CHAPTER 6 SOME BASIC COMPUTATIONAL TOOLS

For simplicity, all homology in this chapter is taken with coefficients in R, and coefficient modules are, therefore, suppressed from the notation.

A. Conditions on Singular Cycles for Computing 𝔏 and ^{#}𝔏. The first result, Theorem 6.1, describes to what extent it is possible to express an intersection class of homology classes in the Conley indices of an isolated invariant set in terms of relative singular cycles supported in an isolating block for the invariant set. Corollary to it is the result mentioned at the end of §2.B; viz., it is possible to express an intersection number of homology classes in the Conley indices of an isolated invariant set directly in terms of a pair of relative singular cycles supported in a block modulo some minor restrictions on how the chains are situated within the block. The following hypotheses on the homology classes and representing chains in the block will be assumed:

Hypotheses CB.

(1) B is an isolating block for S;

(2) α ∈ H̃_{i}C (S) and γ ∈ H̃_{j}C∗(S);

(3) c^{+} is a singular cycle in B modulo B^{+} representing α and c^{−} is a singular cycle in B modulo B^{−} representing γ;

(4) α_{0} := ⟨c^{+}⟩ ∈ H_{i}(|c^{+}|, |∂c^{+}|) and γ_{0} := ⟨c^{−}⟩ ∈ H_{j}(|c^{−}|, |∂c^{−}|).

Note that if |∂ c^{±}| ∩ |c^{∓}| = ∅, then as follows from Definition 2.8(C), α_{0} ⌢_{•} γ_{0} is a well-defined element of Ȟ_{i+j−m}(|c^{+}| ∩ |c^{−}|) where m = dim(M).

6.1 Theorem. Under hypotheses CB, if (i) |c^{+}| ∩ |c^{−}| ∩ (B^{−} ∪ B^{+}) = ∅, then there exists N ∈ ob(AIP(S)) satisfying

(ii) |c^{+}| ∩ |c^{−}| ⊂ N_{1} ∩ N∗_{1} , (iii) N ↪ ((B, B^{+}), (B, B^{−}))

and

(iv) V ∈ N (N_{1} ∩ N∗_{1})  implies  π_{V}^{S} (α ⌢_{•} γ) = π_{V}^{|c^{+}|∩|c^{−}|}(α_{0} ⌢_{•} γ_{0}) ;

hence, for such V, π_{V}^{S} (α ⌢_{•} γ) ≠ 0 implies ∅ ≠ |c^{+}| ∩ |c^{−}| ⊂ B \ (B^{+} ∪ B^{−}).

Taking V := M in (iv) we immediately get the following.

6.2 Corollary. Assume i + j = m. Then (i) implies

#(α ⌢_{•} γ) = ^{#}(α_{0} ⌢_{•} γ_{0}) .

Hence, if ^{#}(α ⌢_{•} γ) ≠ 0, then |c^{+}| ∩ |c^{−}| ≠ ∅ whether or not (i) holds, but if it does, then |c^{+}| ∩ |c^{−}| ⊂ B \ (B^{+} ∪ B^{−}).

Corollary 6.2 together with Proposition 2.13 yields the corresponding result for homology intersection numbers.

6.3 Corollary. Assume M := R^{m} and i + j = m. Then (i) implies

α # γ = α_{0} # γ_{0} .

Hence, if α # γ ≠ 0, then |c^{+}| ∩ |c^{−}| ≠ ∅ whether or not (i) holds, but if it does, then |c^{+}| ∩ |c^{−}| ⊂ B \ (B^{+} ∪ B^{−}).

Proof of Theorem 6.1. Observe that condition (i) in the statement of the theorem can be rewritten in the form

(6.1)  | c^{+}| ∩ B^{+} ∩ | c^{−}| = ∅ = | c^{−}| ∩ B^{−} ∩ | c^{+}| .

From (6.1) and the fact that c^{±} is a relative cycle modulo B^{±} it is immediate that

|∂c^{+}| ∩ | c^{−}| = ∅ = | c^{+}| ∩ |∂c^{−}| .

Hence, by Definition 2.8(C), the intersection class α_{0} ⌢_{•} γ_{0} is well-defined as an element of Ȟ_{i+j−m}(| c^{+}| ∩ | c^{−}|).

The basic idea of the proof is that when (6.1) holds it is possible to construct N ∈ ob(AIP(S)) satisfying

(1) (N_{1}, N_{0}) ⊂ (B, B^{+}) and (N∗_{1}, N∗_{0}) ⊂ (B, B^{−});

(2) c^{+} and c^{−} are relative cycles, the former in N_{1} modulo N_{0}, the latter in N∗_{1}

modulo N∗_{0}

with the caveat that to carry out this construction it may first be necessary to deform within B the chain c^{±} to a chain c_{1}^{±} inconsequentially in the sense that | c^{+}| ∩ | c^{−}| and the boundary chain ∂c^{±} remain pointwise fixed by the deformation and where α_{1} is the class of c_{1}^{+} in H_{∗}(|c^{+}|, |∂c_{1}^{+}|) and γ_{1} is the class of c_{1}^{−} in H_{∗}(|c_{1}^{−}|, |∂c_{1}^{−}|)

(6.2)  α_{0} ⌢_{•} γ_{0} = α_{1} ⌢_{•} γ_{1} .

It follows from (1) and (2) that

S ∪ (| c^{+}| ∩ | c^{−}|) ⊂ N_{1} ∩ N∗_{1} ⊂ B \ (B^{+} ∪ B^{−}) .

From these inclusions and from (1), (2), Lemma 2.10, and Theorem 2.11, it follows that V ∈ N (N_{1} ∩ N∗_{1}) implies

(6.3)  π_{V}^{|c^{+}|∩|c^{−}|}(α_{0} ⌢_{•} γ_{0}) = π_{V}^{S} (α ⌢_{•} γ) .

Thus, if (6.1) holds and π_{V}^{S} (α ⌢_{•} γ) ≠ 0, also 0 ≠ α_{0} ⌢_{•} γ_{0} ∈ Ȟ_{∗}(| c^{+}| ∩ | c^{−}|), whence | c^{+}| ∩ | c^{−}| ≠ ∅ which of course is also true if (6.1) fails.

To prove the existence of N ∈ ob(AIP(S)) with the desired properties there are two cases to consider with the second having two subcases. These cases are determined by the emptiness or non-emptiness of the sets C^{+} and C^{−} defined by C^{±} := | c^{±}| ∩ B^{∓}.

Figure 3. The three cases considered in Theorem 6.1. Each square represents an isolating block with exit set the black edges and entrance set the gray. The black arc corresponds to the relative cycle c^{+}, the gray to c^{−}.

Case I: C^{−} = ∅ = C^{+}. Then

∅ = | c^{+}| ∩ B^{−} = | c^{+}| ∩ τ∗_{B}^{−1}(0)

and

∅ = | c^{−}| ∩ B^{+} = | c^{−}| ∩ τ_{B}^{−1}(0) .

Thus, τ∗_{B} restricted to | c^{+}| and τ_{B} restricted to | c^{−}| are positive continuous functions, each having compact domain, whence each has positive infimum. Therefore, d > 0 can be chosen small enough so that if N equals the AIP(regular) pair of index pairs constructed from B as in the proof of Proposition 4.4 for the non-smooth case,

(| c^{+}|, |∂c^{+}|) ⊂ (N_{1}, N_{0}) ⊂ (B, B^{+})

and

(| c^{−}|, |∂c^{−}|) ⊂ (N∗_{1}, N∗_{0}) ⊂ (B, B^{−}) ,

whence it is immediate that properties (1) and (2) are satisfied. As observed the desired conclusions then follow.

Case II: C^{+} ≠ ∅ or C^{−} ≠ ∅. Essentially, this case can be reduced to the previous case by using the flow to pull c^{±} off of B^{∓} as follows. Note that (6.1) is equivalent to C^{∓} ∩| c^{±}| = ∅. Therefore we can find V^{±} open, satisfying C^{±} ⊂ V^{±} and cl (V^{±}) is disjoint from | c^{∓}| ∪ cl (V^{∓}), and by compactness of C^{±} and continuity of the flow, we can find ε > 0 and W^{±} an open neighborhood of C^{±} satisfying

W^{+} · [0, ε] ⊂ V^{+} and W^{−} · [−ε, 0] ⊂ V^{−} .

Also, choose ρ^{±}: B → [0, 1], continuous, satisfying supp(ρ^{±}) ⊂ W^{±} and ρ^{±}(C^{±}) = 1. Then for each u ∈ B define

h^{+}(u) := ρ^{+}(u) min{ε, τ_{B}(u)} , h^{−}(u) := ρ^{−}(u) min{ε, τ∗_{B}(u)} ,

and for each u ∈ B and s ∈ [0, 1] define

D_{s}^{±}(u) := D^{±}(u, s) := u · ±sh^{±}(u) .

Then D^{±} is a continuous deformation of B into itself that leaves B^{±} pointwise fixed throughout the deformation; in particular D_{1}^{±} is a self-map of the pair (B, B^{±}) inducing the identity isomorphism on homology.

Define

c_{1}^{±} := D_{1 ♯}^{±}c^{±}

where D_{1 ♯}^{±} denotes the map on singular chains induced by D_{1}^{±}. It is immediate that c^{±} and c_{1}^{±} are homologous relative cycles in B modulo B^{±}, whence c_{1}^{+} represents α and c_{1}^{−} represents γ. Further, from the given construction of D^{±},

(6.4)  | c^{+}| ∩ | c^{−}| = | c_{1}^{+}| ∩ | c_{1}^{−}| .

For by the choice of ε, W^{±}, and ρ^{±}, the set theoretic symmetric difference of | c^{±}| and | c_{1}^{±}| lies within V^{±}, whence (6.4) follows because as chosen cl (V^{+}) and cl (V^{−}) are disjoint with cl (V^{±}) disjoint from | c^{∓}|. Hence, with

(X^{±}, A^{±}) := (| c^{±}| ∪ cl (V^{±}) , |∂c^{±}|) ,

the construction of D^{±} shows that its restriction to X^{±}×[0, 1] defines a deformation of X^{±} into itself leaving A^{±} pointwise fixed. Thus, noting that |∂c^{±}| = |∂c_{1}^{±}|, we see that c^{±} and c_{1}^{±} are homologous relative cycles in (X^{±}, A^{±}).

Let α_{2} ∈ H_{i}(X^{+}, A^{+}) be the common homology class of c^{+} and c_{1}^{+} and let γ_{2} ∈ H_{j}(X^{−}, A^{−}) be the common homology class of c^{−} and c_{1}^{−}. Also, let c_{1}^{+} have homology class α_{1} ∈ H_{i}(| c_{1}^{+}|, |∂c_{1}^{+}|) and let c_{1}^{−} have homology class γ_{1} ∈ H_{j}(| c_{1}^{−}|, |∂c_{1}^{−}|). Too, note that ((X^{+}, A^{+}), (X^{−}, A^{−})) satisfies the off-diagonal condition and X^{+} ∩ X^{−} = | c^{+}| ∩ | c^{−}|. It follows from these definitions and observation as well as (6.4) and Lemma 2.10 that

(6.5)  α_{0} ⌢_{•} γ_{0} = α_{2} ⌢_{•} γ_{2} = α_{1} ⌢_{•} γ_{1}

as elements of Ȟ_{i+j−m}(| c^{+}| ∩ | c^{−}|). Thus (6.2) holds.

To complete the proof of Case II it will suffice to construct N ∈ ob(AIP(S)) satisfying the inclusions

(| c_{1}^{+}|, |∂c_{1}^{+}|) ⊂ (N_{1}, N_{0}) ⊂ (B, B^{+}) (6.6a) and

(| c_{1}^{−}|, |∂c_{1}^{−}|) ⊂ (N∗_{1}, N∗_{0}) ⊂ (B, B^{−}) . (6.6b)

For given such an N, let c_{1}^{+} have homology class α_{3} ∈ H_{i}(N_{1}, N_{0}) and let c_{1}^{−} have homology class γ_{3} ∈ H_{j}(N∗_{1}, N∗_{0}). It is immediate from (6.6) and Theorem 2.11 that if V is any neighborhood of N_{1} ∩ N∗_{1}, then

(6.7)  π_{V}^{N_{1}∩N∗_{1}}(α_{3} ⌢_{•} γ_{3}) = π_{V}^{S} (α ⌢_{•} γ) ,

and by (6.5) and Lemma 2.10,

(6.8)  ǐ_{∗}(α_{0} ⌢_{•} γ_{0}) = ǐ_{∗}(α_{1} ⌢_{•} γ_{1}) = α_{3} ⌢_{•} γ_{3}

where ǐ_{∗}: Ȟ_{∗}(|c^{+}| ∩ |c^{−}|) → Ȟ_{∗}(N_{1} ∩ N∗_{1}) is inclusion induced. From (6.7) and (6.8) it is immediate that if V ∈ N (N_{1} ∩ N∗_{1}), then (6.3) holds.

It is convenient to divide the proof of existence of N ∈ ob(AIP(S)) satisfying the inclusions (6.6) into two subcases as follows. Set C_{1}^{±} := | c_{1}^{±}| ∩ B^{∓}.

Subcase (a): C_{1}^{+} = ∅ = C_{1}^{−}. Then their exists an AIP(regular) pair of index pairs satisfying the inclusions (6.6) as an immediate consequence of the argument used to prove Case I.

Subcase (b): C_{1}^{+} ≠ ∅ or C_{1}^{−} ≠ ∅. The proof in this case is more difficult. First, it is straightforward to check that the construction of D^{±} implies that C_{1}^{±} ⊂ B^{±}, and this inclusion implies that | c_{1}^{+}| ∩ B^{−} = | c_{1}^{+}| ∩ B^{−} ∩ B^{+} and | c_{1}^{−}| ∩ B^{+} = | c_{1}^{−}| ∩ B^{+} ∩ B^{−}.

Using the notation of [C], for any compact K ⊂ B, let P (K, B) denote the smallest compact subset of B containing K and positively invariant relative to B, i.e., P (K, B) is the intersection of all the compact and positively invariant relative to B subsets of B containing K, and let P′(K, B) denote the smallest subset of B containing K and positively invariant relative to B, i.e., P′(K, B) is the union over u ∈ K of orbit segments u · [0, t] ⊂ B where t ≥ 0. Also let P ∗(K, B) and P^{′ ∗}(K, B) denote the analogues of P (K, B) and P′(K, B), respectively, when defined relative to the reverse time flow. Then P (K, B) = P′(K, B) whenever A^{−}(B) ⊂ int_{B}(K) [C,III.4.1.C] where in the notation of [C], A^{−}(B) = W^{u}(S; B); hence also P ∗(K, B) = P^{′ ∗}(K, B) whenever A^{+}(B) := W^{s}(S, B) ⊂ int_{B}(K). As B is a block it is immediate that A^{±}(B) and B^{∓} are disjoint closed sets. As a consequence of the lemmas in [C] used to demonstrate the existence of index pairs, in particular [C,III.4.1.D], we can find Ẑ a compact B-neighborhood of A^{−}(B) satisfying P (Ẑ, B) is disjoint from B^{−}. Set Z := Ẑ ∪ | c_{1}^{+}| and define

N_{1} := P (Z, B) , N_{0} := B^{+} ∩ N_{1} .

It follows easily that (N_{1}, N_{0}) is a regular index pair for S satisfying the inclusions (6.6a).

Also,

(6.9)  N_{1} ∩ B^{−} ⊂ B^{−} ∩ B^{+} .

For as Z is a compact B-neighborhood of A^{−}(B), P′(Z, B) = N_{1}. Hence, if ū ∈ N_{1} ∩ B^{−}, then ū = u · t where u ∈ Z, t ≥ 0, and u · [0, t] ⊂ B. By choice of Ẑ, u ∉ Ẑ, and the definition of B^{−} forces t = 0. Thus ū = u ∈ | c_{1}^{+}| ∩ B^{−} = C_{1}^{+}, and as C_{1}^{+} ⊂ B^{+}, the inclusion (6.9) follows.

An argument similar to that of the previous paragraph together with (6.9), the choice of Ẑ, and (6.1) yields that

(6.10)  | c_{1}^{−}| ∩ B^{−} ∩ N_{1} = ∅ .

From (6.9) and because A^{+}(B) is disjoint from B^{+}, we can find U an open neighborhood of A^{+}(B) satisfying U ∩ B is disjoint from B^{+} and also from N_{1} ∩ B^{−} . Again using [C,III.4.1.D], but with respect to the reverse time flow, we can find Ẑ∗ a compact B-neighborhood of A^{+}(B) so that P ∗(Ẑ∗, B) ⊂ U. Set Z∗ := Ẑ∗ ∪| c_{1}^{−}|, and then define

N∗_{1} := P ∗(Z∗, B) , N∗_{0} := B^{−} ∩ N∗_{1} .

It follows easily that (N∗_{1}, N∗_{0}) is a regular index pair for S with respect to the reverse time flow satisfying the inclusions (6.6b).

To complete the proof of Subcase (b) and of the theorem, it only remains to check that ((N_{1}, N_{0}), (N∗_{1}, N∗_{0})) satisfies the off-diagonal condition, whence this pair of index pairs will be AIP(regular) for S.

To the contrary, suppose ū ∈ N_{1} ∩ N∗_{0}. Then ū = u · −t where t ≥ 0, u ∈ Z∗, and u · [−t, 0] ⊂ B. Because N_{1} ∩ B^{−} ⊂ B^{+}, from the choice of Ẑ∗, it follows that u ∈ | c_{1}^{−}|. Also, because ū · [0, t] = u · [−t, 0] ⊂ B and because ū ∈ B^{+}, the definition of B^{+} forces t = 0. Thus ū = u ∈ | c_{1}^{−}| ∩ B^{−} ∩ N_{1} = ∅ by (6.10). Thus

N_{1} ∩ N∗_{0} = ∅.

Again to the contrary, suppose ū ∈ N_{0} ∩ N∗_{1}. Again ū = u · −t where t ≥ 0, u ∈ Z∗ and u · [−t, 0] ⊂ B. Again ū ∈ B^{+}, now because N_{0} ⊂ B^{+}, whence as before u ∈ | c_{1}^{−}| and t = 0. Hence ū = u ∈ | c_{1}^{−}| ∩ B^{+} = C_{1}^{−} ⊂ B^{−}. Thus since

N_{0} ⊂ N_{1}, we get ū ∈ | c_{1}^{−}| ∩ B^{−} ∩ N_{1} = ∅. Hence also N_{0} ∩ N∗_{1} = ∅. □

Let us note that to actually have α_{0} ⌢_{•}γ_{0} = α⌢_{•}γ in the situation of Theorem 6.1 it would be necessary to have |c^{+}| ∩ |c^{−}| = S, but apparently this by itself is not sufficient. The simple albeit quite restrictive hypotheses of the next proposition do ensure the above equality of intersection classes and that result suffices for the needs of the current work.

6.4 Proposition. Under the assumption of hypotheses CB, if |c^{+}| = W^{u}(S; B) and |c^{−}| = W^{s}(S; B), then in Ȟ_{∗}(S),

α_{0} ⌢_{•} γ_{0} = α ⌢_{•} γ .

Remark. Note it is not assumed here that S has hyperbolic structure, nor is it even assumed that the local stable and unstable sets W^{s}(S; B) and W^{u}(S; B) are compact manifolds with boundary.

Proof. We must show that for each V ∈ N (S),

(6.11)  π_{V}^{S} (α_{0} ⌢_{•} γ_{0}) = π_{V}^{S} (α ⌢_{•} γ) .

However, from the nature of an inverse limit we need only consider arbitrarily small V. In particular, we may assume that V is open and V ⊂ int_{M}(B). Because W^{u}(S; B) and W^{s}(S; B) intersect in S, via normality it is simple to find B-open neighborhoods V^{u} ⊃ W^{u}(S; B) and V^{s} ⊃ W^{s}(S; B) satisfying V^{u} ∩ V^{s} = V. From Lemma 3.3(2) of [K1] applied to both the forward and reverse time flows, it follows that there exists T > 0 so that if t ≥ T, then B_{0 t} ⊂ V^{u} and B_{t 0} ⊂ V^{s} where we recall from the proof of Proposition 4.4 that B_{0 t} and B_{t 0} are squeezes of B for time t, the former along the entrance set, the latter along the exit set. Let (L_{1}, L_{0}) be the forward time regular index pair for S consisting of B_{0 T} and its exit set and let (L∗_{1}, L∗_{0}) be the reverse-time index pair for S consisting of B_{T 0} and its entrance set. Then L ∈ ob(AIP(S)) and by choice of T, L_{1} ∩ L∗_{1} ⊂ V. Further, because W^{u}(S; B) is negatively invariant, c^{+} is a relative cycle in L_{1} modulo L_{0} and let α_{L} be its homology class. Analogously, the positive invariance of W^{s}(S; B) implies c^{−} is a relative cycle in L∗_{1} modulo L∗_{0} and let γ_{L} be its homology class. It is immediate from Lemma 2.10 that

(6.12)  ǐ_{L_{1}∩L∗_{1}}^{S}(α_{0} ⌢_{•} γ_{0}) = α_{L} ⌢_{•} γ_{L}

Further, as (L_{1}, L_{0}) ⊂ (B, B^{+}) and because (see [K5, Proposition 3.1]) an inclusion of nested index pairs induces a morphism in C (S) between the corresponding index spaces, it follows that α_{L} represents α because c^{+} as a cycle in B modulo B^{+} does; similarly γ_{L} represents γ. Finally, as L_{1} ∩L∗_{1} ⊂ V by construction, V ∈ N (L_{1} ∩L∗_{1}) and it follows immediately from (6.12) and Theorem 2.11 that (6.11) holds. □

The next theorem provides a reasonable means of evaluating 𝔏 on a tensor product of homology classes each of which is represented by a compact submanifold. Its proof uses Theorem 6.1 and its corollaries and Proposition 6.4 to translate [D, Proposition VIII.13.23] to the Conley index setting and some preliminary discussion is necessary before the result from [D] can even be stated. In particular, we need the following general position hypotheses on the submanifolds:

Hypotheses GP (Part 1).

(1) N^{+} and N^{−} are submanifolds with boundary of M orientable over R and with dim(N^{±}) = n^{±}.

(2) N := N^{+} ∩N^{−} is a connected, closed submanifold of M disjoint from ∂N^{±} with n := dim(N) = n^{+} + n^{−} − m.

Let o_{N}^{N^{±}} ∈ H_{n^{±}}(N^{±}, N^{±} \N) denote the fundamental class along N which exists because N^{±} is R-orientable and N is compact, connected, and disjoint from ∂N^{±}. It follows that o_{N}^{N^{+}} ⌢_{•} o_{N}^{N^{−}} ∈ Ȟ_{n}(N). Moreover, as N is an ENR, the natural homomorphism j_{∗}^{N}: H_{∗}(N) → Ȟ_{∗}(N) is an isomorphism (e.g., see [D, VIII.13.17]). Thus using this isomorphism to identify the Čech and singular homology modules of N, we get

o_{N}^{N^{+}} ⌢_{•} o_{N}^{N^{−}} = μo_{N}

where o_{N} is the fundamental class of N and where μ ∈ R is called the intersection multiplicity (relative to R) of N^{+} and N^{−}. Then [D, Proposition VIII.13.23] states, in part, that μ = ±1 if N^{+} and N^{−} have topologically transverse intersection at some u ∈ N. Note that as stated in [D] the just cited proposition assumes N^{±} to be boundaryless, but as the proof is local to a neighborhood of u, it goes through unchanged since N ∩ ∂N^{±} = ∅. To exploit the proposition in the context of the Conley index, requires the following additional hypotheses on N^{+}, N^{−}, and N:

Hypotheses GP (Part 2).

(3) N^{+} and N^{−} have topologically transverse intersection at u ∈ N and N ⊂ S.

(4) For some isolating block B for S, there are inclusions j^{±}: (N^{±}, ∂N^{±}) ⊂ (B, B^{±}) and N^{±} is compact.

(5) Either ∂N^{+} is a weak deformation retraction of N^{+} \ S or N^{+} ⊂ S, and either ∂N^{−} is a weak deformation retraction of N^{−} \ S or N^{−} ⊂ S.

The R-orientability and compactness of N^{±} imply that there is a fundamental class o_{N^{±}} ∈ H_{n^{±}}(N^{±}, ∂N^{±}) (note ∂N^{±} = ∅ if N^{±} ⊂ S); hence, by GP(4) o_{N^{±}} := b̄_{∗}^{±}j_{∗}^{±}o_{N^{±}} is a class in the appropriate Conley index; viz., o_{N^{+}} ∈ H̃_{n^{+}}C (S) and o_{N^{−}} ∈ H̃_{n^{−}}C∗(S).

6.5 Theorem. Under the assumption of hypotheses GP, the following hold.

(i) If N^{+} = W^{u}(S; B), N^{−} = W^{s}(S; B), and (N^{±}, ∂N^{±}) is a triangulable pair,

then

o_{N^{+}} ⌢_{•} o_{N^{−}} = ±o_{S} .

(ii) If N = {u} ⊂ int(B) (so n = 0) and if either (N^{±}, ∂N^{±}) is a triangulable pair or N^{±} ∩ B^{∓} = ∅, then

#(o_{N^{+}} ⌢_{•} o_{N^{−}}) = ±1 .

Thus, in the case M = R^{m}, o_{N^{+}} # o_{N^{−}} = ±o_{m}.

Proof. First, suppose N^{±} ⊄ S. Then, by hypothesis GP(5), the inclusion ∂N^{±} ⊂ N^{±}\S induces an isomorphism on homology. Hence, application of the Five Lemma to the sequences of the pairs in the inclusion i: (N^{±}, ∂N^{±}) ⊂ (N^{±}, N^{±} \ S) shows that i induces an isomorphism on homology. On the other hand, if N^{±} is a subset of S, then ∂N^{±} = ∅ as a consequence of hypothesis GP(4), whence it is trivial that i induces an isomorphism. In either case, the induced isomorphism i_{∗} maps the fundamental class o_{N^{±}} to o_{S∩N^{±}}^{N^{±}}, the fundamental class of N^{±} along S ∩ N^{±}. Also, the homomorphism induced by the inclusion (N^{±}, N^{±} \ S) ⊂ (N^{±}, N^{±} \ N) sends o_{S∩N^{±}}^{N^{±}} to o_{N}^{N^{±}}. Now as observed, the transversality hypothesis GP(3) implies o_{N}^{N^{+}} ⌢_{•} o_{N}^{N^{−}} = ±o_{N}. Hence from the naturality of intersection pairings with respect to inclusions expressed by Lemma 2.10,

(6.13)  o_{N^{+}} ⌢_{•} o_{N^{−}} = o_{N^{+}∩S}^{N^{+}} ⌢_{•} o_{N^{−}∩S}^{N^{−}} = o_{N}^{N^{+}} ⌢_{•} o_{N}^{N^{−}} = ±o_{N} .

First, assume that N^{+} = W^{u}(S; B), N^{−} = W^{s}(S; B), and (N^{±}, ∂N^{±}) is a polyhedral pair. Then there is a singular chain c^{±} with support N^{±} that is a relative cycle in N^{±} modulo its boundary representing o_{N^{±}}; viz, c^{±} is the sum of singular simplexes one for each geometric simplex of dimension n^{±} in the triangulation of N^{±} with each such singular simplex represented by a homeomorphism of the standard simplex Δ_{n^{±}} onto the geometric simplex. Thus |c^{+}| ∩ |c^{−}| = S ⊂ int(B). It then follows from Proposition 6.4 and (6.13) that

o_{N^{+}} ⌢_{•} o_{N^{−}} = ±o_{S} .

Next, assume N = {u}, whence o_{N} ∈ H_{0}(N) and is represented by u regarded as a 0-simplex. If also (N^{±}, ∂N^{±}) is a triangulable pair, then as above, o_{N^{±}} is represented by a singular chain c^{±} with support N^{±} as a relative cycle in N^{±} modulo ∂N^{±}. Then |c^{+}| ∩ |c^{−}| = {u} ⊂ int(B) so that condition (i) of Theorem 6.1 is satisfied. Because o_{N} is represented by the 0-simplex u, it follows from Corollary 6.2 and (6.13) that

±1 = ±⟨1, i_{M}^{N}o_{N}⟩ = ^{#}(o_{N^{+}} ⌢_{•} o_{N^{−}})

as desired.

If instead of the triagulability assumption we assume that N^{±}∩B^{∓} = ∅, then via the constructions used in the proof of Proposition 4.4, we can find P ∈ ob(AIP(S)) satisfying

((N^{+}, ∂N^{+}), (N^{−}, ∂N^{−})) ↪ P ↪((B, B^{+}), (B, B^{−})) .

In this case, Theorem 2.11, (6.13), and the fact that o_{N} is represented by a 0-simplex ensure that

±1 = ±⟨1, i_{M}^{N}o_{N}⟩ = ±⟨1, π_{M}^{P_{1}∩P ∗_{1}}ǐ_{P_{1}∩P ∗_{1}}^{N}o_{N}⟩ = ^{#}(o_{N^{+}} ⌢_{•} o_{N^{−}})

as desired.

In either case, if M = R^{m}, that o_{N^{+}} # o_{N^{−}} = ±o_{m} then follows from Proposition 2.13. □

In most cases of interest where Theorem 6.5(i) applies, M and the flow are smooth (i.e., at least C^{1}) and S is a smooth invariant submanifold of the flow with hyperbolic structure, whence N^{−} := W^{s}(S; B) and N^{+} := W^{u}(S; B) are smooth transverse submanifolds of M with boundary assuming B is a smooth isolating block with corners as defined in Definition 4.3. Note that with this choice of N^{±}, its boundary ∂N^{±} is in fact a strong deformation retract of N^{±}\S; for example, the map on N^{+}\S×[0, 1] given by (u, s) ↦ u·sτ_{B}(u) is a strong deformation retraction of N^{+} \S onto ∂N^{+}. The local invariant submanifolds W^{s}(S; B) and W^{u}(S; B) are geometric representatives for the classes o_{N^{−}} and o_{N^{+}}, respectively, in the sense that in this smooth setting (N^{±}, ∂N^{±}) is a simplicial pair and the fundamental class o_{N^{±}} (coefficients in R) is represented by a singular n^{±}-chain c^{±} = ∑ σ_{ν} where the summation is indexed by the simplexes in the triangulation of N^{±} of dimension n^{±} and σ_{ν} is a suitably oriented homeomorphism of the standard n^{±}-simplex onto ν.

Analogous remarks apply to the application of Theorem 6.5(ii) except that there typically at least one of N^{+} or N^{−} is a fiber in some fibering of W^{u}(S; B) or W^{s}(S; B) respectively with N^{+} and N^{−} of complementary dimension. Such a situation arises in computing the intersection pairing on the Conley indices of a hyperbolic periodic orbit; see Theorem 7.6 and Lemma 7.7 below.

Let us note that a priori the classes o_{N^{±}} in the statement of Theorem 6.5 might depend on the choice of block B satisfying hypothesis GP(4). However, in those situations where N^{+} and N^{−} are contained in the local invariant sets within the block that is not the case as shown by the following.

6.6 Proposition. Let W be a compact submanifold with boundary of M of dimension k, assume W admits a fundamental class o_{W} ∈ H_{k}(W, ∂W), and for i = 1, 2, where (P_{1 i}, P_{0 i}) is a forward time index pair for S, suppose that there is an inclusion j_{i}: (W, ∂W) ⊂ (P_{1 i}, P_{0 i}) where also W ⊂ W^{u}(S; P_{1 i}). Then

(6.14)  p̄_{1 ∗}^{+}j_{1 ∗}o_{W} = p̄_{2 ∗}^{+}j_{2 ∗}o_{W} ∈ H̃_{k}C (S) .

Proof. First, suppose that there is an inclusion

k: (P_{1 1}, P_{0 1}) ⊂ (P_{1 2}, P_{0 2})

and let k̄ be the induced map on the index spaces. Then clearly

k̄_{∗}p_{1 ∗}^{+}j_{1 ∗} = p_{2 ∗}^{+}j_{2 ∗} .

This, the fact that k̄ is the unique morphism in C (S) from P_{1 1}/P_{0 1} to P_{1 2}/P_{0 2} by [K5, Proposition 3.1], and the fact that H̃_{∗}C (S) is the direct limit of the homology modules of the index spaces in C (S), immediately imply that (6.14) holds.

The general case where there is no inclusion relation between the given index pairs can be reduced to the case where there is as follows from Proposition 1 in the Appendix of [K3] and the constructions given in its proof. In particular, it follows easily from the constructions in the proof just cited that there exist nested index pairs (L_{1}, L_{0}), (N_{1 1}, N_{0 1}), and (N_{1 2}, N_{0 2}) each containing the pair (W, ∂W) and for i = 1, 2, inclusions

(P_{1 i}, P_{0 i}) ↪ (N_{1 i}, N_{0 i}) ↩ (L_{1}, L_{0}) ,

thereby accomplishing the desired reduction. Note it is crucial that W be a subset of W^{u}(S; P_{1 i}), i = 1, 2, in order to obtain from the constructions in [K3] that each of the pairs (L_{1}, L_{0}), (N_{1 1}, N_{0 1}), and (N_{1 2}, N_{0 2}) contains the pair (W, ∂W). □

More generally, Proposition 6.6 allows us to make the following:

6.7 Definition. Let W be a compact, k-dimensional submanifold with boundary of M, assume W admits a fundamental class o_{W} ∈ H_{k}(W, ∂W), and suppose there is an inclusion k: (W, ∂W) ⊂ (P_{1}, P_{0}) for some nested index pair (P_{1}, P_{0}) for an isolated invariant set S of some flow in M with W ⊂ W^{u}(S; P_{1}), then p̄_{∗}^{+}k_{∗}o_{W} ∈ H̃_{k}C (S) is called the canonical image of the fundamental class o_{W}. By Proposition 6.6, the definition of the canonical image does not depend on the choice of nested index pair (P_{1}, P_{0}) satisfying the constraints of the definition. If α ∈ H̃_{k}C (S) and α is the canonical image of o_{W}, then let us say that W geometrically represents α.

B. The Behavior of 𝔏 under Orbit Preserving Maps. The following definition describes the basic conditions under which transformation of the intersection pairing shall be studied.

6.8 Definition. (A) For i = 1, 2, let M_{i} be a manifold and let S_{i} be an isolated invariant set of a flow in M_{i}. If, for i = 1, 2, there exists an open neighborhood U_{i} of S_{i} and a continuous, proper surjection f: U_{1} → U_{2} satisfying (i) S_{1} = f^{−1}(S_{2}) and (ii) each orbit of the flow in U_{1} induced by that in M_{1} is mapped by f onto an orbit of the flow in U_{2} induced by that in M_{2} in such manner that positive semi-orbits go to positive semi-orbits and negative to negative, then, and only then, call S_{1} the f-pre-image of S_{2}. It is not assumed that f preserves time-parameterization.

(B) If in addition to the conditions imposed on f in part (A), also f is required to be a homeomorphism of U_{1} onto U_{2}, then call f a local orbital equivalence of S_{1} and S_{2} and say that S_{1} and S_{2} are locally orbitally equivalent.

The following proposition makes possible the main results of this section. Its proof follows straightforwardly from the definitions and is left to the reader.

6.9 Proposition. Under the assumptions of Definition 6.8, the following statements hold:

(1) if S_{2} has isolating neighborhood N relative to the flow in U_{2}, then S_{1} has isolating neighborhood f^{−1}(N) relative to the flow in U_{1};

(2) if S_{2} has index pair ⟨N_{1}, N_{0}⟩ relative to the flow in U_{2}, then S_{1} has index pair ⟨f^{−1}(N_{1}), f^{−1}(N_{0})⟩ relative to the flow in U_{1}; (3) if P is AIP(thick) for S_{2}, then f^{−1}(P) is AIP(thick) for S_{1} where f^{−1}(P)

is defined componentwise by taking the inverse image under f of the components of P;

(4) if f is a local orbital equivalence of S_{1} and S_{2} and P ∈ ob(AIP(S_{2})) with respect to the flow in U_{2}, then f^{−1}(P) ∈ ob(AIP(S_{1})) with respect to the flow in U_{1}.

6.9.1 Remark. Thus, there exists a contravariant functor assigning to a manifold M′ and flow φ′ in M′ the set Iso(M′, φ′) of isolated invariant sets of φ′ and assigning to a continuous, proper, orbit-preserving surjection f: M′′ → M′ the set map f^{−1}: Iso(M′, φ′) → Iso(M, φ). Further, for each S′ ∈ Iso(M′, φ), there exists a map between Conley indices C (f^{−1}): C (S′) → C (f^{−1}(S′)) so that the Conley index may be viewed as a functor on the sets of isolated invariant sets of flows.

To state the main result of this section, make the following observations. With f, S_{1} and S_{2} as in Definition 6.8(A), application of part (3) of Theorem 5.2 to the map between Conley indices mentioned in Remark 6.9.1 and the singular homology functor yields a functorial homomorphism C (f^{−1})_{∗}: H̃_{∗}C (S_{2}) → H̃_{∗}C (S_{1}). When f is a local orbital equivalence, application of this construction to its inverse yields an isomorphism C (f)_{∗}: H̃_{∗}C (S_{1}) → H̃_{∗}C (S_{2}) whose inverse is C (f^{−1})_{∗}. Analogous remarks apply to the reverse time flow and yield an isomorphism C∗(f)_{∗}: H̃_{∗}C∗(S_{1}) → H̃_{∗}C∗(S_{2}).

6.10 Proposition. For i = 1, 2, let M_{i} be a manifold oriented over the ring R and let S_{i} be an isolated invariant set of a flow in M_{i}. If S_{1} and S_{2} are locally orbitally equivalent and if the homeomorphism h: U_{1} → U_{2} describing the orbital equivalence is orientation preserving, then where 𝔏^{i} is the intersection pairing associated with S_{i},

ȟ^{−1} ^{∘} 𝔏^{2} = 𝔏^{1} ^{∘} (C (h^{−1})_{∗} ⊗ C∗(h^{−1})_{∗}) ,

6.10.1 Remark. Note that the conclusion of the proposition could as well be written in the form

𝔏^{1} = ȟ^{−1} ^{∘} 𝔏^{2} ^{∘} (C (h)_{∗} ⊗ C∗(h)_{∗}) .

suggesting the abbreviation 𝔏^{1} = h^{∗}𝔏^{2}; however, the conclusion of the proposition as written is preferable in the sense that it generalizes to continuous, proper, local surjections; see Proposition 6.13.

Proof. Let P ∈ ob(AIP(S_{1})). Then for each V ∈ M_{Δ̸ o}^{(2)}(P) as a consequence of [D, Corollary VIII.13.7], for each α ∈ H_{∗}(h(V_{1}), h(V_{0})) and γ ∈ H_{∗}(h(V ∗_{1}), h(V ∗_{0})),

h_{∗}((h_{∗})^{−1}α ⌢_{•}^{V} (h_{∗})^{−1}γ) = α ⌢_{•}^{h(V)} γ .

Thus, from the definitions—see diagram (2.6)—it follows that

h_{∗}𝔏_{V_{1}∩V ∗_{1}}^{P} ^{∘} (h_{∗} ⊗ h_{∗})^{−1} = 𝔏_{h(V_{1}∩V ∗_{1})}^{h(P)} ,

whence passage to the limit over the cofinal family of neighborhoods of S_{1} having the form V_{1} ∩ V ∗_{1} for some V ∈ M_{Δ̸ o}^{(2)}(P) yields

(6.15)  ȟ^{∘} 𝔏_{P}^{1} ^{∘} (h_{∗} ⊗ h_{∗})^{−1} = 𝔏_{h(P)}^{2}

because the sets in P are compact so that 𝔏_{P}^{P} = 𝔏_{P}^{1} and 𝔏_{h(P)}^{h(P)} = 𝔏_{h(P)}^{2}. Set Q = h(P) and observe that

C (h^{−1})_{∗} = p̄_{∗}^{+}h_{∗}^{−1}(q̄_{∗}^{+})^{−1}  and  C∗(h^{−1})_{∗} = p̄_{∗}^{−}h_{∗}^{−1}(q̄_{∗}^{−})^{−1} .

From this and (6.15), it follows that

(6.16) ȟ^{∘} 𝔏_{P}^{1} ^{∘} (p̄_{∗}^{+} ⊗ p̄_{∗}^{−})^{−1} ^{∘} (C (h^{−1})_{∗} ⊗ C∗(h^{−1})_{∗}) = 𝔏_{Q}^{2} ^{∘} (q̄_{∗}^{+} ⊗ q̄_{∗}^{−})^{−1} .

As N (S_{1}, AIP) is cofinal in N (S_{1}) by Theorem 2.11, also the family of neighborhoods of S_{2} of the form h(P_{1}) ∩ h(P ∗_{1}) is cofinal in N (S_{2}). Thus, it follows from Theorem 2.11 that passage to the limit in (6.16) over P ∈ ob(AIP(S)) yields the desired result. □

To state and prove the generalization mentioned in Remark 6.10.1 requires a definition of intersection pairing on the tensor product of the Čech homology modules of the Conley indices of S to Ȟ_{∗}(S) and an extension of the notion of homology transfer map to obtain a homomorphism from the Čech homology of a compact pair to the Čech homology of its pre-image under a continuous, proper map between manifolds.

6.11 Definition. The intersection pairing on the Čech modules of the Conley indices can be obtained as follows. Using normality, one can show that if P ∈ ob(AIP(S)), then the family of open sets {V_{1} ∩ V ∗_{1} : V ∈ M_{Δ̸ o}^{(2)}(P)} is cofinal in N (P_{1} ∩ P ∗_{1}). Hence, from the naturality expressed by diagram (2.4), taking inverse limits one can show that there is a pairing

𝔏̌_{P}: Ȟ_{∗}(P_{1}, P_{0}) ⊗ Ȟ_{∗}(P ∗_{1}, P ∗_{0}) → Ȟ_{∗}(P_{1} ∩ P ∗_{1}) .

It is not difficult to show that the family of pairings 𝔏̌_{P}, P ∈ ob(AIP(S)), is natural relative to inclusion morphisms. A proof similar to that of Theorem 2.11 then shows that the analogue of equation (2.13) holds for this family of pairings, whence as in Theorem 2.11 passage to the inverse limit yields a pairing

𝔏̌: ˜ Ȟ_{∗}C (S) ⊗ ˜ Ȟ_{∗}C∗(S) → Ȟ_{∗}(S) ,

called the intersection class pairing on the Čech homology modules of the Conley indices of S.

6.11.1 Remark. It is a simple exercise that there are natural maps

i_{S}^{+}: H̃_{∗}C (S) → ˜ Ȟ_{∗}C (S) and i_{S}^{−}: H̃_{∗}C∗(S) → ˜ Ȟ_{∗}C∗(S)

and that because diagram (2.6) commutes 𝔏 and 𝔏̌ are related by

𝔏 = 𝔏̌^{∘} (i_{S}^{+} ⊗ i_{S}^{−}) .

6.12 Definition. The extension of the notion of homology transfer is obtained as follows. Let f: M_{1} → M_{2} be a continuous, proper map between manifolds and (P_{1}, P_{0}) a compact pair in M_{2}. Then for any open pair (V_{1}, V_{0}) ⊃ (P_{1}, P_{0}) there is a transfer map f_{!}: H_{∗}(V_{1}, V_{0}) → H_{∗}(f^{−1}(V_{1}), f^{−1}(V_{0})) and the family of these is natural relative to inclusions of the open pairs. Passage to the inverse limit over the open neighborhood pairs of (P_{1}, P_{0}) yields the desired extension, denote it f̌_{!}, once it is observed that the family of open sets f^{−1}(V_{i}), V_{i} an open neighborhood of P_{i}, is cofinal in N (f^{−1}(P_{i})) for i = 1, 2. This is an exercise left to the reader.

6.13 Proposition. For i = 1, 2, let M_{i} be a manifold of dimension m_{i} oriented over the PID R, let S_{i} be an isolated invariant set of a flow in M_{i}, and suppose that S_{1} is the f-pre-image of S_{2}. Then f induces graded module homomorphisms of degree m_{1} − m_{2},

C \binom{}{f̌_{!}} : ˜ Ȟ_{∗}C (S_{2}) → ˜ Ȟ_{∗}C (S_{1})  and  C∗\binom{}{f̌_{!}} : ˜ Ȟ_{∗}C∗(S_{2}) → ˜ Ȟ_{∗}C∗(S_{1})

so that for i = 1, 2 where 𝔏̌^{i} is the intersection pairing on the tensor product of the Čech homology modules of the forward and reverse time indices of S_{i} to the Čech homology of S_{i}

(6.17)  f̌_{!} ^{∘} 𝔏̌^{2} = 𝔏̌^{1} ^{∘} (C \binom{}{f̌_{!}} ⊗ C∗\binom{}{f̌_{!}}) .

Proof. Let P be AIP(thick) for S_{2}, and set Q_{i} := f^{−1}(P_{i}) and Q∗_{i} = f^{−1}(P ∗_{i}) for i = 0, 1. By Proposition 6.9, Q is AIP(thick) for S_{1}.

Now define C \binom{}{f̌_{!}} : ˜ Ȟ_{∗}C (S_{2}) → ˜ Ȟ_{∗}C (S_{1}) by

C \binom{}{f̌_{!}} = ¯ q̌_{∗}^{+}f̌_{!}(¯p̌_{∗}^{+})^{−1} .

Arguments similar to those in Proposition 6.6 show that this definition of C \binom{}{f̌_{!}} is independent of the particular choice of P so long as P is AIP(thick). The homomorphism C∗\binom{}{f̌_{!}} : ˜ Ȟ_{∗}C∗(S_{2}) → ˜ Ȟ_{∗}C∗(S_{1}) is defined in an analogous manner.

The remainder of the proof being similar to the corresponding part of the proof of Proposition 6.10 is left to the reader, but now one invokes [D, Proposition VIII.13.6] instead of its corollary. □

6.13.1 Remark. If M is at least C^{1} and the flow is generated by a continuous vectorfield, then S_{i} (i = 1, 2) admits index pairs that are C^{0} manifold with boundary pairs, whence the Čech and singular modules of the Conley indices will be isomorphic. It follows that in (6.17), for i = 1, 2, the pairing 𝔏̌^{i} can be replaced by the pairing 𝔏^{i} if C \binom{}{f̌_{!}} and C∗\binom{}{f̌_{!}} are replaced by C (f_{!}) := (i_{S_{1}}^{+})^{−1}C \binom{}{f̌_{!}} i_{S_{2}}^{+} and C∗(f_{!}) := (i_{S_{1}}^{−})^{−1}C∗\binom{}{f̌_{!}} i_{S_{2}}^{−} respectively. It is not difficult to show using Corollary VIII.13.7 of [D] that if f is an orientation preserving local orbital equivalence of S_{1} and S_{2} then C (f_{!}) = C (f^{−1})_{∗} and C∗(f_{!}) = C∗(f^{−1})_{∗}. Thus, Proposition 6.13 implies Proposition 6.10 when the minimal smoothness conditions are met.