form with each eigenvalue of E having positive real part and each eigenvalue of C having negative real part and with the quadratic forms x_{1} ↦ ⟨Ex_{1} | x_{1}⟩ and x_{2} ↦ ⟨−Cx_{2} | x_{2}⟩ defined for x_{1} ∈ R^{P} and for x_{2} ∈ R^{N} being positive definite so that the vectorfield ξ_{σ} on R × R^{P} × R^{N} defined by ξ_{σ} := σ^{∗}ξ has local representative ξ̂_{σ} of the form

\begin{aligned}ξ̂_{σ 1}(θ, x_{1}, x_{2}) & = 1 & + Θ(θ, x_{1}, x_{2}) & ∈ R \\ ξ̂_{σ 2}(θ, x_{1}, x_{2}) & = Ex_{1} & + R_{1}(θ, x_{1}, x_{2}) & ∈ R^{P} \\ ξ̂_{σ 3}(θ, x_{1}, x_{2}) & = Cx_{2} & + R_{2}(θ, x_{1}, x_{2}) & ∈ R^{N} \end{aligned}  for (θ, x_{1}, x_{2}) ∈ R × R^{P} × R^{N}

where Θ, R_{1} and R_{2} are 2τ-periodic in θ and vanish at all points (θ, 0, 0) ∈ R × R^{P} × R^{N} as do the partial derivatives D_{k}R_{j}, for k = 2, 3 and j = 1, 2.

Using the notation of Lemma 7.17, the following notions occur in the next lemma. A vectorfield χ on Ũ is j-invariant if j^{∗}χ = χ; a subset C̃ of Ũ is j-invariant if j(C̃) = C̃; a flow φ̃ in C̃ is j-invariant if j ^{∘} φ̃_{t} ^{∘} j = φ̃_{t} for each t ∈ R. Also, if f: V′ → V is a covering map, ψ′ a flow in V′, and ψ a flow in V, then ψ′ covers ψ if f ^{∘} ψ′_{t} = ψ_{t} ^{∘} f for each t ∈ R.

7.18 Lemma. Using the conclusions and notation of Lemma 7.17 if χ̂_{σ} is a self-map of R × R^{P} × R^{N} satisfying

χ̂_{σ}(θ + 2τ, x_{1}, x_{2}) = χ̂_{σ}(θ, x_{1}, x_{2}) for (θ, x_{1}, x_{2}) ∈ R × R^{P} × R^{N},

and if χ_{σ} is the vectorfield on R × R^{P} × R^{N} whose local representative is χ̂_{σ} then the following hold:

(1) there exists a unique vectorfield χ_{2} on Ũ satisfying χ_{σ} = (/2τ)^{∗}χ_{2} and the flow of χ_{σ} covers the flow of χ_{2};

(2) χ_{2} = σ_{2}^{∗}χ for some (necessarily unique) vectorfield χ on U if, and only if, j^{∗}χ_{2} = χ_{2};

(3) if χ_{2} is j-invariant, then the flow it generates is j-invariant and covers the flow generated by χ;

(4) if χ_{2} is transverse to the codimension one, C^{1} submanifold Σ̃ of Ũ, then j^{∗}χ_{2} is transverse to the codimension one, C^{1} submanifold j(Σ̃);

(5) if χ_{2} is j-invariant and transverse to the codimension one j-invariant C^{1} submanifold Σ_{2}, then Σ := σ_{2}(Σ_{2}) is a codimension one submanifold of U and χ is transverse to Σ;

(6) if χ_{2} is j-invariant, if C̃ is a j-invariant isolated invariant set of the flow generated by χ_{2}, and if ⟨P_{1}, P_{0}⟩ is an index pair for C̃ relative to that flow, then the definition (Q_{1}, Q_{0}) := (P_{1}∩j(P_{1}), Q_{1}∩(P_{0}∩j(P_{0}))) makes (Q_{1}, Q_{0}) a nested, j-invariant index pair relative to that flow and (σ_{2}(Q_{1}), σ_{2}(Q_{0})) a nested index pair for C := σ_{2}(C̃) relative to the flow generated by χ; in particular, if C̃ is isolated by the block K, then N := K ∩ j(K) is a j-invariant isolating block for C̃ with N^{±} = (K^{±} ∩ j(K)) ∪ (j(K^{±}) ∩ K) and B := σ_{2}(N) is a block isolating C relative to the flow generated by χ with B^{±} = σ_{2}(N^{±}) and with B a C^{r} block with corners if N is; (7) conversely, if χ_{2} is j-invariant and B is an isolating block relative to the

flow of χ isolating C, then σ_{2}^{−1}(B) is a j-invariant isolating block isolating C̃ := σ_{2}^{−1}(C) which is j-invariant.

Proof. In the main, the statements of the lemma follow straightforwardly from the definitions. Note that uniqueness of the integral curve of χ_{2} through a given initial point implies that the flow of χ_{σ} covers that of χ_{2}, and a similar argument implies that the flow of χ_{2} covers that of χ when χ_{2} is j-invariant. Also, when χ_{2} is j-invariant and K is a block for C̃ as in statement (6), it is easy to prove that j(K) isolates C̃ as a consequence of the j-invariance of the flow. The j-invariance also implies that τ_{j(K)} and τ∗_{j(K)} are lower semi-continuous because τ_{K} and τ∗_{K} are; upper semi-continuity is a simple consequence of the compactness of j(K). Thus, j(K) is a block whence K ∩ j(K) is a j-invariant block since as noted in Definition 1.4(B), the intersection of two blocks is always a block. Proofs of the remaining statements are left as an exercise for the reader. □

Let ξ_{σ} be the vectorfield defined in statement (5) of Lemma 7.17.

7.19 Corollary. There exists a unique j-invariant vectorfield ξ_{2} on Ũ satisfying the relations ξ_{2} = σ_{2}^{∗}ξ and ξ_{σ} = (/2τ)^{∗}ξ_{2}.

Proof. The existence and uniqueness of ξ_{2} satisfying ξ_{σ} = (/2τ)^{∗}ξ_{2} is immediate from statement (1) of the lemma. As ξ_{σ} := σ^{∗}ξ = (/2τ)^{∗}σ_{2}^{∗}ξ, the uniqueness of ξ_{2} forces ξ_{2} = σ_{2}^{∗}ξ. Hence, ξ_{2} is j-invariant by statement (2) of the lemma. □

Let E ∈ Gl(P; R) and C ∈ Gl(N; R) be the matrices of statement (5) of Lemma 7.17 and let Fξ_{σ} and F^{⊥}ξ_{σ} be the vectorfields on R × R^{P} × R^{N} whose respective local representatives Fξ̂_{σ} and F^{⊥}ξ̂_{σ} are given as follows:

Fξ̂_{σ}(θ, x_{1}, x_{2}) := (1, Ex_{1}, Cx_{2}) ;
F^{⊥}ξ̂_{σ}(θ, x_{1}, x_{2}) := (0, Ex_{1}, Cx_{2}) .

7.20 Corollary. There exists a unique j-invariant vectorfield Fξ_{2} on Ũ satisfying Fξ_{σ} = (/2τ)^{∗}Fξ_{2} and a unique vectorfield Fξ on U satisfying Fξ_{2} = σ_{2}^{∗}Fξ and Fξ_{σ} = σ^{∗}Fξ. An analogous statement holds for the vectorfield F^{⊥}ξ_{σ}.

Proof. The existence of a unique Fξ_{2} satisfying Fξ_{σ} = (/2τ)^{∗}Fξ_{2} is immediate from statement (1) of the lemma. Next, note that the involution j lifts to a self-homeomorphism j_{σ} of R × R^{P} × R^{N} given explicitly by j_{σ}(θ, x_{1}, x_{2}) = (θ +τ, L_{+}(θ)x_{1}, L_{−}(θ)x_{2}) and characterized by the relation /2τ^{∘}j_{σ} = j^{∘}/2τ. From this relation and the fact that j^{∗}ξ_{2} = ξ_{2}, it follows that

T (/2τ)^{∘} ξ_{σ} = T (/2τ)^{∘} T j_{σ} ^{∘} ξ_{σ} ^{∘} j_{σ} .

This equality and the uniqueness of Taylor series expansions forces the first-order Taylor polynomials of the local representative ξ̂_{σ} at (θ, 0, 0) and of the local representative of T j_{σ} ^{∘} ξ_{σ} ^{∘} j_{σ} at (θ, 0, 0) to be equal. Because Fξ̂_{σ} is just the first-order Taylor polynomial of ξ̂_{σ} at (θ, 0, 0) with the linear term in the first (i.e., θ) component deleted, the just cited equality of first-order Taylor polynomials allows one to compute that

T (/2τ)^{∘} Fξ_{σ} = T (/2τ)^{∘} T j_{σ} ^{∘} Fξ_{σ} ^{∘} j_{σ} .

From this and the characterizations T (/2τ)^{∘}Fξ_{σ} = Fξ_{2}^{∘}/2τ and /2τ^{∘}j_{σ} = j^{∘}/2τ, it readily follows that

Fξ_{2} ^{∘} /2τ = T j ^{∘} Fξ_{2} ^{∘} j ^{∘} /2τ ;

i.e., Fξ_{2} is j-invariant. By statement (2) of the lemma, there exists a vectorfield Fξ on the tubular neighborhood U that is characterized by the equation σ_{2}^{∗}Fξ = Fξ_{2}.

The existence of F^{⊥}ξ_{2} is again immediate from statement (1) of the lemma and it is j-invariant as an immediate consequence of the j-invariance of Fξ_{2}. Hence, that F^{⊥}ξ satisfies the required relations is immediate from statement (2) of the lemma. □

In the following theorem we want to distinguish between bundles and fiber bundles especially as regards usage of “subbundle” and “fiber subbundle” and will do so following [Hus] and [St]. A bundle η is a triple, η = (E, p, X) where p: E → X is a continuous surjective map: E is called the total space, p the projection, and X the base space of the bundle. For x ∈ X, p^{−1}(x) is the fiber at x. No further structure is implied. Then ω := (E′, p′, X′) is a subbundle of η if, and only if, X′ = X, E′ ⊂ E, and p′ = p|E′. Note that there is a category of bundles where a morphism is essentially a fiber preserving map of the total spaces. Hence an equivalence between two bundles is essentially a fiber preserving homeomorphism of their total spaces. See [Hus, p. 14] for details and precise definitions.

A fiber bundle is a bundle with additional structure. The reader is referred to [St,§§2.1–4] for the definition as it is used here of a fiber bundle η with fiber Y and topological structure group G. Given Z ⊂ Y, if the action of G on Y induces an action of G on Z, i.e., if Z is invariant under the action of G on Y, then Steenrod defines—see [St, p. 24]—the subbundle of η determined by the invariant subspace Z of Y, let us denote it by η_{Z}, that is a fiber bundle with fiber Z and topological structure group H = G/G_{Z} where G_{Z} := {g ∈ G : ∀z ∈ Z, g · z = z}. Note that G_{Z} is trivial and H ≡ G if G acts effectively on Z. Call a fiber bundle ω with fiber Z and structure group H a fiber subbundle of a fiber bundle η with fiber Y and structure group G if, and only if, Z ⊂ Y, ω = η_{Z}, and H ≡ G/G_{Z}. We shall not need to mention explicitly the notion of equivalence of fiber bundles.

The need for the distinction between the notions of subbundle and of fiber subbundle arises because we will necessarily consider k-disk bundles that do not admit O(k) as natural structure group even though the disk bundle is a subbundle of a kplane bundle. These somewhat anomalous k-disk bundles have as natural structure group a subgroup 𝔊_{k} of Homeo(R^{k}) that is an isomorphic image of Gl(k; R) but not by the inclusion homomorphism. Specifically, for A a linear automorphism of R^{k} and where ∥ · ∥ denotes the Euclidean norm on R^{k}, let g_{A} be the homeomorphism of R^{k} with itself defined by

g_{A}(x) := \{ \begin{matrix}0 & if x = 0 \\ \frac{∥x∥}{∥Ax∥}Ax & if x ≠ 0. \end{matrix}

Clearly, g_{A} is norm-preserving and homogeneous of degree one, i.e., ∥g_{A}(x)∥ = ∥x∥ and g_{A}(tx) = tg_{A}(x) for x ∈ R^{k} and t ∈ R. However, g_{A} is additive if, and only if, A is orthogonal in which case g_{A} = A. It is also straightforward to show that g_{A} ^{∘} g_{B} = g_{AB} for A and B both linear automorphisms of R^{k}. It follows that

𝔊_{k} := {g_{A} : A ∈ Gl(k; R)}

is a group of norm-preserving homeomorphisms of R^{k} and is therefore admissible as a structure group for a k-disk bundle. Note that O(k) is a subgroup of 𝔊_{k} acting in the usual manner (i.e., matrix multiplication) on elements of R^{k} as a consequence of our observation that g_{A} = A for A ∈ O(k).

To simplify the statement of the theorem, let us identify N_{S}, the normal bundle of S, with its realization in the tubular neighborhood U so that each point of U lies on a unique fiber of N_{S}. Note too that N_{S} = N_{S}^{u} ⊕ N_{S}^{s}. Also, the sphere bundle associated to a disk bundle is denoted below by placing a dot over the symbol used to represent the disk bundle; e.g., if ω is a disk bundle, ω̇ is its associated sphere bundle.

7.21 Theorem. S is a hyperbolic closed orbit of Fξ and is a circle of degenerate critical points of F^{⊥}ξ. The tubular neighborhood U of S with tubular map p: U → S contains a family B_{c}, 0 < c < c_{0}, of C^{r}, codimension zero, compact manifolds with corners exhibiting the following properties:

(1) each B_{c} is an isolating block for S for the flows generated by ξ, Fξ, and F^{⊥}ξ and is the total space of a fiber bundle β_{c} over S with fiber D_{c}^{P} × D_{c}^{N} and structure group 𝔊_{P} × 𝔊_{N} that is also a subbundle of the tubular neighborhood (U, p, S);

(2) B_{c}^{±} is the total space of a fiber subbundle β_{c}^{±} of β_{c} with β_{c}^{+} the fiber subbundle determined by S_{c}^{P −1} × D_{c}^{N} and β_{c}^{−} the fiber subbundle determined by D_{c}^{P} × S_{c}^{N−1};

(3) for 0 < c′ < c < c_{0}, β_{c′} is the fiber subbundle of β_{c} determined by the 𝔊_{P} × 𝔊_{N}-invariant subset D_{c′}^{P} × D_{c′}^{N} of D_{c}^{P} × D_{c}^{N};

(4) for ν = u, s, W_{F^{⊥}ξ}^{ν}(S; B_{c}) = W_{Fξ}^{ν}(S; B_{c}) and W_{Fξ}^{u}(S; B_{c}) and W_{Fξ}^{s}(S; B_{c}) are the respective total spaces of fiber subbundles ω_{Fξ c}^{u} and ω_{Fξ c}^{s} of β_{c}, the former the P -disk subbundle determined by D_{c}^{P} × {0}, the latter the N-disk subbundle determined by {0} × D_{c}^{N};

(5) for ν = u, s, ω_{Fξ c}^{ν} is a subbundle of N_{S}^{ν} and is either the 0-disk bundle over S or is bundle equivalent to the unit disk bundle of N_{S}^{ν} relative to any norm;

(6) the splitting N_{S} = N_{S}^{u} ⊕ N_{S}^{s} induces splittings

β_{c} ≡ ω_{Fξ c}^{u} ⊕ ω_{Fξ c}^{s} , β_{c}^{+} ≡ ω̇_{Fξ c}^{u} ⊕ ω_{Fξ c}^{s} , β_{c}^{−} ≡ ω_{Fξ c}^{u} ⊕ ω̇_{Fξ c}^{s};

(7) W_{ξ}^{u}(S; B_{c}) and W_{ξ}^{s}(S; B_{c}) are the respective total spaces of subbundles ω_{ξ c}^{u} and ω_{ξ c}^{s} of β_{c}, the former a P -disk bundle, the latter an N-disk bundle, and for ν = u, s, with the definitions W_{ξ c}^{ν} := W_{ξ}^{ν}(S; B_{c}) and W_{Fξ c}^{ν} := W_{Fξ}^{ν}(S; B_{c}), for sgn u = + and sgn s = −, the pair (W_{ξ c}^{ν}, ∂W_{ξ c}^{ν}) is isotopic in (B_{c} , B_{c}^{sgn ν}) to the pair (W_{Fξ c}^{ν}, ∂W_{Fξ c}^{ν}) via an isotopy whose restriction to the fiber pair of (ω_{ξ c}^{ν}, ω̇_{ξ c}^{ν}) above a point u ∈ S is an isotopy in the fiber pair of (β_{c}, β_{c}^{sgn ν}) above u to the fiber pair of (ω_{Fξ c}^{ν}, ω̇_{Fξ c}^{ν}) above u;

(8) for ν = u, s, for u ∈ S, W_{ξ}^{νν}(u; B_{c}) coincides with the fiber at u of ω_{ξ c}^{ν}; (9) for ν = u, s, the pair (W_{Fξ c}^{ν}, ∂W_{Fξ c}^{ν}) is a strong deformation retract of the

pair (B_{c}, B_{c}^{sgn ν}) and the pair (W_{ξ c}^{ν}, ∂W_{ξ c}^{ν}) is a weak deformation retract of the pair (B_{c}, B_{c}^{sgn ν});

(10) B_{c}/B_{c}^{+} is basepoint homotopic to the Thom space of N_{S}^{u};

(11) B_{c}/B_{c}^{−} is basepoint homotopic to the Thom space of N_{S}^{s}.

Proof. Let Fφ^{σ} denote the flow generated by Fξ_{σ}. Because Fφ_{t}^{σ}(θ, x_{1}, x_{2}) = (θ+t, e^{tE}x_{1}, e^{tC}x_{2}), it is immediate from (5) of Lemma 7.17 and (1) of Lemma 7.18 that the vectorfields Fξ_{2} and ξ_{2} when restricted to S̃ := S^{1} × {0} × {0} ⊂ Ũ are tangent to S̃, are in fact equal on S̃, and integrate to the same uniform rotation of S̃ making S̃ a closed orbit with fundamental period 2τ for both vectorfields. Further, it is clear that collectively the eigenvalues of E and C form a complete set of Floquet exponents of S̃ as a closed orbit of either Fξ_{2} or of ξ_{2}, whence S̃ is a hyperbolic closed orbit of both vectorfields. As σ_{2} is a double cover and since the flow generated by Fξ_{2} covers that generated by Fξ, it follows that S is a closed orbit of Fξ of prime period τ. Thus, where Fφ denotes the flow generated by Fξ and where Fφ^{2} that generated by Fξ_{2}, if ũ ∈ S̃ and u := σ_{2}(ũ), it follows that λ is an eigenvalue of ^{C}T_{u}Fφ_{τ}, the complexification of T_{u}Fφ_{τ}, with eigenvector v if, and only if, λ^{2} is an eigenvalue of ^{C}TũFφ_{2τ}^{2} with eigenvector ṽ = ^{C}(Tũσ_{2})^{−1}v. It follows that S is a hyperbolic closed orbit of Fξ because it is one for Fξ_{2}. Also, as F^{⊥}ξ_{2} is just Fξ_{2} minus the rotational component, clearly each point of S̃ is a degenerate critical point for F^{⊥}ξ_{2}. It follows that each point of S is a degenerate critical point for F^{⊥}ξ.

From statement (5) of Lemma 7.17 it follows that the hypotheses of Proposition 7.9 are satisfied for χ̂ := ξ̂_{σ}, for χ̂ := Fξ̂_{σ}, and for χ̂ := F^{⊥}ξ̂_{σ}. Thus, application of that proposition yields that S̃ := S^{1} × {0} × {0} is an isolated invariant set in Ũ relative to the flows generated by ξ_{2}, Fξ_{2}, and F^{⊥}ξ_{2}. Further, the proposition yields c_{0} > 0 so that if 0 < c < c_{0} then

K_{c} := S^{1} × D_{c}^{P} × D_{c}^{N}

is an isolating block for S̃ relative to each of these flows that relative to all three has common exit set K_{c}^{+} and common entrance set K_{c}^{−} with

K_{c}^{+} = S^{1} × S_{c}^{P −1} × D_{c}^{N}  and  K_{c}^{−} = S^{1} × D_{c}^{P} × S_{c}^{N−1} .

By Lemma 7.18(6), a family of j-invariant blocks for S̃ relative to the flows of ξ_{2}, Fξ_{2}, and F^{⊥}ξ_{2} is obtained by setting

N_{c} := K_{c} ∩ j(K_{c}) for 0 < c < c_{0},

and relative to those flows N_{c} has common exit set N_{c}^{+} and common entrance set N_{c}^{−}.

For 0 < c < c_{0}, N_{c} is the total space of a trivializable fiber bundle β_{2 c} over S^{1} with fiber D_{c}^{P} × D_{c}^{N}, and β_{2 c} has fiber subbundles β_{2 c}^{+} and β_{2 c}^{−} with respective total spaces N_{c}^{+} and N_{c}^{−} and with respective fibers S_{c}^{P −1} × D_{c}^{N} and D_{c}^{P} × S_{c}^{N−1}. To see that these claims are true, first observe that as D_{c}^{P} and D_{c}^{N} are Euclidean disks centered at the origin and as L_{+} and L_{−} are 2τ-periodic C^{r} maps from R into the space of linear automorphisms of R^{P} and R^{N} respectively, L_{+}(θ)D_{c}^{P} and L_{−}(θ)D_{c}^{N} are, for each θ ∈ R, strictly convex topological balls of dimensions P and N respectively centered at the origin. It follows that for each θ ∈ R the intersections

D_{c θ_{+}}^{P} := D_{c}^{P} ∩ L_{+}(θ + τ)D_{c}^{P}  and  D_{c θ_{−}}^{N} := D_{c}^{N} ∩ L_{−}(θ + τ)D_{c}^{N} ,

are strictly convex topological balls with piecewise smooth boundary (i.e., manifolds with corners) of dimension P and N respectively varying smoothly with θ, and from (4) of Lemma 7.17, the definition of N_{c}, and (6) of Lemma 7.18, it follows that

(7.9)  N_{c} = ⋃_{0≤θ<2τ}{e^{iθπ/τ} } × D_{c θ_{+}}^{P} × D_{c θ_{−}}^{N} ,

that

(7.10)  N_{c}^{+} = ⋃_{0≤θ<2τ}{e^{iθπ/τ} } × ∂D_{c θ_{+}}^{P} × D_{c θ_{−}}^{N} ,

and that

(7.11)  N_{c}^{−} = ⋃_{0≤θ<2τ}{e^{iθπ/τ} } × D_{c θ_{+}}^{P} × ∂D_{c θ_{−}}^{N} .

It follows that N_{c} is a manifold with corners and that each smooth piece of the boundary lies entirely in either N_{c}^{+} or N_{c}^{−} and in a local transversal to the flow.

It is clear from (7.9) that N_{c} is the total space of a bundle β_{2 c} over S_{1} whose fiber at e^{iθπ/τ} is D_{c θ_{+}}^{P} × D_{c θ_{−}}^{N}. Let h_{c θ} be the homeomorphism of D_{c}^{P} onto D_{c θ_{+}}^{P} obtained by linearly contracting (when necessary) each radius of D_{c}^{P} onto the radius of D_{c θ_{+}}^{P} which it contains; i.e., for s ∈ [0, 1] and x ∈ S_{c}^{P −1}, let h_{c θ}(sx) := st_{c θ}(x)x where t_{c θ}(x) is the unique positive number in [0, 1] satisfying t_{c θ}(x)x ∈ ∂D_{c θ_{+}}^{P}. Similarly, define a homeomorphism k_{c θ} of D_{c}^{N} onto D_{c θ_{−}}^{N}. The smoothness of L_{±} ensures that the map of S^{1} × D_{c}^{P} × D_{c}^{N} onto N_{c} defined by (e^{iθπ/τ} , u_{1}, u_{2}) ↦ (e^{iθπ/τ} , h_{c θ}(u_{1}), k_{c θ}(u_{2})) is a homeomorphism and determines an equivalence of β_{2 c} with the trivial fiber bundle over S^{1} with fiber D_{c}^{P} × D_{c}^{N}. It then follows from (7.10) and (7.11) that N_{c}^{±} is the total space of a fiber subbundle β_{2 c}^{±} of β_{2 c} with β_{2 c}^{+} the fiber subbundle determined by S_{c}^{P −1} × D_{c}^{N} and with β_{2 c}^{−} the fiber subbundle determined by D_{c}^{P} × S_{c}^{N−1}.

Suppose now that 0 < c′ < c < c_{0}; let us see that β_{2 c′} is the fiber subbundle of β_{2 c} determined by the subset D_{c′}^{P} × D_{c′}^{N} of D_{c}^{P} × D_{c}^{N}. Clearly it suffices to show that h_{c′ θ} is the restriction of h_{c θ} to D_{c′}^{P} and that k_{c′ θ} is the restriction of k_{c θ} to D_{c′}^{N}. Now it follows straightforwardly from the definitions that h_{c θ}(S_{c′}^{P −1}) = ∂D_{c′ θ_{+}}^{P}, whence also h_{c θ}(D_{c′}^{P}) = D_{c′ θ_{+}}^{P}. In particular, if x′ ∈ S_{c′}^{P −1}, then

h_{c θ}(x′) = \frac{c′}{c} h_{c θ}(\frac{c}{c′} x′) = t_{c θ}(\frac{c}{c′} x′)x′ ∈ ∂D_{c′ θ_{+}}^{P} .

Hence, by definition of t_{c′ θ}, necessarily

t_{c θ}(\frac{c}{c′} x′) = t_{c′ θ}(x′)

yielding thereby that h_{c′ θ}(x′) = h_{c θ}(x′), which is sufficient to show that h_{c′ θ} is the restriction of h_{c θ} to D_{c′}^{P}. Similarly, k_{c′ θ} is the restriction of k_{c θ} to D_{c′}^{N}.

Define B_{c} := σ_{2}(N_{c}) for 0 < c < c_{0}. By (6) of Lemma 7.18 it is immediate that B_{c} is an isolating block with corners for S relative to the flows generated by ξ, Fξ, and F^{⊥}ξ. Also, relative to these flows B_{c} has common exit set B_{c}^{+} and common entrance set B_{c}^{−} because the analogous statement is true of N_{c} relative to the flows of ξ_{2}, Fξ_{2}, and F^{⊥}ξ_{2} which cover the flows of ξ, Fξ, and F^{⊥}ξ respectively.

A proof of the rest of statement (1) and of statements (2) and (3) are obtained as follows. Because j permutes the fibers of β_{2 c} and because j restricted to S̃ = S^{1}×{0}×{0} is just the antipodal map, j can be regarded as defining a bundle selfmorphism of β_{2 c} over the antipodal map of S^{1}. Hence, identifying u with j(u) for each u ∈ N_{c} and thereby identifying z with −z for each z ∈ S^{1}, we obtain from β_{2 c} a fiber bundle β̃_{c} := (B̃_{c}, p̃_{1}, RP^{1} ≡ S^{1}) with fiber D^{P} × D^{N} and structure group 𝔊_{P} ×𝔊_{N}. Specifically, for each p = [±z] ∈ RP^{1}, there is a connected neighborhood V_{p} evenly covered relative to the double covering S̃ → RP^{1} induced by j. Let V_{0} and V_{1} be the disjoint opens mapped by the double covering onto V_{p} so that writing z_{θ} := e^{iθπ/τ} for θ ∈ R, we have V_{0} = {z_{θ} : θ ∈ J_{0}} for some open interval J_{0} and V_{1} = {z_{θ} : θ ∈ J_{0} + τ}. There are two local trivializations of β̃_{c}|V_{p} obtained by identifying V_{p} with V_{i} (i = 0, 1) and mapping V_{i} × D_{c}^{P} × D_{c}^{N} onto p̃_{1}^{−1}(V_{p}) by sending (z_{θ}, x_{1}, x_{2}) to [(z_{θ}, h_{θ}(x_{1}), k_{θ}(x_{2}))] where the square brackets denote an equivalence class under the identification of u and j(u). The transition functions between these two coordinate descriptions are the maps on V_{p} into 𝔊_{P} × 𝔊_{N} given by

[±z_{θ}] ↦ g_{L_{+}(θ)} × g_{L_{−}(θ)} ;

i.e., in going from V_{0} ×D_{c}^{P} ×D_{c}^{N} to V_{1} ×D_{c}^{P} ×D_{c}^{N}, the transition in the fiber at z_{θ}, θ ∈ J_{0} is given by the homeomorphism g_{L_{+}(θ)} × g_{L_{−}(θ)}, but in going in the opposite direction, the transition in the fiber at z_{θ+τ}, θ ∈ J_{0}, is the inverse homeomorphism g_{L_{+}(θ+τ)} × g_{L_{−}(θ+τ)}. Thus β̃_{c} has structure group 𝔊_{P} × 𝔊_{N}. Also, for 0 < c′ < c, because β_{2 c′} is the fiber subbundle of β_{2 c} determined by D_{c′}^{P} × D_{c′}^{N}, it is immediate from the construction that β̃_{c′} is the fiber subbundle of β̃_{c} determined by D_{c′}^{P} ×D_{c′}^{N}.

Let j̃: β_{2 c} → β̃_{c} be the bundle map defined by the identification of u with j(u). As j is the involution of the double cover σ_{2}: Ũ → U, it follows that B_{c} is homeomorphic to B̃_{c} with the homeomorphism being given by j̃^{∘}σ_{2}^{−1}. It then follows from statement (3) of Lemma 7.17 that this homeomorphism allows us to regard B_{c} as the total space of a subbundle β_{c} of the tubular neighborhood, β_{c} := (B_{c}, p|B_{c}, S), and also allows us to regard β_{c} as having fiber D_{c}^{P} × D_{c}^{N} with structure group 𝔊_{P} × 𝔊_{N}. In addition, it follows from this construction of β_{c} that for 0 < c′ < c the fiber subbundle of β_{c} determined by D_{c′}^{P} × D_{c′}^{N} is β_{c′} since the same relation holds between β̃_{c′} and β̃_{c′′}. Also, from the descriptions of B_{c}^{±} and N_{c}^{±} above and in particular the descriptions (7.9)–(7.11), it is immediate that B_{c}^{+} and B_{c}^{−} are the respective total spaces of fiber subbundles β_{c}^{+} and β_{c}^{−} of β_{c}, the former the fiber subbundle determined by S_{c}^{P −1} × D_{c}^{N}, the latter that determined by D_{c}^{P} × S_{c}^{N−1}.

To prove statements (4)–(6), let us begin by looking at the local invariant manifolds of S̃ within N_{c} relative to the flow of Fξ_{2}, and for simplicity in writing what follows, for ν = u, s, set W_{ξ_{2} c}^{ν} := W_{ξ_{2}}^{ν}(S̃; N_{c}), set W_{Fξ_{2} c}^{ν} := W_{Fξ_{2}}^{ν}(S̃; N_{c}), and analogously define W_{ξ c}^{ν} = W_{ξ}^{ν}(S; B_{c}) and W_{Fξ c}^{ν} = W_{Fξ}^{ν}(S; B_{c}). For ν = u, s, as W_{Fξ_{2} c}^{ν} = W_{Fξ_{2}}^{ν}(S̃; K_{c})∩W^{ν}Fξ_{2}(S̃; j(K_{c})) and because W_{Fξ_{2}}^{ν}(S̃; j(K_{c})) = j(W_{Fξ_{2}}^{ν}(S̃; K_{c})) as a consequence of Lemma 7.18(3), it follows from (7.9) and the definition of Fξ_{2} that

(7.12a)  W_{Fξ_{2} c}^{u} = ⋃_{0≤θ<2τ}{e^{iθπ/τ} } × D_{c θ_{+}}^{P} × {0}

and

(7.12b)  W_{Fξ_{2} c}^{s} = ⋃_{0≤θ<2τ}{e^{iθπ/τ} } × {0} × D_{c θ_{−}}^{N}

so that W_{Fξ_{2} c}^{u} is the total space of a P -disk subbundle ω_{Fξ_{2} c}^{u} of β_{2 c} and W_{Fξ_{2} c}^{s} is the total space of an N-disk subbundle ω_{Fξ_{2} c}^{s} of β_{2 c} and both are trivializable.

Because W_{Fξ_{2} c}^{ν} is j-invariant for ν = u, s, it follows that the same method used to construct β_{c} from β_{2 c} when applied to ω_{Fξ_{2} c}^{ν} yields fiber subbundles ω_{Fξ c}^{u} and ω_{Fξ c}^{s} of β_{c} with respective total spaces W_{Fξ c}^{u} and W_{Fξ c}^{s} and with ω_{Fξ c}^{u} a P -disk bundle and ω_{Fξ}^{s} an N-disk bundle. Also, because W_{F^{⊥}ξ_{2} c}^{ν} = W_{Fξ c}^{ν} for ν = u, s as a consequence of (7.12) and the definition of F^{⊥}ξ_{2}, it follows that W_{F^{⊥}ξ c}^{ν} = W_{Fξ c}^{ν} for ν = u, s.

Note too that from (7.9) and (7.12) it is immediate that

β_{2 c} = ω_{Fξ_{2} c}^{u} ⊕ ω_{Fξ_{2} c}^{s} ,

and from (7.11–13) it follows that

β_{2 c}^{+} = ω̇_{Fξ_{2} c}^{u} ⊕ ω_{Fξ_{2} c}^{s}  and  β_{2 c}^{−} = ω_{Fξ_{2} c}^{u} ⊕ ω̇_{Fξ_{2} c}^{s} .

Further each of these splittings is induced by the hyperbolic splitting of the normal bundle N_{S} as an immediate consequence of the constructions made. In particular, the total space of N_{S}^{u} is σ(R × R^{P} × {0}), which equals σ_{2}(S^{1} × R^{P} × {0}), whence from the construction of ω_{Fξ c}^{u} it is a subbundle of N_{S}^{u}. Similarly, the total space of N_{S}^{s} is σ_{2}(S^{1} × {0} × R^{N}) and N_{S}^{s} has ω_{Fξ c}^{s} as subbundle.

We now show that either ω_{Fξ c}^{u} is either the 0-disk bundle or is bundle equivalent to the unit disk bundle of N_{S}^{u} relative to the norm of some Euclidean metric for N_{S}^{u}. An analogous argument establishes the analogous result for ω_{Fξ c}^{s} and N_{S}^{s}. If P = 0, clearly ω_{Fξ c}^{u} is the 0-disk bundle over S. Suppose, therefore, P ≥ 1. Choose a Euclidean metric in N_{S}^{u}. As σ_{2}: S^{1} × R^{P} × {0} → N_{S}^{u} is a double cover, by Lemma 7.17(3) the Euclidean metric pulls back to one in the bundle N u ; S̃ := (S^{1} × R^{P} × {0}, p_{1}, S^{1}) and let ∥ · ∥_{θ} denote the resulting norm in the fiber R^{P} × {0} ≡ R^{P} above the point e^{iθπ/τ}. Note that necessarily this norm is j-invariant; i.e.,

∥x∥_{θ} = ∥L_{+}(θ)x∥_{θ+τ} for x ∈ R^{P} and θ ∈ R.

Choose c_{1} > 0 so large that

D_{c θ_{+}}^{P} ⊂ {x ∈ R^{P} : ∥x∥_{θ} ≤ c_{1}} for θ ∈ R and 0 < c < c_{0}.

For each x ∈ ∂D_{c θ_{+}}^{P} there exists a unique t(x) ≥ 1 so that ∥t(x)x∥_{θ} = c_{1}. Note that t(x) varies continuously with x ∈ ∂D_{c θ_{+}}^{P} and that t(L_{+}(θ)x) = t(x). It follows that the formula

f(e^{iθπ/τ} , sx, 0) = (e^{iθπ/τ} , s t(x)x, 0) for x ∈ ∂D_{c θ_{+}}^{P} and s ∈ [0, 1]

defines a j-invariant bundle equivalence of ω_{Fξ_{2} c}^{u} and D̃(c_{1}), the disk bundle of radius c_{1} in N u ; S̃ defined using the norm of the pulled back Euclidean metric, and as a consequence of its j-invariance thereby induces an equivalence of ω_{Fξ}^{u} and D(c_{1}), the disk bundle of radius c_{1} in the chosen Euclidean metric on N_{S}^{u}. It follows that ω_{Fξ}^{u} is bundle equivalent to D(1). This completes the proof of statements (4)–(6).

As in the proof of statement (5) of Theorem 7.12, standard proofs of the invariant manifold theorems, e.g., see [Ab-R, Lemma 28.2], imply that decreasing c_{0} if necessary there exists a C^{r} function G: S^{1} × int(D_{c_{0}}^{P}) → R^{N} so that for 0 < c < c_{0}, if G_{c} is the restriction of G to S^{1} ×D_{c}^{P} then the graph of G_{c} is W_{ξ_{2}}^{u}(S̃; K_{c}). Further, the proof of the invariant manifold theorems given in [HPS], shows that the graph of the restriction of G_{c} to {e^{iθπ/τ} } × int(D_{c_{0}}^{P}) is W_{ξ_{2}}^{uu}(e^{iθπ/τ}; K_{c}).

Note that in fact G_{c} has image in D_{c}^{N} since W_{ξ_{2}}^{u}(S̃; K_{c}) ⊂ K_{c} = S^{1} × D_{c}^{P} × D_{c}^{N}. For 0 < c < c_{0} and θ ∈ R, set

Ŵ_{ξ_{2} c θ}^{u} := {e^{iθπ/τ} } × D_{c θ_{+}}^{P}  and  Ŵ_{ξ_{2} c}^{u} := ⋃_{θ∈R} Ŵ_{ξ_{2} c θ}^{u}

and define

Ĝ_{c} := G_{c}|Ŵ_{ξ_{2} c}^{u} ,  and  Ĝ_{c θ} := G_{c}|W_{ξ_{2} c θ}^{u} .

Then the graph of Ĝ_{c} equals W_{ξ_{2} c}^{u} and the graph of Ĝ_{c θ} equals W_{ξ_{2}}^{uu}(z̄_{θ}, N_{c}) where z̄_{θ} := (z_{θ}, 0, 0). To prove the statements of the last sentence it suffices, as follows from the observation (7.9), to show that

(7.13)  G_{c}({e^{iθπ/τ} } × D_{c θ_{+}}^{P}) ⊂ D_{c θ_{−}}^{N}

and to note that W_{ξ_{2}}^{uu}(z̄_{θ}, N_{c}) coincides with the local unstable manifold within N_{c} at z̄_{θ} of the diffeomorphism φ_{2τ}^{2} . The inclusion (7.13) follows easily from the equality W_{ξ_{2} c}^{u} = W_{ξ_{2}}^{u}(S̃; K_{c}) ∩ j(W_{ξ_{2}}^{u}(S̃; K_{c})) which is proved in the same way that the analogous statement obtained by replacing ξ_{2} with Fξ_{2} was proved earlier in this demonstration. In particular, the inclusion W_{ξ_{2} c}^{u} ⊂ W_{ξ_{2}}^{u}(S̃; K_{c}) implies that (i) G_{c}({e^{iθπ/τ} }×D_{c θ_{+}}^{P}) is a subset of D_{c}^{N} while the inclusion W_{ξ_{2} c}^{u} ⊂ j(W_{ξ_{2}}^{u}(S̃; K_{c})) implies that (ii) G_{c}({e^{iθπ/τ} } × D_{c θ_{+}}^{P}) is a subset of L_{−}(θ + τ)D_{c}^{N}, and (i) and (ii) together yield the inclusion (7.13).

Now, by the observation (7.9) we may regard N_{c} as the total space of an N-disk bundle over Ŵ_{ξ_{2} c}^{u}, and we may therefore regard the graph map of Ĝ_{c} as defining a section of this bundle with image W_{ξ_{2} c}^{u}. Note that the zero section of this bundle is precisely W_{Fξ_{2} c}^{u}. Thus, the formula

Z_{t}(e^{iθπ/τ} , x, 0) = (e^{iθπ/τ} , x, tG_{c}(e^{iθπ/τ} , x)) for t ∈ [0, 1]

defines an isotopy in N_{c} from W_{Fξ_{2} c}^{u} to W_{ξ_{2} c}^{u} and note that Z_{t} is j-invariant; i.e., j ^{∘} Z_{t} = Z_{t} ^{∘} j. Also, it follows from (7.10) that the restriction of the graph map of Ĝ_{c} to ∂Ŵ_{ξ_{2} c}^{u} = ⋃_{θ∈R}{e^{iθπ/τ} } × ∂D_{c θ_{+}}^{P} has image in N_{c}^{+}. Hence, in fact, Z_{t} is an isotopy from the pair (W_{Fξ_{2} c}^{u}, ∂W_{Fξ_{2} c}^{u}) to the pair (W_{ξ_{2} c}^{u}, ∂W_{ξ_{2} c}^{u}) in the pair (N_{c}, N_{c}^{+}). Further, throughout the isotopy the image at stage t of the fiber pair of (ω_{Fξ_{2} c}^{u}, ω̇_{Fξ_{2} c}^{u}) above any point of S lies within the fiber pair of (β_{2 c}, β_{2 c}^{+}) above that point. It follows that W_{ξ_{2} c}^{u} is the total space of a trivializable P -disk bundle ω_{ξ_{2} c}^{u} that is a subbundle of β_{2 c} with the end map of the isotopy defining a bundle equivalence from ω_{Fξ_{2} c}^{u} to ω_{ξ_{2} c}^{u}. Once again, application of the method of constructing β_{c} from β_{2 c} to its j-invariant subbundle ω_{ξ_{2} c}^{u} yields that W_{ξ c}^{u} is the total space of a bundle ω_{ξ c}^{u} with fiber D_{c}^{P}, base S, and structure group 𝔊_{P}, and the fiber of ω_{ξ c}^{u} at u ∈ S coincides with W^{uu}(u, B_{c}). The j-invariance of N_{c}, W_{ξ_{2} c}^{u}, and W_{Fξ_{2} c}^{u} imply that σ_{2} ^{∘} Z_{t} ^{∘} σ_{2}^{−1} is a well-defined isotopy of the pair (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) to the pair (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) in the pair (B_{c}, B_{c}^{+}) in such manner that throughout the isotopy the image of the fiber pair of (ω_{Fξ c}^{u}, ω̇_{Fξ c}^{u}) above a point u ∈ S lies in the fiber pair of (β_{c} , β_{c}^{+}) above u from which it follows that ω_{ξ c}^{u} has structure group 𝔊_{P}.

Let us now show that (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) is a strong deformation retract of (B_{c}, B_{c}^{+}) and that (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) is a weak deformation retract of (B_{c}, B_{c}^{+}). First, note from (7.12) and (7.9) it follows that the fibers at e^{iθπ/τ} of ω_{Fξ_{2} c}^{u} and β_{2 c} are respectively {e^{iθπ/τ} } × D_{c θ_{+}}^{P} × {0} and {e^{iθπ/τ} } × D_{c θ_{+}}^{P} × D_{c θ_{−}}^{N}. Thus, for θ ∈ R, contraction of D_{c θ_{−}}^{N} linearly along radii to 0 while leaving D_{c θ_{+}}^{P} pointwise fixed, defines a j-invariant strong deformation retraction D_{t} of the pair (N_{c}, N_{c}^{+}) onto the pair (W_{Fξ_{2} c}^{u}, ∂W_{Fξ_{2} c}^{u}) that at each point of S^{1} deforms the fiber of β_{2 c} above that point within itself. Therefore, the composition σ_{2} ^{∘} D_{t} ^{∘} σ_{2}^{−1} is a well-defined strong deformation retraction of the pair (B_{c}, B_{c}^{+}) onto (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) that at each point of S deforms the fiber of β_{c} above that point within itself. As the pair (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) is isotopic in (B_{c}, B_{c}^{+}) to the pair (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) and because the latter pair is a strong deformation retract of (B_{c}, B_{c}^{+}), the inclusion (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) ⊂ (B_{c}, B_{c}^{+}) is a homotopy equivalence as desired.

It is immediate from the previous paragraph that B_{c}/B_{c}^{+} is basepoint homotopic to W_{Fξ c}^{u}/∂W_{Fξ c}^{u} which is basepoint homotopic to T(N_{S}^{u}) as an immediate consequence of statement (5) of the theorem already proved and Definition 7.1.

Arguments analogous to those just given show that W_{ξ c}^{s} and W_{Fξ c}^{s} are the total spaces of subbundles ω_{ξ c}^{s} and ω_{Fξ c}^{s} of β_{c} with the latter the fiber subbundle determined by {0} × D_{c}^{N} and with the former having W_{ξ}^{ss}(u; B_{c}) as its fiber at u ∈ S, that the pairs (W_{ξ c}^{s}, ∂W_{ξ c}^{s}) and (W_{Fξ c}^{s}, ∂W_{Fξ c}^{s}) are isotopic in (B_{c}, B_{c}^{−}) via an isotopy whose restriction to the fiber pair of (ω_{ξ c}^{s}, ω̇_{ξ c}^{s}) above a point u ∈ S is an isotopy in the fiber pair of (β_{c} , β_{c}^{−}) above u to the fiber pair of (ω_{Fξ c}^{s}, ω̇_{Fξ c}^{s}) above u, that ω_{Fξ c}^{s} is, as N = 0 or as N ≥ 1, either the 0-disk bundle or equivalent to the unit disk bundle of N_{S}^{s}, that therefore W_{Fξ c}^{s}/∂W_{Fξ c}^{s} is homeomorphic to the Thom space T(N_{S}^{s}), that (W_{Fξ c}^{s}, ∂W_{Fξ c}^{s}) is a strong and (W_{ξ c}^{s}, ∂W_{ξ c}^{s}) a weak deformation retract of (B_{c}, B_{c}^{−}), whence the index space B_{c}/B_{c}^{−} is basepoint homotopic to T(N_{S}^{s}). □

The following result will be used below to help identify generators of the homology modules of the Conley indices of a hyperbolic periodic orbit and is corollary to the proofs of Proposition 7.9 and Theorem 7.21. The assumptions and notation are the same as those for Theorem 7.21 and its proof.

7.22 Lemma. If 0 < c′ < c′′ < c_{0}, then the total space of the fiber subbundle of β_{c′′} determined by the 𝔊_{P} ×𝔊_{N}-invariant subset cl (D_{c′′}^{P} \ D_{c′}^{P})×D_{c′′}^{N} of D_{c′′}^{P} × D_{c′′}^{N} is a subset CB_{c′′}^{+} of B_{c′′} with the properties

(1) (B_{c′′}, CB_{c′′}^{+}) is a regular index pair for S relative to the flows generated by ξ, Fξ, and F^{⊥}ξ;

(2) if c′ ≤ λ ≤ c′′, then there is an inclusion of nested index pairs j_{λ}: (B_{λ}, B_{λ}^{+}) ⊂ (B_{c′′}, CB_{c′′}^{+}).

Proof. Let Cβ_{c′′}^{+} be the fiber subbundle of of β_{c′′} determined by cl (D_{c′′}^{P} \ D_{c′}^{P})× D_{c′′}^{N}. Let CK_{c′′}^{+} := S^{1} × cl (D_{c′′}^{P} \ D_{c′}^{P}) × D_{c′′}^{N}. Then ⟨K_{c′′}, CK_{c′′}^{+}⟩ is a regular index pair for S̃ relative to the flows generated by ξ_{2}, Fξ_{2}, and F^{⊥}ξ_{2}. One only has to verify that CK_{c′′}^{+} is positively invariant relative to K_{c′′} with respect to the three flows, for as K_{c′′} and K_{c′} are blocks for S̃ relative to all three flows, orbits would then necessarily enter the collar CK_{c′′}^{+} along K_{c′}^{+} and exit along K_{c′′}^{+}. However, the positive invariance relative to all three flows is an immediate consequence of the inequalities (7.5_{1}) and (7.5_{2}) in the proof of Proposition 7.9 as they apply to the vectorfields ξ_{2}, Fξ_{2}, and F^{⊥}ξ_{2}. With the definitions

CN_{c′′}^{+} := N_{c′′} ∩ (CK_{c′′}^{+} ∪ j(CK_{c′′}^{+}))  and  CB_{c′′}^{+} := σ_{2}(CN_{c′′}^{+}) ,

it is immediate from Lemma 7.18(6) that ⟨N_{c′′}, CN_{c′′}^{+}⟩ and ⟨B_{c′′}, CB_{c′′}^{+}⟩ are nested index pairs, the former a j-invariant index pair for S̃ with respect to the flows generated by ξ_{2}, Fξ_{2} and F^{⊥}ξ_{2} and the latter an index pair for S with respect to the flows generated by ξ, Fξ, and F^{⊥}ξ. From the construction of β_{c′′} in the proof of Theorem 7.21, it follows easily that CB_{c′′}^{+} is the total space of Cβ_{c′′}^{+}. The inclusion (B_{λ}, B_{λ}^{+}) ⊂ (B_{c′′}, CB_{c′′}^{+}) holds for c′ ≤ λ ≤ c′′ as an immediate consequence of statements (2) and (3) of Theorem 7.21 and the inclusion S_{λ}^{P −1} × D_{λ}^{N} ⊂ cl (D_{c′′}^{P} \ D_{c′}^{P}) × D_{c′′}^{N}, which clearly holds for c′ ≤ λ ≤ c′′. □

Throughout the rest of this section, C (S) and C∗(S) will denote the Conley indices of S relative to the flow generated by ξ under the assumption of hypotheses HCO.

7.23 Theorem. Under the assumption of the hypotheses HCO and the conclusions of Theorem 7.21, the following hold:

(1) [C (S)] = [T(N_{S}^{u})]; in particular, either P = 0 and [C (S)] = [S^{1} ⨿ {∗}] or P ≥ 1 and, as N_{S}^{u} is or is not orientable over Z, either [C (S)] = [S^{P +1} ∨S^{P}] or [C (S)] = [S^{P −1}RP^{2}]; (2) [C∗(S)] = [T(N_{S}^{s})]; in particular, either N = 0 and [C∗(S)] = [S^{1} ⨿ {∗}] or

N ≥ 1 and, as N_{S}^{s} is or is not orientable over Z, either [C∗(S)] = [S^{N+1}∨S^{N}] or [C∗(S)] = [S^{N−1}RP^{2}].

Proof. By Theorem 7.21(9), for 0 < c < c_{0} the index space B_{c}/B_{c}^{+} is basepoint homotopic to T(N_{S}^{u}); hence, by the connected simple system properties of a Conley index, each index space in C (S) is basepoint homotopic to T(N_{S}^{u}). Similarly, by Theorem 7.21(10), each index space in C∗(S) is basepoint homotopic to T(N_{S}^{s}).

If P = 0, then N_{S}^{u} is the vector bundle of rank 0 over S so that T(N_{S}^{u}) ≃ S^{1}⨿{∗} as a consequence of Definition 7.1. Similarly, when N = 0, T(N_{S}^{s}) ≃ S^{1}⨿{∗}. In the remaining cases N_{S}^{u} is a P -plane bundle with P ≥ 1 and N_{S}^{s} is an N-plane bundle with N ≥ 1, whence the theorem is an immediate consequence of Lemma 7.24 following. □

7.24 Lemma. For each integer k ≥ 1, if γ is a k-plane bundle over S^{1}, then as γ is or is not orientable over Z its Thom space has the pointed homotopy type of S^{k+1} ∨ S^{k} or of S^{k−1}RP^{2}.

Lemma 7.24 is a simple consequence of the following two propositions. The first of these is well-known and follows quite simply by representing the closed Möbius band as a two-cell with one-pair of edges identified after a half-twist.

7.25 Proposition. A closed Möbius band with its bounding circle identified to a point and taken as the basepoint is homeomorphic as a pointed space to RP^{2} with any point of RP^{2} as basepoint.

Let S̃^{2} denote the quotient space obtained from S^{2} by identifying (0, 0, 1) and (0, 0, −1) to a single point taken as the basepoint. S̃^{2} is often called the dimpled two-sphere.

7.26 Proposition. S^{2} ∨ S^{1} and S̃^{2} are basepoint homotopic.

Proof. The basepoint of S^{2} is taken to be its north pole, i.e., the point with Cartesian coordinates (0, 0, 1), and the basepoint of S^{1} is taken to be the point with Cartesian coordinates (1, 0). Let q: S^{2} → S̃^{2} be the quotient map. For k = 1, 2, let j_{k}: S^{k} → S^{2} ∨ S^{1} be the natural embedding. Use will be made of spherical and polar coordinates to define the required homotopy equivalences. Specifically, let s: [0, π] × [0, 2π] → S^{2} be defined by s(ϕ, θ) := (sin(ϕ) cos(θ), sin(ϕ) sin(θ), cos(ϕ)) and let p: [0, 2π] → S^{1} be defined by p(ψ) := (cos(ψ), sin(ψ)). It is also convenient to set J_{i} := [\frac{i}{3}, \frac{i+1}{3}], Φ_{i} := πJ_{i}, and Ψ_{i} := 2πJ_{i} for i = 0, 1, 2, and to set I_{i} := [\frac{i}{2}, \frac{i+1}{2}] for i = 0, 1.

To define f: S̃^{2} → S^{2} ∨ S^{1} it suffices to define f̃: S^{2} → S^{2} ∨ S^{1} such that f̃(0, 0, ±1) = ∗ where ∗ is the basepoint of S^{2} ∨ S^{1}, for then take f = f̃^{∘} q^{−1}. Define f̃ as follows:

f̃^{∘} s(ϕ, θ) := \{ \begin{matrix}j_{2} ^{∘} s(3ϕ, θ) & if ϕ ∈ Φ_{0} \\ j_{2} ^{∘} s(2π − 3ϕ, 0) & if ϕ ∈ Φ_{1} \\ j_{1} ^{∘} p(6ϕ − 4π) & if ϕ ∈ Φ_{2} \end{matrix}

and note that f̃ maps the north and south poles of S^{2} to the basepoint of S^{2} ∨ S^{1} as required.

The map g: S^{2} ∨ S^{1} → S̃^{2} is defined by defining it separately on S^{2} ≡ j_{2}(S^{2}) and S^{1} ≡ j_{1}(S^{1}) as follows:

g|S^{2} := q  and  g|S^{1} ^{∘} p(ψ) = q^{∘} s(ψ/2, 0) .

Note that both g|S^{2} and g|S^{1} map the basepoint of their domains to the basepoint of S̃^{2}.

The composite g^{∘}f is homotopic to 1S̃^{2} relative to the basepoint via the homotopy H_{t} defined by

H_{t} ^{∘} q^{∘} s(ϕ, θ) := \{ \begin{matrix}q^{∘} s(3ϕ, θ) & if t ∈ I_{0}, ϕ ∈ Φ_{0} \\ q^{∘} s(π + (1 − 2t)(π − 3ϕ), 0) & if t ∈ I_{0}, ϕ ∈ Φ_{1} \\ q^{∘} s(2π(3t − 1) + 3ϕ(1 − 2t), 0) & if t ∈ I_{0}, ϕ ∈ Φ_{2} \\ q^{∘} s((5 − 4t)ϕ, θ) & if t ∈ I_{1}, ϕ ∈ Φ_{0} \\ q^{∘} s((2t − 1)ϕ + (2 − 2t)π, θ) & if t ∈ I_{1}, ϕ ∈ Φ_{1} ∪ Φ_{2}. \end{matrix}

The composite f ^{∘} g is homotopic to 1_{S^{2}∨S^{1}} relative to the basepoint via the homotopy K_{t} defined by

K_{t}|S^{2} ^{∘} s(ϕ, θ) := \{ \begin{matrix}j_{2} ^{∘} s(3ϕ, θ) & if t ∈ J_{0} ∪ J_{1}, ϕ ∈ Φ_{0} \\ j_{2} ^{∘} s(2π − 3ϕ, 0) & if t ∈ J_{0}, ϕ ∈ Φ_{1} \\ j_{1} ^{∘} p((1 − 3t)(6ϕ − 4π)) & if t ∈ J_{0}, ϕ ∈ Φ_{2} \\ j_{2} ^{∘} s(π − (2 − 3t)(3ϕ − π), 0) & if t ∈ J_{1}, ϕ ∈ Φ_{1} \\ j_{2} ^{∘} s((3t − 1)π, 0) & if t ∈ J_{1}, ϕ ∈ Φ_{2} \\ j_{2} ^{∘} s((7 − 6t)ϕ, θ) & if t ∈ J_{2}, ϕ ∈ Φ_{0} \\ j_{2} ^{∘} s((3t − 2)ϕ + (3 − 3t)π, θ) & if t ∈ J_{2}, ϕ ∈ Φ_{1} ∪ Φ_{2} \end{matrix}

and

K_{t}|S^{1} ^{∘} p(ψ) := \{ \begin{matrix}j_{2} ^{∘} s((1 − 2t)\frac{3}{2}ψ, 0) & if t ∈ I_{0}, ψ ∈ Ψ_{0} \\ j_{2} ^{∘} s((1 − 2t)(2π − \frac{3}{2}ψ), 0) & if t ∈ I_{0}, ψ ∈ Ψ_{1} \\ j_{1} ^{∘} p(3ψ − 4π) & if t ∈ I_{0}, ψ ∈ Ψ_{2} \\ j_{1} ^{∘} p((2t − 1)ψ) & if t ∈ I_{1}, ψ ∈ Ψ_{0} ∪ Ψ_{1} \\ j_{1} ^{∘} p((2 − 2t)(3ψ − 4π) + (2t − 1)ψ) & if t ∈ I_{1}, ψ ∈ Ψ_{2}. □ \end{matrix}

7.27 Proof of Lemma 7.24. Let ε_{X}^{ℓ} denote the trivial ℓ-plane bundle over a space X. As noted in [Sw, p. 229], for η any k-plane bundle over a space X and x_{0} ∈ X, the vector bundle η ⊕ ε_{X}^{ℓ} is naturally identified with the product bundle η × ε_{{x_{0}}}^{ℓ}. It follows that

T(η ⊕ ε_{X}^{ℓ}) ≃ T(η) ∧ T(ε_{{x_{0}}}^{ℓ}) ≃ T(η) ∧ S^{ℓ} ≃ S^{ℓ}T(η)

because the Thom space of a product bundle is the smash product of the Thom spaces of the factors—e.g., see [Sw, Proposition 2.28]—and because the Thom space of an ℓ-plane bundle over a point is obviously homeomorphic to S^{ℓ}.

Now, as a consequence of the classification theorem for fiber bundles over spheres, [St, Theorem 18.5], for each k ≥ 1, there are exactly two equivalence classes of kplane bundles over S^{1}. One of these is the equivalence class of the trivial k-plane bundle over S^{1} which is orientable over Z; the other is the class of the k-plane bundles not orientable over Z and, in particular, as k = 1 or as k > 1, is the equivalence class of the canonical line bundle γ_{1}^{1} over RP^{1} ≡ S^{1} or of the k-plane bundle γ_{1}^{1} ⊕ ε_{S^{1}}^{k−1} over S^{1}.

First, suppose γ is orientable over Z. If γ is equivalent to the trivial line bundle over S^{1}, then T(γ) ≃ S^{1}×D^{1}/S^{1}×S^{0} and the latter is clearly a homeomorph of S̃^{2}, whence the desired result holds for k = 1 in the orientable case as a consequence of Proposition 7.26. For k > 1, γ must be bundle equivalent to ε_{S^{1}}^{k} ≃ ε_{S^{1}}^{1} ⊕ ε_{S^{1}}^{k−1}, whence by the observations of the first paragraph of the proof

T(γ) ≃ S^{k−1}T(ε_{S^{1}}^{1}) ≃ S^{k−1}(S^{2} ∨ S^{1}) ≃ S^{k+1} ∨ S^{k}

yielding the desired result for k > 1 in the orientable case.

In the non-orientable case because γ is either equivalent to γ_{1}^{1} when k = 1 or to γ_{1}^{1} ⊕ ε_{S^{1}}^{k−1} when k > 1 and because T(γ_{1}^{1}) is clearly a homeomorph of the closed Möbius band with its bounding circle identified to a point taken as the basepoint, Proposition 7.25 and a computation analogous to that in the orientable case yields that T(γ) ≃ S^{k−1}RP^{2} as desired. □

For simplicity in stating the next several lemmas, let us continue to identify N_{S} := N_{S}^{u} ⊕ N_{S}^{s}, the normal bundle of S, with the tubular neighborhood U. Then, where E_{S}^{ν} is the total space of N_{S}^{ν} (ν = u, s), E_{S}^{ν} ⊂ U and E_{S 0}^{ν}, the complement in E_{S}^{ν} of the zero section of N_{S}^{ν}, coincides with E_{S}^{ν} \ S. Also, let W_{Fξ c u}^{ν} denote the fiber at u ∈ S of ω_{Fξ c}^{ν}, and where ω̇_{Fξ c}^{ν} is the sphere bundle associated to ω_{Fξ c}^{ν}, let ∂W_{Fξ c u}^{ν} denote its fiber at u ∈ S. Similarly, for ν = u, s, W_{ξ c u}^{ν} and ∂W_{ξ c u}^{ν} denote respectively the fiber at u of ω_{ξ c}^{ν} and the fiber at u of the associated sphere bundle ω̇_{ξ c}^{ν}.

7.28 Lemma. Under the assumption of hypotheses HCO and conclusions of Theorem 7.21, if τ^{u} is an R-orientation of the vector bundle N_{S}^{u}, then for 0 < c < c_{0}, for each u ∈ S, there exist unique fundamental classes

o_{Fξ c u}^{u} ∈ H_{P} (W_{Fξ c u}^{u}, ∂W_{Fξ c u}^{u}; R)  and  o_{ξ c u}^{u} ∈ H_{P} (W_{ξ c u}^{u}, ∂W_{ξ c u}^{u}; R)

that together satisfy

(1) under the inclusion induced homomorphisms o_{Fξ c u}^{u} and o_{ξ c u}^{u} have common image in H_{P} (B_{c}, B_{c}^{+}; R) that generates that R-module;

(2) ⟨i_{c u}^{u}^{∗}τ^{u}, o_{Fξ c u}^{u}⟩ = 1 where i_{c u}^{u}: (W_{Fξ c u}^{u}, ∂W_{Fξ c u}^{u}) → (E_{S}^{u}, E_{S 0}^{u}) is inclusion.

An analogous statement holds for an R-orientation τ^{s} of N_{S}^{s} and for 0 < c < c_{0}, for each u ∈ S, the existence of fundamental classes o_{Fξ c u}^{s} and o_{ξ c u}^{s} for the fibers W_{Fξ c u}^{s} and W_{ξ c u}^{s} respectively satisfying the analogues of conditions (1) and (2) obtained by replacing B_{c}^{+} with B_{c}^{−} and by replacing the expanding normal bundle with the contracting one.

Proof. Let 0 < c < c_{0}. The inclusion i_{c}^{u}: (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) ⊂ (E_{S}^{u}, E_{S 0}^{u}) induces isomorphisms on cohomology, for it has the factorization

(W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) ⟶^{e_{2}} (W_{Fξ c}^{u}, W_{Fξ c}^{u} \ S) ⟶^{e_{1}} (E_{S}^{u}, E_{S 0}^{u}) ,

and e_{1} induces isomorphisms by excision and e_{2} induces isomorphisms as a consequence of the Five Lemma and the fact that ∂W_{Fξ c}^{u}, the total space of the sphere bundle ω̇_{Fξ c}^{u} associated to the disk bundle ω_{Fξ c}^{u}, is a strong deformation retract of W_{Fξ c}^{u} \ S, the complement of the zero section in the total space of ω_{Fξ c}. Thus, it is immediate that i_{c}^{u ∗}τ^{u} is an orientation class for the P -disk bundle ω_{Fξ c}^{u}.

The fundamental class o_{Fξ c u}^{u} of W_{Fξ c u}^{u} (u ∈ S) can now be determined as follows. Let u ∈ S. Because the inclusion i_{c u}^{u} has the factorization

(W_{Fξ c u}^{u}, ∂W_{Fξ c u}^{u}) ⟶^{j_{c u}^{u}} (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) ⟶^{i_{c}^{u}} (E_{S}^{u}, E_{S 0}^{u}) .

it is immediate that i_{c u}^{u ∗}τ^{u} generates H^{P} (W_{Fξ c u}^{u}, ∂W_{Fξ c u}^{u}; R) as an R-module. Thus, the fundamental class o_{Fξ c u}^{u} is uniquely specified by requiring it to be the dual of i_{c u}^{u ∗}τ^{u} relative to the Kronecker pairing; i.e., choose o_{Fξ c u}^{u} so that condition (2) is satisfied.

The fundamental class o_{ξ c u}^{u} of W_{ξ c u}^{u} (u ∈ S) is determined from the fundamental class o_{Fξ c u}^{u} as follows. Let u ∈ S. By Theorem 7.21(7), the fiber pair (W_{Fξ c u}^{u}, ∂W_{Fξ c u}^{u}) is isotopic in (B_{c}, B_{c}^{+}) (in fact in the fiber pair over u) to the fiber pair (W_{ξ c u}^{u}, ∂W_{ξ c u}^{u}). Hence, the image of o_{Fξ c u}^{u} under the isomorphism on homology between these fiber pairs induced by the end map of such an isotopy is a fundamental class of W_{ξ c u}^{u}, call it o_{ξ c u}^{u}, which by the manner of its definition has the same image under the homomorphism on homology induced by the inclusion k_{c u}^{u}: (W_{ξ c u}^{u}, ∂W_{ξ c u}^{u}) ⊂ (B_{c}, B_{c}^{+}) as does o_{Fξ c u}^{u} under the homomorphism on homology induced by the inclusion k̇_{c u}^{u}: (W_{Fξ c u}^{u}, ∂W_{Fξ c u}^{u}) ⊂ (B_{c}, B_{c}^{+}).

This specifies o_{ξ c u}^{u} uniquely once we know that k̇_{c u ∗}^{u}o_{Fξ c u}^{u} is not the zero class in H_{P} (B_{c}, B_{c}^{+}; R). However, the inclusion k̇_{c u}^{u} has the factorization

(W_{Fξ c u}^{u}, ∂W_{Fξ c u}^{u}) ⟶^{j_{c u}^{u}} (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) ⟶^{k̇_{c}^{u}} (B_{c}, B_{c}^{+}) ,

and the inclusion k̇_{c}^{u} induces isomorphisms on homology as a consequence of Theorem 7.21(9). Thus, k̇_{c u ∗}^{u}o_{Fξ c u}^{u} is non-zero if, and only if, j_{c u ∗}^{u}o_{Fξ c u}^{u} is non-zero. Now not only is j_{c u ∗}^{u}o_{Fξ c u}^{u} non-zero, in fact it generates H_{P} (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}; R) because j_{c u}^{u} induces an isomorphism on homology in dimension P (the proof will be given momentarily). Hence, k̇_{c u ∗}^{u}o_{Fξ c u}^{u} generates H_{P} (B_{c}, B_{c}^{+}; R) as desired.

That j_{c u}^{u} induces an isomorphism on homology in dimension P is a consequence of the Thom isomorphism theorem and naturality of cap products. Specifically, the diagram

H_{P} (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}; R) ⟶^{i_{c}^{u ∗}τ^{u}⌢} H_{0}(W_{Fξ c}^{u}; R) ⟶^{p_{∗}} H_{0}(S; R)
j_{c u ∗}^{u}↑  ^{j_{c u ∗}^{u}}↑  ↑
H_{P} (W_{Fξ c u}^{u}, ∂W_{Fξ c u}^{u}; R) ⟶^{i_{c u}^{u ∗}τ^{u}⌢} H_{0}(W_{Fξ c u}^{u}; R) ⟶^{p_{∗}} H_{0}({u}; R)

is commutative because the vertical arrows are all inclusion induced and the arrows in the bottom row are the restrictions of those in the top row. Specifically, the left-hand rectangle commutes by naturality of cap products. Since the inclusions S ⊂ W_{Fξ c}^{u} and {u} ⊂ W_{Fξ c u}^{u} are strong deformation retracts, the horizontal arrows in the right-hand rectangle are isomorphisms. As the rightmost vertical arrow is also an isomorphism, it follows that the middle vertical arrow is too. Hence, also j_{c u ∗}^{u} is an isomorphism in dimension P because the rows of the diagram are isomorphisms by the Thom isomorphism theorem; e.g., see [Sp, Theorem 5.7.10]. □

7.29 Lemma. Under the assumption of hypotheses HCO and conclusions of Theorem 7.21, if τ^{u} is an R-orientation of the vector bundle N_{S}^{u} and if o_{S} ∈ H_{1}(S; R) is a fundamental class, then for 0 < c < c_{0} there exist unique fundamental classes

o_{Fξ c}^{u} ∈ H_{P +1}(W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}; R)  and  o_{ξ c}^{u} ∈ H_{P +1}(W_{ξ c}^{u}, ∂W_{ξ c}^{u}; R)

that together satisfy

(1) under the inclusion induced isomorphisms o_{Fξ c}^{u} and o_{ξ c}^{u} have common image in H_{P +1}(B_{c}, B_{c}^{+}; R) generating that R-module;

(2) p_{∗}(i_{c}^{u}^{∗}τ^{u} ⌢ o_{Fξ c}^{u}) = o_{S} where i_{c}^{u}: (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) → (E_{S}^{u}, E_{S 0}^{u}) is inclusion. An analogous statement holds for an R-orientation τ^{s} of N_{S}^{s}, a fundamental class

o_{S} of S, and for 0 < c < c_{0}, the existence of fundamental classes

o_{Fξ c}^{s} ∈ H_{P +1}(W_{Fξ c}^{s}, ∂W_{Fξ c}^{s}; R)  and  o_{ξ c}^{s} ∈ H_{P +1}(W_{ξ c}^{s}, ∂W_{ξ c}^{s}; R)

satisfying the analogues of conditions (1) and (2) obtained by replacing B_{c}^{+} with B_{c}^{−} and by replacing the expanding normal bundle with the contracting one.

Proof. The Thom isomorphism theorem establishes that for 0 < c < c_{0},

e ↦ p_{∗}(i_{c}^{u}^{∗}τ^{u} ⌢ e)

defines an isomorphism from H_{P +1}(W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}; R) onto H_{1}(S). Choose o_{Fξ c}^{u} to be the unique element whose image under this isomorphism is o_{S}—thus, condition (2) is satisfied—whence it follows that o_{Fξ c}^{u} is a fundamental class for W_{Fξ c}^{u}. As in the proof of the previous lemma, the fundamental class o_{ξ c}^{u} of W_{ξ c}^{u} is defined as the image under the isomorphism on homology induced by the end map of an isotopy from the pair (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) to the pair (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) which at each stage of the isotopy has image in the pair (B_{c}, B_{c}^{+}). Where k_{c}^{u} and k̇_{c}^{u} are the inclusions

k_{c}^{u}: (W_{ξ c}^{u}, ∂W_{ξ c}^{u}) ⊂ (B_{c}, B_{c}^{+})  and  k̇_{c}^{u}: (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}) ⊂ (B_{c}, B_{c}^{+}) ,

it follows that in H_{P +1}(B_{c}, B_{c}^{+}; R),

k̇_{c ∗}^{u}o_{Fξ c}^{u} = k_{c ∗}^{u}o_{ξ c}^{u} .

Further, this common image generates because both k̇_{c}^{u} and k_{c}^{u} induce isomorphisms on homology by Theorem 7.21(9). □

7.30 Definition. Let τ^{u} and τ^{s} be R-orientations of N_{S}^{u} and N_{S}^{s} respectively and let o_{S} be a fundamental class of S.

(A) The families of fundamental classes {o_{Fξ c u}^{u} : 0 < c < c_{0}, u ∈ S} and {o_{ξ c u}^{u} : 0 < c < c_{0}, u ∈ S} determined by τ^{u} according to the prescription of Lemma 7.28 are called respectively the compatible family of fundamental classes determined by τ^{u} for the fibers of the bundles ω_{Fξ c}^{u} (0 < c < c_{0}) and the compatible family of fundamental classes determined by τ^{u} for the fibers of the bundles ω_{ξ c}^{u} (0 < c < c_{0}). A similar designation is made for the families of fundamental classes {o_{Fξ c u}^{s} : 0 < c < c_{0}, u ∈ S} and {o_{ξ c u}^{s} : 0 < c < c_{0}, u ∈ S} determined by τ^{s}.

(B) The families of fundamental classes {o_{Fξ c}^{u}}_{0<c<c_{0}} and {o_{ξ c}^{u}}_{0<c<c_{0}} determined by τ^{u} and o_{S} according to the prescription of Lemma 7.29 are called respectively the compatible family of fundamental classes determined by τ^{u} and o_{S} for the family of submanifolds W_{Fξ}^{u} := {W_{Fξ c}^{u}}_{0<c<c_{0}} and the compatible family of fundamental classes determined by τ^{u} and o_{S} for the family of submanifolds W_{ξ}^{u} := {W_{ξ c}^{u}}_{0<c<c_{0}}. A similar designation is made for the family of fundamental classes {o_{Fξ c}^{s}}_{0<c<c_{0}} for the family of submanifolds W_{Fξ}^{s} := {W_{Fξ c}^{s}}_{0<c<c_{0}} and for the family of fundamental classes {o_{ξ c}^{s}}_{0<c<c_{0}} for the family of submanifolds W_{ξ}^{s} := {W_{ξ c}^{s}}_{0<c<c_{0}}.

7.31 Theorem. Under the assumption of the hypotheses HCO and the conclusions of Theorem 7.21 the following hold:

(1) if N_{S}^{u} is R-orientable, then (a) H̃_{k}C (S; R) ≃ 0 for k ≠ P, P + 1;

(b) H̃_{P} C (S; R) ≃ R, and in particular an R-orientation τ^{u} of N_{S}^{u} determines a unique generator o_{ξ P}^{u} ∈ H̃_{P} C (S; R) characterized by the property that it is the common canonical image of all members of the compatible family of fundamental classes determined by τ^{u} for the fibers of the bundles ω_{ξ c}^{u} (0 < c < c_{0});

(c) H̃_{P +1}C (S; R) ≃ R, and in particular an R-orientation τ^{u} of N_{S}^{u} and a fundamental class o_{S} of S together determine a unique generator o_{ξ P +1}^{u} ∈ H̃_{P +1}C (S; R) characterized by the property that it is the common canonical image of all members of the compatible family of fundamental classes determined by τ^{u} and o_{S} for the family of submanifolds W_{ξ}^{u};

(2) if N_{S}^{s} is R-orientable, then (a) H̃_{k}C∗(S; R) ≃ 0 for k ≠ N, N + 1;

(b) H̃_{N} C∗(S; R) ≃ R, and in particular an R-orientation τ^{s} of N_{S}^{s} determines a unique generator o_{ξ N}^{s} ∈ H̃_{N} C∗(S; R) characterized by the property that it is the common canonical image of all members of the compatible family of fundamental classes determined by τ^{s} for the fibers of the bundles ω_{ξ c}^{s} (0 < c < c_{0});

(c) H̃_{N+1}C∗(S; R) ≃ R, and in particular an R-orientation τ^{s} of N_{S}^{s} and a fundamental class o_{S} of S together determine a unique generator o_{ξ N+1}^{s} ∈ H̃_{N+1}C∗(S; R) characterized by the property that it is the common canonical image of all members of the compatible family of fundamental classes determined by τ^{s} and o_{S} for the family of submanifolds W_{ξ}^{s};

(3) given R-orientations τ^{u} of N_{S}^{u} and τ^{s} of N_{S}^{s} and given a fundamental class o_{S} of S, then where o_{ξ P}^{u} and o_{ξ P +1}^{u} are determined by τ^{u} and o_{S} as in (1) and where o_{ξ N}^{s} and o_{ξ N+1}^{s} are determined by τ^{s} and o_{S} as in (2),

#(o_{ξ P +1}^{u} ⌢_{•} o_{ξ N}^{s}) = σ , ^{#}(o_{ξ P}^{u} ⌢_{•} o_{ξ N+1}^{s}) = (−1)^{P} σ ,

and

o_{ξ P +1}^{u} ⌢_{•} o_{ξ N+1}^{s} = σ o_{S} .

Proof. The notation established for the conclusions and proof of Theorem 7.21, Lemmas 7.28 and 7.29, and Definition 7.30 will be used throughout this proof.

Let τ^{u} and τ^{s} be R-orientations of N_{S}^{u} and N_{S}^{s} respectively, let o_{S} ∈ H_{1}(S; R) be a fundamental class, and temporarily fix c, 0 < c < c_{0}.

Application of the Thom isomorphism theorem to the P -disk bundle ω_{Fξ c}^{u} over S yields

(7.14)  H_{k}(W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}; R) ≃ 0  for k ≠ P, P + 1.

There are two consequences of statement (7.14) with reference to the homology of C (S) of which the second leads to the definition of the sought for generator o_{ξ P}^{u}. First, Theorem 7.21(8) implies that H_{k}(B_{c}, B_{c}^{+}; R) ≃ 0 for the same range of indices as in (7.14). Hence, also

H̃_{k}C (S; R) ≃ 0 , for k ≠ P, P + 1.

Second, since in particular (7.14) holds for k = P − 1, by the universal coefficient theorem H^{P} (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}; R) ≃ Hom(H_{P} (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}; R); R), whence there is a unique element o_{P c}^{u} ∈ H_{P} (W_{Fξ c}^{u}, ∂W_{Fξ c}^{u}; R) satisfying

⟨i_{c}^{u ∗}τ^{u}, o_{P c}^{u}⟩ = 1 .

This definition of o_{P c}^{u}, the factorization i_{c u}^{u} = i_{c}^{u} ^{∘} j_{c u}^{u}, and the characterization of o_{Fξ c u}^{u} in statement (2) of Lemma 7.28 imply that

j_{c u ∗}^{u}o_{Fξ c u}^{u} = o_{P c}^{u} for each u ∈ S.

This equality, the factorization k̇_{c u}^{u} = k̇_{c}^{u} ^{∘} j_{c u}^{u}, and statement (1) of Lemma 7.28 imply that the canonical image of o_{ξ c u}^{u} in H̃_{P} C (S; R) is independent of u ∈ S and cyclically generates H̃_{P} C (S; R) over R; i.e., where b̄_{c ∗}^{+}: H_{∗}(B_{c}, B_{c}^{+}; R) → H̃_{∗}C (S; R) is the canonical isomorphism and where

o_{ξ P c}^{u} := b̄_{c ∗}^{+}k̇_{c ∗}^{u}o_{P c}^{u} = b̄_{c ∗}^{+}k_{c u ∗}^{u}o_{ξ c u}^{u} ,

r ↦ ro_{ξ P c}^{u} defines an isomorphism R ≃ H̃_{P} C (S; R). Note that o_{ξ P c}^{u} can be characterized as the unique generator of H̃_{P} C (S; R) that is the canonical image of the Kronecker dual of i_{c}^{u ∗}τ^{u}.

In a similar manner, one obtains a generator o_{ξ N c}^{s} of H̃_{N} C∗(S; R) that can be characterized as the unique generator of that module that is the canonical image of the Kronecker dual of i_{c}^{s ∗}τ^{s} where i_{c}^{s} is the inclusion of pairs (W_{Fξ c}^{s}, ∂W_{Fξ c}^{s}) ⊂ (E_{S}^{s}, E_{S 0}^{s}).

For ν = u, s, for i = P when ν = u but for i = N when ν = s, let o_{ξ c}^{ν} ∈ H_{i+1}(W_{ξ c}^{ν}, ∂W_{ξ c}^{ν}; R) be the fundamental class determined by τ^{ν} and o_{S} according to the prescription of Lemma 7.29. Thus, k_{c ∗}^{u}o_{ξ c}^{u} generates H_{P +1}(B_{c}, B_{c}^{+}; R) and k_{c ∗}^{s}o_{ξ c}^{s} generates H_{N+1}(B_{c}, B_{c}^{−}; R). It follows that o_{ξ P +1 c}^{u} := b̄_{c ∗}^{+}k_{c ∗}^{u}o_{ξ c}^{u} generates H̃_{P +1}C (S; R) and that o_{ξ N+1 c}^{s} := b̄_{c ∗}^{−}k_{c ∗}^{s}o_{ξ N+1 c}^{s} generates H̃_{N+1}C∗(S; R).

The generating classes o_{ξ P c}^{u}, o_{ξ P +1 c}^{u}, o_{ξ N c}^{s}, and o_{ξ N+1 c}^{s} are, in fact, independent of our choice of c ∈ ]0, c_{0} [. The proof of this is in the main analogous to the proof of independence of c given in the course of proving Theorem 7.15(2). There are two main structural elements in that proof: the first of these is diagram (7.7); the second is the family of isotopies H_{t}^{c,c̄} from W_{dξ c}^{u} to W_{dξ c̄}^{u} defined for 0 < c, c̄ < c_{0} using the flow of dξ. The analogues of these structural elements in the current situation are obtained as follows.

Diagram (7.7) is used as is, but the blocks B_{c′} and B_{c′′} where 0 < c′ < c′′ < c_{0} are now those provided by Theorem 7.21 rather than those provided by Theorem 7.12, and the set CB_{c′′}^{+} is now defined by Lemma 7.22. By that lemma (B_{c′′}, CB_{c′′}^{+}) is a regular index pair relative to the flows generated by ξ, Fξ, and F^{⊥}ξ, and the inclusion j_{λ}: (B_{λ}, B_{λ}^{+}) → (B_{c′′}, CB_{c′′}^{+}) is defined for c′ ≤ λ ≤ c′′. As in the proof of Theorem 7.15(2), all arrows in the diagram except the left vertical arrow are the homotopy classes of either inclusions, inclusion induced maps on quotients, or quotient maps. Similarly, to define the left vertical arrow we first must define a family of isotopies H^{c c′} from W_{Fξ c}^{u} to W_{Fξ c′}^{u} for 0 < c, c′ < c_{0}.

We proceed to define an isotopy from W_{Fξ c′}^{u} to W_{Fξ c′′}^{u} whenever 0 < c′ ≤ c′′ < c_{0} making use of the fact that W_{Fξ c}^{u} = W_{F^{⊥}ξ c}^{u} for 0 < c < c_{0}. In particular, the isotopy is defined using the flow generated by F^{⊥}ξ which will be denoted by the map (u, t) ↦ u ∗ t, and all exit time maps are computed relative to this flow. For 0 ≤ t ≤ 1 define

H_{t}^{c′ c′′}(u) := \{ \begin{matrix}u & if u ∈ S \\ u ∗ tτ_{B_{c′′}} (u ∗ τ_{B_{c′}} (u)) & if u ∈ W_{Fξ c′}^{u} \ S. \end{matrix}

The proof that each H^{c′ c′′} is continuous and defines the required isotopy is essentially the same as in the proof of Theorem 7.15. Note that continuity could not be proved using the flow of Fξ instead of F^{⊥}ξ.

To define the left vertical arrow [h] in diagram (7.7) in the current context, note that as each point of the tubular neighborhood U lies in a unique fiber of the normal bundle as realized in the tubular neighborhood there are continuous projections π_{ν}: U → E_{S}^{ν} for ν = u, s. We write u = u_{1} ⊕ u_{2} for the unique u ∈ U for which u_{1} = π_{u}(u) and u_{2} = π_{s}(u) whenever u_{1} ∈ E_{S}^{u}, u_{2} ∈ E_{S}^{s} and p(u_{1}) = p(u_{2}). Also, note that the restriction of π_{ν} to B_{c} has image in W_{Fξ c}^{ν} for ν = u, s, for 0 < c < c_{0}. Let ℓ_{c′ c′′} be the inclusion π_{s}(B_{c′}) ⊂ π_{s}(B_{c′′}). Then define h: (B_{c′}, B_{c′}^{+}) → (B_{c′′}, B_{c′′}^{+}) by

h(u) = H_{1}^{c′ c′′} ^{∘} π_{u}(u) ⊕ ℓ_{c′ c′′} ^{∘} π_{s}(u) .

The left-hand vertical arrow in diagram (7.7) is [h], the homotopy class of the map h. Note that j_{c′′} ^{∘}h is homotopic to j_{c′} via the homotopy H_{t}^{c′ c′′} ^{∘}π_{u} ⊕ℓ_{c′ c′′} ^{∘}π_{s}, whence it follows that diagram (7.7) is commutative in the current context.

The proof that o_{ξ P c}^{u} is in fact independent of c is now left to the reader since it is entirely analogous to the computations carried out in the proof of Theorem 7.15(2). Therefore, the canonical image of o_{P c}^{u} is independent of c ∈ ]0, c_{0} [, denote it o_{ξ P}^{u}, and generates H̃_{P} C (S; R). That is, the canonical image of o_{ξ c u}^{u} in H̃_{P} C (S; R) is independent of c ∈ ]0, c_{0} [ and u ∈ S. In a similar manner, one shows that the canonical image of the class o_{ξ c u}^{s} in H̃_{N} C∗(S; R) is independent of c ∈ ]0, c_{0} [ and u ∈ S and generates H̃_{N} C∗(S; R); let o_{ξ N}^{s} denote this canonical image.

The proof that o_{ξ P +1 c}^{u} is independent of c is also similar to the proof of Theorem 7.15(2); however, we make one additional observation on the classes o_{Fξ c}^{u}, 0 < c < c_{0}, before leaving the additional details to the reader. Specifically, to carry out the proof of independence in a manner analogous to that used in the proof of Theorem 7.15 it must be shown that H_{1 ∗}^{c c̄}o_{Fξ c}^{u} = o_{Fξ c̄}^{u}. This follows from the definitions of o_{Fξ c}^{u} and o_{Fξ c̄}^{u} in terms of the Thom isomorphism and the classes τ^{u} and o_{S} according to the prescription of Lemma 7.29 as well as naturality of cap products, the fact that i_{c}^{u} is homotopic to iu ; c̄ ^{∘} H_{1}^{c c̄}, and the fact that H_{1}^{c c̄} is the identity on S. These details and the remaining details of the proof of independence of c are left as an exercise for the reader. Once the exercise is carried out, it follows that the canonical image of o_{ξ c}^{u} in H̃_{P +1}C (S; R) is independent of c ∈ ]0, c_{0} [, call the canonical image o_{ξ P +1}^{u}, and generates H̃_{P +1}C (S; R). In a similar manner, one shows that the canonical image of the class o_{ξ c}^{s} in H̃_{N+1}C∗(S; R) is independent of c ∈ ]0, c_{0} [, call the canonical image o_{ξ N+1}^{s}, and generates H̃_{N+1}C∗(S; R).

Finally, statement (3) holds as an immediate consequence of Lemma 7.7 using the flow of Fξ. □

7.32 The Proof of Theorem 7.6. Under the assumption of hypotheses HCO, conclusions (1)–(6) of Theorem 7.4 follow from the earlier results of this subsection: Conclusion (1) from Theorem 7.21(1), conclusion (2) (and also conclusions (1) and (2) of Theorem 7.6) from Theorem 7.23, and conclusions (3)–(6) (and therefore also conclusion (3) of Theorem 7.6) are restatements of the results of Theorem 7.31 which in the case of conclusion (6) also requires using conclusion (8) of Theorem 7.21. □