CHAPTER 3 STATEMENT OF THE CONTINUATION

RESULTS AND EXAMPLES

Throughout this chapter {φ_{λ}}_{λ∈Λ} is a continuous family of C^{s} flows on the C^{r} manifold M, 0 ≤ s ≤ r ≤ ∞, and S(φ) is the corresponding space of isolated invariant sets described in Definition 1.1(A) above. By a path β in S(φ) is meant a continuous map β: [0, 1] → S(φ). As shown in [K3] and as described below in Chapter 5, functorially associated to the homotopy class (endpoints fixed) of a path β in S(φ) are continuation isomorphisms on homology (coefficients in any R-module)

H̃_{∗}C (S_{β}(0)) ⟶^{C(β)_{∗}} H̃_{∗}C (S_{β}(1)) and H̃_{∗}C∗(S_{β}(0)) ⟶^{C∗(β)_{∗}} H̃_{∗}C∗(S_{β}(1)) .

(By “functorially associated” we mean that H̃_{∗}C and H̃_{∗}C∗ can be regarded as functors on the fundamental groupoid of S(φ) to the category of graded R-modules.)

A. Invariance of Intersection Numbers under Continuation. The general result for the invariance of intersection numbers under continuation is the following result whose proof is given in §5.11.

3.1 Theorem. Let β be a path (or a path class with endpoints fixed) in S(φ). Then for k = 0, 1, …, m, for α ∈ H̃_{m−k}C (S_{β}(0); R) and γ ∈ H̃_{k}C∗(S_{β}(0); R),

(3.1)  ^{#}(α ⌢_{•} γ) = ^{#}(C (β)_{∗}α ⌢_{•} C∗(β)_{∗}γ) .

When M = R^{m}, in H_{m}(R^{m}, R^{m} \ {0}; R),

(3.2)  α # γ = C (β)_{∗}α # C∗(β)_{∗}γ .

3.1.1 Remark. For s ∈ [0, 1] and β as in the statement of Theorem 3.1, define the path β^{s} in S(φ) by β^{s}(t) := β(st) for 0 ≤ t ≤ 1. Application of Theorem 3.1 to β^{s} for each s ∈ [0, 1] shows that ^{#}(C (β^{s})_{∗}α ⌢_{•} C∗(β^{s})_{∗}γ) does not vary with s.

From the construction of C (β) reviewed below, it will be seen that Theorem 3.1 is a simple consequence of the following result proved in §5.10.

3.2 Theorem. Assume [0, 1] ⊂ W ⊂ Λ ⊂ R with W open in R, let σ: W → S(φ) be a local section of π_{φ}: S(φ) → Λ, and let β be the restriction of σ to [0, 1]. Then equality (3.1) holds for each α ∈ H̃_{m−k}C (S_{σ}(0); R) and γ ∈ H̃_{k}C∗(S_{σ}(0); R). If M = R^{m}, then equality (3.2) also holds.

Theorem 3.2 will first be proved under additional assumptions of smoothness where the main idea of the proof is not obscured by technical difficulties. Specifically, the first continuation result proved is the following. The proof appears in §5.8.

3.3 Theorem. In addition to the hypotheses of Theorem 3.2, assume M is C^{r} (r ≥ 1), give the space of C^{0} vectorfields on M the compact-open topology, and for each λ ∈ Λ, assume φ_{λ} is generated by a continuous vectorfield X_{λ} with λ ↦ X_{λ} continuous. Then equality (3.1) holds for each α ∈ H̃_{m−k}C (S_{σ}(0); R) and γ ∈ H̃_{k}C∗(S_{σ}(0); R). If M = R^{m}, then equality (3.2) also holds.

The following discussion and proposition give a useful criterion for establishing continuity of a map g: Y → S(φ) that will ease the reader’s understanding of some examples presented afterward illustrating the above continuation theorems and the analogue of these results for the intersection class pairing.

It follows straightforwardly from the definitions that a map g: Y → S(φ) is continuous at y_{0} if, and only if, λ_{g} is continuous at y_{0} and for each N that λ_{g}(y_{0})- isolates S_{g}(y_{0}), there exists W an open neighborhood of y_{0} so that for each y ∈ W, N also λ_{g}(y)-isolates S_{g}(y). In fact it suffices to consider a single isolating neighborhood of S_{g}(y_{0}) as a consequence of the following:

3.4 Proposition. A map g: Y → S(φ) is continuous at y_{0} if, and only if, λ_{g} is continuous at y_{0} and there exist W and N that together satisfy W is an open neighborhood of y_{0} and N is a λ_{g}(y)-isolating neighborhood of S_{g}(y) for each y ∈ W.

Proof. Proof of necessity of the continuity condition follows straightforwardly from the definitions and is left to the reader. To prove sufficiency, assume N and W exist and suppose g(y_{0}) ∈ σ_{N′}(U) where U is open in Λ(N′) and N′ is a compact subset of M. Then S_{g}(y_{0}) = S(N′, λ_{g}(y_{0})); i.e., N and N′ λ_{0}-isolate the same invariant set where λ_{0} := λ_{g}(y_{0}). Set K := N ∩ N′ so that K also λ_{0}-isolates S_{g}(y_{0}) and K ⊂ N and K ⊂ N′. Now if two compact sets one containing the other λ_{0}-isolate the same invariant set, then by [K3, Lemma 1.5] they also λ-isolate the same invariant set for λ in a neighborhood of λ_{0}. Thus, there exists U′ an open neighborhood of λ_{0} satisfying U′ ⊂ Λ(N) ∩ Λ(K) ∩ U so that K, N and N′ λ-isolate the same invariant set for each λ ∈ U′. Then continuity of λ_{g} at y_{0} implies that we can choose W′ an open neighborhood of y_{0} with W′ ⊂ W so that λ_{g}(W′) ⊂ U′. Then on the one hand, by choice of U′ and W′, the compact sets N and N′ both λ_{g}(y)-isolate the same invariant set for each y ∈ W′, and on the other, by definition of N and W and as W′ ⊂ W, necessarily N λ_{g}(y)-isolates S_{g}(y) for each y ∈ W′. Thus S_{g}(y) = S(N′, λ_{g}(y)) for each y ∈ W′ and λ_{g}(W′) ⊂ U′, i.e., g(W′) ⊂ σ_{N′}(U′), which yields continuity of g at y_{0}. □

3.5 Example. Let a be a continuous map of some open interval W ⊃ [0, 1] into the open interval ]0, 1[. For each λ ∈ Λ := W, call the planar system (differentiation is with respect to a variable τ and is denoted by a dot over a variable)

(3.3_{λ})  \begin{aligned}v̇ & = w \\ ẇ & = −v(1 − v)(v − a(λ)) \end{aligned}

the fast system at λ associated to the scalar second-order equation

ε^{2}v′′ + v(1 − v)(v − a(x)) = 0 .

Figure 1

This scalar equation considered on the interval 0 ≤ x ≤ 1 with Neumann endpoint conditions is the equilibrium equation of the Fisher equation

(3.4)  v_{t} = ε^{2}v_{xx} + v(1 − v)(v − a(x))

on the infinite half-strip [0, 1] × R^{+} satisfying Neumann conditions on the infinite edges that is a model for the evolution of a diploid population in a genetically isolated non-homogeneous finite one-dimensional habitat in the heterozygote inferior case where v is the frequency of one of the alleles and x ∈ [0, 1] represents position in the habitat; i.e., with a and A corresponding to the two alleles, the heterozygote genotype Aa is observed to be inferior to either of the homozygote genotypes aa or AA and v is the ratio of the number of alleles A to the total number of alleles of both types in the population at a given point in space and time [Fisher,Ha].

Sketched in Figure 1 are three phase portraits which qualitatively exhaust the possibilities for system (3.3_{λ}). These were sketched by drawing, for fixed λ, level curves of the Hamiltonian

H(v, w, λ) := \frac{1}{2}w^{2} + V (v, λ) where V (v, λ) := ∫_{0}^{v} u(1 − u)(u − a(λ)) du .

The technique for sketching such level curves from the graph of V for fixed λ via graphical integration is described, for example, in [A]. The phase portraits in Figures 1a, b, c correspond, respectively to the cases (i) 1/2 < a(λ) < 1, (ii) a(λ) = 1/2, (iii) 0 < a(λ) < 1/2. In each case, there is a largest compact invariant subset S_{a}(λ) of the flow φ_{λ} generated by system (3.3_{λ}), which in Figures 1a, b, c is the union of the shaded region (a closed 2-cell) together with any critical point outside that region (either (0, 0) or (1, 0), each represented by a heavy dot), whose analytic description is:

S_{a}(λ) := \{ \begin{matrix}\{(v, w) : (v, w) = (0, 0) or v ∈ ]0, 1] and H(v, w, λ) ≤ V (1, λ)\} \\ if 1 > a(λ) > 1/2; \\ \{(v, w) : v ∈ [0, 1] and H(v, w, λ) ≤ 0\} if a(λ) = 1/2; \\ \{(v, w) : (v, w) = (1, 0) or v ∈ [0, 1[ and H(v, w, λ) ≤ V (0, λ)\} \\ if 0 < a(λ) < 1/2. \end{matrix}

Using Proposition 3.4, the reader should have little difficulty showing that σ_{a}: W → S(φ) defined by σ_{a}(λ) := (S_{a}(λ), λ) is a continuous local section of π_{φ}. It may help the reader to note that S_{a}(λ) varies upper semi-continuously with λ and is the largest compact invariant subset of the flow φ_{λ}.

Let β be the path in S(φ) that is the restriction of σ_{a} to [0, 1]. Our goal in this example is to analyze Theorem 3.1 as it applies to the path β and to discern what it tells us about the intersection of the local invariant manifolds that are geometric representatives vis-à-vis a suitable isolating block of generating classes of the homology of the forward and reverse time Conley indices of the isolated invariant sets at the beginning and end of β. The point to be made is best illustrated if we assume henceforth that (a(0) − 1/2)(a(1) − 1/2) < 0, and for definiteness let us assume that a(0) > 1/2 > a(1). Thus, S_{a}(0) qualitatively corresponds to the S_{a}(λ) of Figure 1a and S_{a}(1) to the S_{a}(λ) of Figure 1c.

Drawn in Figures 1a, c, respectively covering the cases a(λ) > 1/2 and a(λ) < 1/2, are the bounding curves of an isolating block for S_{a}(λ) with two components, both closed 2-cells and each a block in its own right. The exit set consists of the thick black line segments of slope minus one, the entrance set consists of the thick dark grey line segments of slope plus one, and the rest of the boundary of the block consists of the thinner light grey curve segments that are in fact orbit segments of the flow φ_{λ}. One component block, call it C(λ), contains that part of S_{a}(λ) that is an invariant closed 2-cell bounded by a homoclinic orbit, the other, call it B(λ), contains the saddle point in S_{a}(λ) not in the invariant 2-cell. Our computation of the intersection number pairing on the Conley indices of S_{a}(λ) will be based on using geometric representatives of homology classes of the index pairs formed from these blocks and their exit and entrance sets, and for this purpose the graphical description of these blocks provided by Figure 1 will suffice. However, explicit analytic descriptions of these blocks and their exit and entrance sets is given below in the course of proving Theorem A.1 in Appendix A, an existence result for the Fisher equation (3.4) about which more is said in the last paragraph of this example. It is left as an easy exercise for the interested reader as it is not needed here that from these blocks one can compute that for a(λ) ≠ 1/2 each index space for S_{a}(λ) in both forward and reverse time has the homotopy type of a pointed one-sphere; hence by invariance of the homotopy type of index spaces under continuation, the same is true for a(λ) = 1/2. Thus, taking all coefficients in R (R = Z_{2} is a good choice) we must show that the Conley indices of S_{a}(λ) in both time directions have one-dimensional homology isomorphic to R while the homology modules must vanish in all other dimensions.

The necessary computations follow from the following sequence of observations throughout which it is assumed that a(λ) ≠ 1/2. First, H_{∗}(C(λ), C(λ)^{±}) ≡ 0 as follows from the long exact singular homology sequence of the pair since C(λ) is a 2-cell and C(λ)^{±} is an arc in its boundary. Second, H_{1}(B(λ), B(λ)^{±}) ≃ R, but H_{q}(B(λ), B(λ)^{±}) ≃ 0 for all q ≠ 1. Again this follows from the long exact sequence of the pair using reduced singular homology since the pair (B(λ), B(λ)^{±}) is homeomorphic to the pair consisting of the standard 2-cell and one pair of parallel edges. Further, an examination of the connecting map of the sequence from the 1-level to the 0-level shows that any path in B(λ) starting in one component of B(λ)^{±} and ending in the other when regarded as a relative 1-cycle in B(λ) modulo B(λ)^{±} has homology class that generates H_{1}(B(λ), B(λ)^{±}). Third, an index pair for S_{a}(λ) consisting of the block K(λ) := B(λ) ∪ C(λ) together with its exit or entrance set has one-dimensional singular homology isomorphic to R but vanishes in all other dimensions; hence, the indices C (S_{a}(λ)) and C∗(S_{a}(λ)) have one-dimensional homology isomorphic to R, but have vanishing homology in all other dimensions. For from the Meyer-Vietoris sequence for singular homology and the first observation it follows that

H_{∗}(B(λ), B(λ)^{±}) ≃ H_{∗}(K(λ), K(λ)^{±})

where the isomorphism is inclusion induced, and this isomorphism followed by the canonical homomorphism from H_{∗}(K(λ), K(λ)^{±}) to the singular homology of the Conley index in the appropriate time direction is an isomorphism as follows from the observations made in Definitions 1.4(B) and 1.5(A)—also see Proposition 4.2 below. The third observation then follows from the second.

Let α_{λ} generate H̃_{1}C (S_{a}(λ)), let γ_{λ} generate H̃_{1}C∗(S_{a}(λ)), and let s_{λ} denote the saddle point in B(λ) for a(λ) ≠ 1/2. Then since as observed any path in a 2-cell starting in one edge of a parallel pair and ending in the other edge of the parallel pair is a relative 1-cycle generating the homology of the 2-cell modulo the parallel pair of edges, in particular, the local unstable manifold of s_{λ} within B(λ) is an arc that can be regarded as a geometric generator of H_{1}(B(λ), B(λ)^{+}), and therefore by the isomorphisms described in the previous paragraph, it can be regarded as a geometric representative of α_{λ}. Similarly, the local stable manifold of s_{λ} within B(λ) is an arc that can be regarded as a geometric representative of γ_{λ}. Finally, because these local invariant manifolds are transverse as is easily computed from the linearization of system (3.3_{λ}) about s_{λ}, it follows from Theorem 6.5 below—also see statements (2) and (3) of Theorem 7.5—that

(3.5a)  ^{#}(α_{λ} ⌢_{•} γ_{λ}) = ±1

or, equivalently, because we are working with a planar flow that

(3.5b)  α_{λ} # γ_{λ} = ±o_{2} .

By Theorem 3.1 (or 3.2 or 3.3),

(3.6)  C (β)_{∗}α_{0} # C∗(β)_{∗}γ_{0} = α_{0} # γ_{0} .

This equality is certainly non-trivial in the present context, for it asserts that there is a functorial identification of the intersection of the local invariant manifolds of the saddle point (0, 0) of system (3.3_{0}) with the intersection of the local invariant manifolds of the saddle point (1, 0) of system (3.3_{1}) even though the first saddle arises from the v = 0 branch of the zero set of v(1 − v)(v − a(x)) and the second arises from the v = 1 branch of the zero set. Also, see Theorem 3.7 below and the paragraph preceding it.

Before leaving this example let us make one final observation. Because C∗(β)_{∗} is an isomorphism necessarily C∗(β)_{∗}γ_{0} = ±γ_{1}. This equality combined with (3.5) and (3.6) yields that

(3.7)  C (β)_{∗}α_{0} # γ_{1} ≠ 0 .

One way to interpret (3.7) is that any arc representing α_{0} relative to some forward time index pair for the saddle point s_{0} continues to an arc that intersects any arc that represents γ_{1} relative to some reverse time index pair for the saddle point s_{1}. This interpretation together with the main result of [K4] (which in part was motivated by this interpretation) yields Theorem A.1 of Appendix A which states in part that for 0 < ε ≪ 1 there exists a family v_{ε} of non-constant equilibrium solutions for the Fisher equation (3.4) over the interval [0, 1] satisfying Neumann endpoint conditions.

B. Continuation of 𝔏 over a Path of Isolated Invariant Sets. To discuss continuation results for the intersection class pairing, a topological space of Čech homology classes of isolated invariant sets will be introduced.

3.6 Definition and Proposition. (A) For each k ∈ N, for any R-module G′′, define Ȟ_{k}S(φ; G′′) to be the set of ordered triples (ζ̌, S, λ) satisfying (S, λ) ∈ S(φ) and ζ̌ ∈ Ȟ_{k}(S; G′′) and define Ȟ_{∗}S(φ; G′′) := ⋃{Ȟ_{k}S(φ; G′′) : k ∈ N}. Also, for k ∈ N, let πȞ_{k}: Ȟ_{k}S(φ; G′′) → S(φ) be given by projection of the last two coordinates.

(B) For k ∈ N, for (ζ̌, S, λ) ∈ Ȟ_{k}S(φ; G′′), for N a λ-isolating neighborhood of S, and for U open in Λ(N) with λ ∈ U, define N (ζ̌, S, λ, N, U) to be the set of points (ζ̌′, S′, λ′) ∈ Ȟ_{k}S(φ; G′′) satisfying λ′ ∈ U, S′ = S(N, λ′), ζ̌′ ∈ Ȟ_{k}(S′; G′′) and π_{N}^{S} ζ̌ = π_{N}^{S′} ζ̌′ where π_{N}^{S} and π_{N}^{S′} are respectively the canonical projections of Ȟ_{∗}(S; G′′) and Ȟ_{∗}(S′; G′′) to H_{∗}(N; G′′).

For k ∈ N, define B_{k} to be the set consisting of sets of the form N (ζ̌, S, λ, N, U). (C) B_{k} is a basis for a topology on Ȟ_{k}S(φ) (call it the fine topology) and

πȞ_{k}: Ȟ_{k}S(φ; G′′) → S(φ) is continuous.

Proof. Clearly B_{k} covers Ȟ_{k}S(φ; G′′). If (ζ̌′′, S′′, λ′′) ∈ N (ζ̌_{i}, S_{i}, λ_{i}, N_{i}, U_{i}) for i = 1, 2, then by [K3, Lemma 1.5] for some open neighborhood U′′ of λ′′, U′′ ⊂ Λ(N_{1}) ∩ Λ(N_{2}) ∩ Λ(N_{1} ∩ N_{2}), and λ′ ∈ U′′ implies S(N_{i}, λ′) = S(N_{1} ∩ N_{2}, λ′) for i = 1, 2. Claim

N (ζ̌′′, S′′, λ′′, N_{1} ∩ N_{2}, U′′) ⊂ ⋂_{i=1}^{2} N (ζ̌_{i}, S_{i}, λ_{i}, N_{i}, U_{i}) .

For suppose (ζ̌′, S′, λ′) ∈ N (ζ̌′′, S′′, λ′′, N_{1} ∩ N_{2}, U′′). Then π_{N_{1}∩N_{2}}^{S′} ζ̌′ = π_{N_{1}∩N_{2}}^{S′′} ζ̌′′, whence, for i = 1, 2,

π_{N_{i}}^{S′′} ζ̌′′ = i_{N_{i}}^{N_{1}∩N_{2}} ^{∘} π_{N_{1}∩N_{2}}^{S′′} ζ̌′′
= i_{N_{i}}^{N_{1}∩N_{2}} ^{∘} π_{N_{1}∩N_{2}}^{S′} ζ̌′
= π_{N_{i}}^{S′} ζ̌′ .

From this and because (ζ̌′′, S′′, λ′′) ∈ N (ζ̌_{i}, S_{i}, λ_{i}, U_{i}) for i = 1, 2, it follows that also (ζ̌′, S′, λ′) ∈ N (ζ̌_{i}, S_{i}, λ_{i}, U_{i}) for i = 1, 2, whence B_{k} is a basis.

Continuity of πȞ_{k} at each point of Ȟ_{k}S(φ; G′′) is immediate from the inclusion πȞ_{k}(N (ζ̌, S, λ, N, U)) ⊂ σ_{N}(U) holding for each N (ζ̌, S, λ, N, U) ∈ B_{k}. □

Remark. Unless Ȟ_{∗}(S; G′′) ≃ H_{∗}(S; G′′) for each (S, λ) ∈ S(φ) (for example, if S is always a manifold), the “projection” πȞ_{k} need not be an open map. However, if the Čech and singular homologies are isomorphic at each point of S(φ), then it is an easy exercise to construct local sections of Ȟ_{k}S(φ; G′′) making πȞ_{k} a local homeomorphism.

The next theorem is an algebraic statement of the continuity along a continuous path of isolated invariant sets of the intersection of the stable and unstable sets of the invariant sets on the path. For its statement, assume G and G′ are R-modules. Its proof is given in §5.14.

3.7 Theorem. For β a path in S(φ), for α ∈ H̃_{i}C (S_{β}(0); G), and for γ ∈ H̃_{j}C∗(S_{β}(0); G′), set k := i+j −m and define the path ℓ(β, α, γ) in Ȟ_{k}S(φ; G⊗G′) by

ℓ(β, α, γ)(s) := (C (β^{s})_{∗}α ⌢_{•} C∗(β^{s})_{∗}γ , β(s)) for s ∈ [0, 1].

Then relative to the fine topology on Ȟ_{k}S(φ; G ⊗ G′), ℓ(β, α, γ) is a continuous lift of β with initial point (α ⌢_{•} γ, β(0)).

3.8 Definition. (A) For G′′ an R-module, if (S, λ) ∈ S(φ) and ˇ0_{k} is the zero element of Ȟ_{k}(S; G′′), call (ˇ0_{k}, S, λ) a singular zero point of Ȟ_{k}S(φ; G′′) if, and only if, for each neighborhood Z of (ˇ0_{k}, S, λ) there exists (ζ̌, S′, λ′) ∈ Z with ζ̌ not the zero element of Ȟ_{k}(S′; G′′). Call (S, λ) ∈ S(φ) a G′′-singular point of S(φ) if, and only if, for some k ∈ N, (ˇ0_{k}, S, λ) is a singular zero point of Ȟ_{k}S(φ; G′′).

(B) Call a G′′-singular point (S, λ) ∈ S(φ) a G′′-bifurcation point of S(φ) if, and only if, for some N that λ-isolates S, for each open U satisfying λ ∈ U ⊂ Λ(N), there exist λ_{1}, λ_{2} ∈ U \ {λ} so that

Ȟ_{∗}(S_{σ_{N}} (λ_{1}); G′′) ≄ Ȟ_{∗}(S_{σ_{N}} (λ_{2}); G′′) .

The following example shows that, in general, a lift of β having the form ℓ(β,α, γ) with initial point not in the zero section of Ȟ_{k}S(φ) can have a singular zero point in its image whose projection to S(φ) is a bifurcation point. Of course, as a consequence of Theorem 3.1, necessarily k > 0 where α ⌢_{•} γ ∈ Ȟ_{k}(S_{β}(0)).

3.9 Example. Consider the one-parameter family of autonomous systems in R^{3} (λ ∈ Λ := R)

(3.8_{λ})

u̇ = u^{2} − v^{2} − w^{2} − λ
v̇ = −2uv + min{0, λ}w + max{0, λ − u^{2}}
ẇ = −2uw − min{0, λ}v

where the dots over u, v, and w indicate differentiation with respect t ∈ R. This family of systems is discussed extensively in [K4, Example 5.1] in conjunction with the behavior for 0 < ε ≪ 1 of solutions to initial value problems for the system εu′ = F(u, x) where u := (u, v, w) ∈ R^{3}, differentiation is with respect to x ∈ R, and F(u, λ) is given by the right hand side of (3.8_{λ}). Here it is examined in terms of the peculiarities of the intersection number and class pairings it points out. The reader is reminded that in this work when given a block B, its exit (entrance) set is B^{+} (respectively, B^{−}) whereas in [K4] the opposite signs are used to designate exit and entrance sets.

Let φ_{λ} be the flow in R^{3} generated by system (3.8_{λ}) so that φ_{λ}, λ ∈ R is a continuous family of flows in R^{3}. For λ < 0 let S(λ) be the circle locus u = 0, v^{2} + w^{2} = −λ, for λ = 0 set S(0) := {(0, 0, 0)}, and for λ > 0 let S(λ) be the doubleton with elements (±\sqrt{λ}, 0, 0). Then S(λ) is the maximal isolated invariant set of system (3.8_{λ}) for each λ ∈ R, and as discussed in [K4, Example 5.1], the section σ of π_{φ}: S(φ) → Λ defined by σ(λ) = (S(λ), λ) is continuous. For λ < 0, S(λ) is a hyperbolic periodic orbit with two-dimensional stable and unstable manifolds, S(0) is a degenerate critical point, and for λ > 0, S(λ) consists of two critical points, one having two-dimensional unstable manifold and one-dimensional stable, the other having the dimensions of the invariant manifolds reversed. Typical phase portraits for the three cases including an isolating block are sketched in Figure 2. Call the blocks in Figures 2a, 2b and 2c, respectively T (λ, δ) (the case λ < 0), B(δ) (the case λ = 0), and K(λ, δ) (the case λ > 0) where δ is chosen to lie in ]0, \sqrt{−λ}[ when λ < 0 and in ]0, \frac{1}{2} \sqrt{λ}[ when λ > 0, and is an arbitrary positive number for λ = 0. T (λ, δ) is a solid torus of revolution (about the u-axis) with square cross-section of diagonal length 2δ centered on its point of intersection with the periodic orbit with one pair of opposite edges of the cross-section generating the exit set and the other pair the entrance set. Hence the exit set of T (λ, δ) consists of two disjoint annuli in the bounding torus as does the entrance set, the two sets of two annuli being complementary up to closure. B(δ) is a solid ball of radius δ centered at the origin with exit set consisting of a cap centered at the north pole (i.e., u = δ, v = w = 0) and a band extending downward from the equator while the entrance set consists of the closure of the complement relative to the bounding sphere of the exit set and consists of a cap centered at the south pole together with a band extending upward from the equator. K(λ, δ) is the union of two disjoint isolating blocks each a bent solid cylinder of square cross-section with diagonal of length 2δ. Let K_{±}(λ, δ) be the cylinder containing (±\sqrt{λ}, 0, 0). Then the exit set of K(λ, δ) consists of the top and bottom of K_{+}(λ, δ) and the lateral surface of K_{−}(λ, δ) while the entrance set consists of the top and bottom of K_{−}(λ, δ) and the lateral surface of K_{+}(λ, δ). The reader is referred to [K4, Example 5.1] for a precise analytic description of the blocks and their exit and entrance sets.

It is easy to compute that the index spaces K(λ, δ)/K(λ, δ)^{±} have the homotopy type of S^{1} ∨ S^{2}; hence by invariance of the Conley homotopy index under continuation along any continuous curve in S(φ) the same is true of B(δ)/B(δ)^{±} and T (λ, δ)/T (λ, δ)^{±}. It follows that H̃_{j}C (S(λ); Z) and H̃_{j}C∗(S(λ); Z) are isomorphic to Z when j = 1, 2 but are otherwise the trivial group.

A compatible choice of generators for the non-zero groups is made as follows where we continue to assume Z coefficients for the rest of this example. Choose a generator α_{j,−1} of H̃_{j}C (S(−1)) and a generator γ_{j,−1} of H̃_{j}C∗(S(−1)) for j = 1, 2. Set λ_{β}(t) := 2t − 1 for t ∈ [0, 1], define the path β in S(φ) by β(t) := σ^{∘} λ_{β}(t), and for 0 < s ≤ 1 define the auxiliary path β^{s} in S(φ) as in Remark 3.1.1. For each λ ∈ [−1, 1] set s_{λ} := \frac{1}{2}(λ + 1) and for j = 1, 2 set

α_{j λ} := C (β^{s_{λ}})_{∗}α_{j,−1} , γ_{j λ} := C∗(β^{s_{λ}})_{∗}γ_{j,−1} .

Then α_{j λ} generates H̃_{j}C (S(λ)) and γ_{j λ} generates H̃_{j}C∗(S(λ)) (j = 1, 2). In [K4] there is a lengthy discussion on the various possible choices of geometric representatives for α_{j λ} and an analogous discussion applies to the reverse time indices Figure 2. (a) The block T (−1, δ) and local stable and unstable manifolds of S(−1). (b) The block B(δ) (shaded portions of the bounding sphere indicate the exit set) and local stable and unstable sets of S(0). Note that the positive (negative) u-axis is part of the unstable (stable) set of S(0). (c) The block K(1, δ) := K_{−}(1, δ)∪K_{+}(1, δ) and local stable and unstable manifolds of S(1). and the choice of geometric representatives for γ_{j λ}. In particular, for λ ∈ [−1, 0[, the local unstable (stable) manifold of S(λ) within T (λ, δ) is an annulus with one bounding circle in each annular component of the exit (entrance) set and is a geometric representative of α_{2 λ} (respectively, γ_{2 λ}). Also, the annulus that is the local unstable or stable manifold within T (λ, δ) can be fibered by arcs with each arc having an endpoint in each of the bounding circles of the annulus, and such an arc in the local unstable (stable) manifold represents α_{1 λ} (respectively, γ_{1 λ}).

Because, when −1 ≤ λ < 0, any local stable and unstable manifolds of S(λ) intersect transversely in the periodic orbit S(λ), it is immediate from Theorem 6.5 that α_{2 λ} ⌢_{•} γ_{2 λ} generates Ȟ_{1}(S(λ)) ≃ Z and also that

(3.9)  ^{#}(α_{1 λ} ⌢_{•} γ_{2 λ}) = ±1 and ^{#}(α_{2 λ} ⌢_{•} γ_{1 λ}) = ±1 .

Further, by the invariance of intersection numbers under continuation, it follows that (3.9) holds for all λ ∈ [−1, 1]. This is interesting as regards the example under discussion because S(λ) undergoes bifurcation as λ increases through zero from a degenerate critical point to two non-degenerate ones (here, “non-degenerate” means that the local stable and unstable sets are topological cells that are topologically transverse) and for λ > 0 the two intersection numbers in (3.9) are associated to different critical points. Specifically, from the analysis of the family of systems (3.8_{λ}) undertaken in [K4, Example 5.1] and from a similar analysis of these systems with time reversed, it follows for λ > 0 and 0 < δ ≤ \frac{1}{2} \sqrt{λ} that α_{2 λ} ⌢_{•} γ_{1 λ} is represented by the intersection of the local unstable and stable manifolds of (−\sqrt{λ}, 0, 0) within K_{−}(λ, δ) while α_{1 λ} ⌢_{•} γ_{2 λ} is represented by the intersection of the local stable and unstable manifolds of (\sqrt{λ}, 0, 0) within K_{+}(λ, δ).

As S(0) is a single point and S(λ) for λ > 0 is two points, Ȟ_{1}(S(λ)) = 0 for λ ≥ 0. Thus

C (β^{s_{λ}})_{∗}α_{2,−1} ⌢_{•} C∗(β^{s_{λ}})_{∗}γ_{2,−1} = ˇ0_{1} ∈ Ȟ_{1}(S(λ)) for λ ≥ 0.

Hence, (ˇ0_{1}, S(0), 0) is a Z-singular zero point in the image of ℓ(β, α_{2,−1}, γ_{2,−1}), and also (S(0), 0) is a Z-bifurcation point of S(φ).

Definition 3.8 and Example 3.9 raise the question as to what are sufficient conditions on the manifold M, the family of flows φ, and the path β ∈ S(φ) to ensure that β has no singular points or bifurcation points in its image. For example if M and φ are smooth and S_{β}(λ) is a non-degenerate hyperbolic set for λ ∈ Λ a smooth manifold, then transversality theory yields that in general there will be no bifurcation points on β. A more difficult question is under what circumstances are bifurcation points unavoidable; i.e., if for a given family of flows φ and path β into S(φ) there is a bifurcation point on β, then is it the case that whenever φ is embedded in a larger family of flows Φ and β is regarded as a path in S(Φ), then any path β′ sufficiently close to β also has a bifurcation point in its image.

Neither of the questions just raised shall be explored here. However, we would like to point out that homotopy theoretic methods and obstruction theory probably have a role to play in the answers. To see this let us note that in some circumstances, it is possible to put a different topology on Ȟ_{k}S(φ) so that ℓ(β, α, γ) is a continuous lift of β implies its first component is always a zero class or never a zero class.

For example, with coefficients in the PID R, it is possible to define an alternate topology on Ȟ_{k}S(φ; R) if we assume that for each (S, λ) ∈ S(φ), S has arbitrarily small λ-isolating neighborhoods N satisfying H_{k}(N; R) is a free R-module. Call this assumption Hypothesis τ_{k}(R).

The point of Hypothesis τ_{k}(R) is that for each λ-isolated invariant set S it ensures the existence of arbitrarily small λ-isolating neighborhoods N of S with the property that for 0 ≠ ζ ∈ H_{k}(N; R), there exists ψ_{ζ} ∈ Hom(H_{k}(N; R), R) satisfying ψ_{ζ}(ζ) ≠ 0—just choose ψ_{ζ} from a suitable dual basis. The Universal Coefficient Theorem for Cohomology then yields the existence of z^{ζ N} ∈ H^{k}(N; R) so that ⟨z^{ζ N}, ζ⟩ = 1.

A sub-basis for a topology on Ȟ_{k}S(φ; R) can then be defined as follows. For each compact N ⊂ M, for each 0 ≠ z ∈ H^{k}(N; R), and for each open V ⊂ Λ(N), define ⟨z, N, V ⟩ to be the set of triples (ζ̌, S, λ) satisfying (i) λ ∈ V and (S, λ) ∈ S(φ), (ii) S = S(N, λ), (iii) ζ̌ ∈ Ȟ_{k}(S; R) and ⟨z, ζ_{N}⟩ ≠ 0 where ζ_{N} is the image of ζ̌ under the canonical projection of Ȟ_{∗}(S; R) to H_{∗}(N; R). Also for the zero element 0^{(k)} ∈ H^{k}(N; R) define ⟨0^{(k)}, N, V ⟩ to be the set of triples (ˇ0, S, λ) where λ and S satisfy (i) and (ii) and ˇ0 is the zero element of Ȟ_{k}(S; R). To see that the collection B̌_{k} of sets of the form ⟨z, N, V ⟩ covers Ȟ_{k}S(φ; R) and can therefore be taken as a sub-basis for a topology make the following observation: If (S, λ) ∈ S(φ) and ˇ0_{k} ≠ ζ̌ ∈ Ȟ_{k}(S; R), then ζ_{U} ≠ 0 for some neighborhood U of S, whence necessarily ζ_{U′} ≠ 0 for each neighborhood U′ of S with U′ ⊂ U. In particular, by Hypothesis (τ_{k}), there exists a λ- isolating neighborhood N of S with N ⊂ U so that ζ_{N} ≠ 0 and H_{k}(N; R) is a free R-module. Hence, Hypothesis (τ_{k}) implies that (ζ̌, S, λ) ∈ ⟨z^{ζ N}, N, Λ(N)⟩.

Call the topology on Ȟ_{k}S(φ; R) generated by the just described sub-basis the weak topology. Each sub-basic open set with non-zero elements is open in the fine topology since N (ζ̌, S, λ, N, V) ⊂ ⟨z, N, V ⟩ if (ζ̌, S, λ) ∈ ⟨z, N, V ⟩ and ζ̌ ≠ 0.

By fiat, Ȟ_{k}S(φ; R) has the zero section as one component relative to this weak topology, whence with the weak topology it is impossible to continue a non-zero class to a zero class. However, this topology is very weak—it is not even Hausdorff on stalks of πȞ_{k} although this is easily remedied if R is an ordered ring or admits a non-trivial absolute value—and it does not seem readily possible to prove useful results of the type that would give sufficient conditions on when a path β in S(φ) has a continuous lift to Ȟ_{∗}S(φ) with non-zero initial point. From the presentation of this alternate topology on Ȟ_{k}S(φ; R), the reader might surmise that it would be better to deal directly with cohomology classes. That is, rather than examining continuations of α ⌢_{•} γ, instead examine continuations of x ⌣ y where x and y are Poincaré duals of α and γ, respectively. As just formulated this does not quite work, although a modification obtained by composing the cup product at the isolating block level with a Lefschetz duality isomorphism on cohomology (assuming field coefficients) can be made to work. Essentially, that is the viewpoint taken by Montgomery in §§6–7 of [M]. In particular, the reader should compare Example 3.9 above with [M, Example 7.8]. However, in many examples it seems easier to directly compute intersection numbers and intersection pairings rather than cup products.

Duality of Conley indices, its relation to 𝔏, and by implication the continuability of cup products are examined in Chapter 9.