CHAPTER 9 THE CAP PRODUCT REPRESENTATION OF 𝔏

AND THE NON-SINGULARITY OF ^{#}𝔏

The main result of this chapter establishes analogues of the formulae (2.5) for the pairing 𝔏 by representing it in a cofinal family of manifold isolating blocks for an isolated invariant set. An immediate corollary of this representation is the non-singularity of ^{#}𝔏. The last section, §9.D, briefly examines how the results of this work can be extended to the case where M is a manifold with boundary and S ∩ ∂M ≠ ∅.

A. The Cap Product Representation and Corollaries.

9.1 Definition. For each isolated invariant set S of the flow define 𝔅(S) by B ∈ 𝔅(S) if, and only if, B is an isolating block for S and a compact, codimension zero, C^{0} submanifold with boundary of M possessing the properties:

(1) ∂B = B^{+} ∪ B^{−} and B^{±}, if non-void, is a codimension zero submanifold with boundary of ∂B;

(2) there exist submanifolds Σ^{+} and Σ^{−} of M and d > 0 so that B^{±} ⊂ Σ^{±}, the flow embeds Σ^{±} × ]−3d, 3d[ as an open submanifold of M, and Σ^{+} ·

[−d, d] ∩ B = τ_{B}^{−1}([0, d]) and Σ^{−} · [−d, d] ∩ B = (τ∗_{B})^{−1}([0, d]);

(3) if c_{B} := B^{+} ∩ B^{−} ≠ ∅, then ∂B^{+} = c_{B} = ∂B^{−} and ∂B^{±} is bicollared in Σ^{±}.

9.2 Proposition. If M is C^{r} (r ≥ 1) and the flow in M is generated by a continuous vectorfield, then 𝔅(S) is cofinal in N (S), the neighborhoods of S directed by reverse inclusion.

Proof. If U ∈ N (S), then by [WY, Theorem 2.4] there exists B ⊂ U an isolating block with corners for S; see Definition 4.3. Then B ∈ 𝔅(S): property (2) holds because via the implicit function theorem it holds locally at each point of Σ^{±} and because B^{±} is compact; property (3) holds by using the unit disk subbundle of any tubular neighborhood of c_{B} for the bicollar. □

R. Churchill in [Ch] has constructed isolating blocks for isolated invariant sets of continuous flows in compact metric spaces and in fact his construction can be generalized to flows in locally compact Hausdorff spaces. The construction is quite analogous to that in [WY]. However, the author has not investigated whether or not Churchill’s construction in the setting of a continuous flow in a C^{0} manifold can be made to yield blocks that are C^{0} manifolds with boundary satisfying the criteria defining 𝔅(S); if so then 𝔅(S) would be cofinal in N (S) in the C^{0} case also.

For the rest of this chapter, coefficients of homology and cohomology modules are generally suppressed and unless noted otherwise are assumed taken in an Rmodule G where as usual M is assumed oriented over the PID R. Also, throughout, S is a fixed isolated invariant set of some flow in M.

When 𝔅(S) ≠ ∅, the Conley indices of S satisfy a duality property: with coefficients taken in any R-module

(9.1) H̃^{k}C∗(S) ≃ H̃_{m−k}C (S) and H̃^{k}C (S) ≃ H̃_{m−k}C∗(S)  for k ∈ N.

As mentioned in the Introduction, existence of the isomorphisms of (9.1) (and generalizations that take into account flows on manifolds with boundary and isolated invariant sets that may or may not intersect the boundary—see Theorem 9.20 below) is shown in [Mc2]. The existence of these isomorphisms and generalizations will be referred to as Poincaré duality of Conley indices. The implications of the general result in the context of intersection pairings on Conley indices will be discussed briefly in §9.D. So as to be able to present the main result of this chapter, Theorem 9.4 below, the statement of a slight variant of the core lemma of [Mc2] follows but only for the special case of a flow on a manifold without boundary; cf. [Mc2, Lemma 2.3]. In the course of proving Theorem 9.4, a proof of Lemma 9.3 falls out that is quite different than that given in [Mc2]; see Lemma 9.16 below.

9.3 Lemma. If B ∈ 𝔅(S) and o_{B} ∈ H_{m}(B, ∂B; R) is the fundamental class of B, then the R-module homomorphisms

H^{∗}(B, B^{+}) ⟶^{⌢o_{B}} H_{m−∗}(B, B^{−})  and  H^{∗}(B, B^{−}) ⟶^{⌢o_{B}} H_{m−∗}(B, B^{+})

are isomorphisms.

Note that the excisiveness conditions needed for the cap product homomorphisms in the above lemma to be well-defined are met because either B^{+} and B^{−} are disjoint and so are both open and closed in ∂B or because they intersect in c_{B} which is bicollared in ∂B.

Let α⊗γ ∈ H̃_{∗}C (S; G)⊗H̃_{∗}C∗(S; G′) for some R-modules G and G′, assume B ∈ 𝔅(S), let α_{B} ∈ H_{∗}(B, B^{+}; G) represent α, and let γ_{B} ∈ H_{∗}(B, B^{−}; G′) represent γ. By Lemma 9.3 there are unique cohomology classes x_{B} ∈ H^{∗}(B, B^{−}; G) and y_{B} ∈ H^{∗}(B, B^{+}; G′) satisfying α_{B} = x_{B} ⌢ o_{B} and γ_{B} = y_{B} ⌢ o_{B}. The main result of this chapter is the following:

9.4 Theorem. (α ⌢_{•} γ)_{B} = (x_{B} ⌣ y_{B}) ⌢ o_{B} = x_{B} ⌢ γ_{B}.

Note that if B, B′ ∈ 𝔅(S) with B ⊃ B′, then as a consequence of Theorem 2.11 defining 𝔏,

i_{B}^{B′}(α ⌢_{•} γ)_{B′} = (α ⌢_{•} γ)_{B} for B, B′ ∈ 𝔅(S) and B ⊃ B′.

Thus, when 𝔅(S) is cofinal in N (S), Theorem 9.4 yields by passage to the limit over 𝔅(S) the following:

9.5 Corollary. If 𝔅(S) is cofinal in N (S), then

(α ⌢_{•} γ) = \underleftarrow{lim}(x_{B} ⌢ γ_{B}) .

Perhaps the most important consequence of Theorem 9.4 is the following:

9.6 Corollary. If 𝔅(S) is non-empty and if H̃_{∗}C (S; R) and H̃_{∗}C∗(S; R) are torsion free, then the intersection number pairing ^{#}𝔏 defines a non-singular dual pairing of H̃_{k}C (S; R) and H̃_{m−k}C∗(S; R).

Proof. Let B ∈ 𝔅(S). From Theorem 9.4 it is clear that

#(α ⌢_{•} γ) = ⟨1, (x_{B} ⌣ y_{B}) ⌢ o_{B}⟩ = (−1)^{k(m−k)}⟨y_{B}, x_{B} ⌢ o_{B}⟩ .

Since x_{B} ↦ x_{B} ⌢ o_{B} is an isomorphism of H^{k}(B, B^{−}) onto H_{m−k}(B, B^{+}) and since the Kronecker pairing is a non-singular dual pairing of H^{k}(B, B^{−}) and H_{k}(B, B^{−}), it follows that

γ ↦ ^{#}𝔏(− ⊗ γ)

is an isomorphism of H̃_{k}C∗(S) onto hom[H̃_{m−k}C (S) , R]. □

A third corollary is Theorem 8.10 under the additional assumption that for the factor invariant sets S_{1} and S_{2}, the family 𝔅(S_{i}) is cofinal in N (S_{i}), i = 1, 2. One need only perform the computation of Lemma 8.8 with a block-exit set pair (B_{1 j}, B_{1 j}^{+}) replacing (V_{1 j}, V_{0 j}) and the corresponding block-entrance set pair (B_{1 j}, B_{1 j}^{−}) replacing (V ∗_{1 j}, V ∗_{0 j}). The details of the precise formulation and its proof are left to the reader.

B. Some Technical Propositions on Poincaré Duality Isomorphisms and Čech Cap Products. The proof of Theorem 9.4 is eventually reduced to several diagram chases. However, the construction of the diagrams and the proof that they are commutative requires considerable work. In particular, five technical propositions relating various Poincaré duality isomorphisms will be needed in the proofs of commutativity and these are now listed. Surely all are well-known, but as the author could find no adequate reference, their proofs save for the first which is a quite simple exercise are given in Appendix B. In all of these propositions, any open submanifold of a manifold oriented over R is given the induced orientation over R. Also, whenever X appears in the statement of one of the following five propositions it is assumed to be an m-dimensional manifold oriented over R, and for simplicity coefficients are suppressed, but unless stated otherwise are assumed taken in an R-module G.

9.7 Proposition. Suppose (K, L) is a closed pair in M with L cobounded in K, and suppose U is open in M and contains the pair (K, L). Then, for k ∈ N, the diagram

Ȟ^{k}(K, L) H_{m−k}(U \ L, U \ K) _{excision}_{≃}_{⌢ o^{U}}  H_{m−k}(M \ L, M \ K)_{⌢ o^{M}}

commutes.

The proof, left to the reader, is a straightforward exercise in the definition of the Čech cap product and the use of some obvious excision isomorphisms.

To state the next proposition, note that whenever V is open in a manifold X there is an inclusion induced homomorphism Ȟ_{c}^{∗}(V) → Ȟ_{c}^{∗}(X) obtained by passage to the direct limit over compact K ⊂ V of the composite homomorphisms

Ȟ^{∗}(V, V \ K) ⟶^{(e_{K}^{∗})^{−1}} Ȟ^{∗}(X, X \ K) → Ȟ_{c}^{∗}(X)

where e_{K}^{∗} is the excision isomorphism and the unlabeled arrow is the canonical map into the direct limit; on page 289 of [D] see statement (6.23) and its proof.

9.8 Proposition. Suppose V is open in X. Then, where the unlabeled arrows are inclusion induced, for i ∈ N, the diagram

Ȟ_{c}^{i}(V) ⟶ Ȟ_{c}^{i}(X)
⌢ o^{V} ↓  ↓_{⌢ o^{X}}
H_{m−i}(V) ⟶ H_{m−i}(X)

is commutative.

The next proposition is an analogue of Proposition 9.7 where Čech cohomology is replaced by Čech cohomology with compact supports.

9.9 Proposition. Let V be open in X, let (K_{1}, L_{1}) be a closed (relative to V) pair in V, and let (K_{2}, L_{2}) be a closed pair in X. Assume that (K_{1}, L_{1}) ⊂ (K_{2}, L_{2}), that this inclusion is an excision, and that V \ L_{1} ⊂ X \ L_{2}. Then V \ K_{1} ⊂ X \ K_{2} and for each i ∈ N,

Ȟ_{c}^{i}(K_{1} \ L_{1})  Ȟ_{c}^{i}(K_{2} \ L_{2})
⌢ o^{V \L_{1}} ↓  ↓_{⌢ o^{X\L_{2}}}
H_{m−i}(V \ L_{1}, V \ K_{1}) ⟶ H_{m−i}(X \ L_{2}, X \ K_{2})

is a commutative diagram of isomorphisms where the unlabeled arrow is an excision isomorphism.

The last two of the five propositions show that standard identities of singular theory relating cap and cross products and cup and cap products carry over to a setting where the cap product is the Čech cap product of a Čech cohomology class and a singular homology class as defined on pp. 292–3 of [D]. The cup product and cross product of Čech cohomology classes are defined in the obvious manner by taking direct limits of singular cup products and singular cross-products of suitable open pairs and therefore require no excisiveness conditions; see [D, §VIII.6.21, p. 288] for some detail. For (K, L) and (K, L′) closed pairs in some manifold, the cup product of a Čech class x̌ ∈ Ȟ^{∗}(K, L) with a Čech class y̌ ∈ Ȟ_{c}^{∗}(K, L′) having compact support can then be defined by passage to the direct limit over elements x̌ ⌣ y̌_{ω} ∈ Ȟ(K, L ∪ ω) where ω lies in the family of subsets of K that contain L′ and are locally compact and cobounded in K and y̌_{ω} ↦ y̌ in the canonical map to the direct limit. So defined x̌ ⌣ y̌ ∈ Ȟ_{c}^{∗}(K, L ∪ L′) because the set of subsets of the form L ∪ ω is cofinal in the family of subsets of K that contain L ∪ L′ and are locally compact and cobounded in K.

9.10 Proposition. Suppose K_{ℓ} is closed in the manifold M_{ℓ} of dimension m_{ℓ} and that M_{ℓ} is R-oriented over K_{ℓ} with fundamental class o_{ℓ} along K_{ℓ}, ℓ = 1, 2. Then, for i_{1}, i_{2} ∈ N, the diagram

H_{i_{1}}(M_{1}, M_{1} \ K_{1}) ⊗ H_{i_{2}}(M_{2}, M_{2} \ K_{2})_{×}  Ȟ_{c}^{m_{1}−i_{1}}(K_{1}) ⊗ Ȟ_{c}^{m_{2}−i_{2}}(K_{2})^{⌢ o_{1} ⊗ ⌢ o_{2}}  _{×} H_{i_{1}+i_{2}}(M_{1} × M_{2}, M_{1} × M_{2} \ K_{1} × K_{2})  Ȟ_{c}^{m_{1}+m_{2}−i_{1}−i_{2}}(K_{1} × K_{2})_{(−1)^{m1(m2−i2)} ⌢ o_{1} × o_{2}}

commutes.

9.11 Proposition. Suppose (K, L) is a closed pair in X. Then the diagram

Ȟ(K, L) ⊗ Ȟ_{c}(K)_{1^{∗} ⊗ ⌢ o^{X}}  Ȟ_{c}(K, L)_{⌢ o^{X}}^{⌣} Ȟ(K, L) ⊗ H(X, X \ K) _{⌢} H(X \ L, X \ K)

commutes.

C. Results Leading to the Proof of Theorem 9.4. The first goal of this section is to define an R-module homomorphism

D_{k}: H̃_{k}C (S) → H̃^{m−k}C∗(S)

that will later be seen to be an isomorphism when 𝔅(S) ≠ ∅ thereby giving a new proof of Poincaré duality of Conley indices for the case S ∩ ∂M = ∅. The following definitions are needed first.

For k ∈ N, for P ∈ ob(AIP(S)) define D_{P k} to be the composition of the homomorphisms in the sequence

H_{k}(P_{1}, P_{0}) ⟶^{i_{∗}} H_{k}(M \ P ∗_{0}, M \ P ∗_{1}) ⟶^{(⌢ o^{M})^{−1}} Ȟ^{m−k}(P ∗_{1}, P ∗_{0})

where i_{∗} is inclusion induced and ⌢ o^{M} is the relative Poincaré-Alexander duality isomorphism. Also, define D̄_{P k} to be the composition of the homomorphisms in the sequence

H̃_{k}C (S) ⟶^{(p̄_{∗}^{+})^{−1}} H_{k}(P_{1}, P_{0}) ⟶^{D_{P k}} Ȟ^{m−k}(P ∗_{1}, P ∗_{0}) ⟶^{(¯p̌^{− ∗})^{−1}} ˜ Ȟ^{m−k}C∗(S) .

9.12 Proposition. D̄_{P k} = D̄_{Q k} for all P, Q ∈ ob(AIP(S)).

Proof. Define an equivalence relation D in ob(AIP(S)) by P ∼^{D} Q if, and only if, D̄_{P k} = D̄_{Q k}. As already noted in Chapter 2, relative Poincaré-Alexander duality isomorphisms are natural with respect to inclusions. It follows immediately that if P, Q ∈ ob(AIP(S)) with P ↪ Q, then P ∼^{D} Q; call this invariance of D under inclusions. The proof then proceeds in analogy with that of Theorem 4.7. That is,

(1) as each P in ob(AIP(S)) includes into an AIP(regular) pair P′, by invariance of D under inclusion, each pair of admissible index pairs for S is equivalent to an AIP(regular) pair of index pairs for S; (2) as in the proof of Claim II in the proof of Theorem 4.7, Lemma 4.6 and invariance of D under inclusions show that any two AIP(regular) pairs of index pairs for S are equivalent. □

9.13 Definition. Define D_{k}: H̃_{k}C (S) → ˜ Ȟ^{m−k}C∗(S) to be the unique Rmodule homomorphism which for each P ∈ ob(AIP(S)) equals D̄_{P k}. Define the R-module homomorphism D∗_{k}: H̃_{k}C∗(S) → ˜ Ȟ^{m−k}C∗(S) analogously by regarding S as an isolated invariant set of the time-reversed flow.

The proof that D_{k} is an isomorphism will utilize parts of the following lemma. The remaining parts will be used in the construction and proofs of commutativity of several diagrams used to prove Theorem 9.4.

9.14 Lemma. Given B ∈ 𝔅(S), there exist P ∈ ob(AIP(S)), V ∈ M_{Δ̸ o}^{(2)}(P), and W, Q, Q^{+}, Q^{−} relatively compact, open subsets of M so that W, Q^{±} ⊂ Q and additionally:

(1) P ↪ ((B, B^{+}), (B, B^{−})) ↪ ((Q, Q^{+}), (Q, Q^{−}));

(2) V_{i} = W \ cl_{W} (V ∗_{1−i}) and V ∗_{i} = W \ cl_{W} (V_{1−i}) (i = 0, 1);

(3) cl_{M} (V_{1} \ V_{0}) ∪ cl_{M} (V ∗_{1} \ V ∗_{0}) ⊂ B^{∘} := int(B);

(4) P ∗_{1} ⊂ Q \ Q^{+} ⊂ P ∗_{1} ∪ V ∗_{0} and P_{1} ⊂ Q \ Q^{−} ⊂ P_{1} ∪ V_{0};

(5) V_{1} and V ∗_{1} are weak deformation retracts of W;

(6) P_{i} is a strong deformation retract of V_{i} and of W \ V ∗_{1−i} (i = 0, 1);

(7) P ∗_{i} is a strong deformation retract of V ∗_{i} and of W \ V_{1−i} (i = 0, 1);

(8) W \ V_{i} is a weak deformation retract of Q \ V_{i} (i = 0, 1);

(9) W \ V ∗_{i} is a weak deformation retract of Q \ V ∗_{i} (i = 0, 1);

(10) B is a strong deformation retract of Q and B^{±} is a strong deformation retract of Q^{±};

(11) B^{−} (respectively B^{+}) is a strong deformation retract of Q\V_{1} (respectively

Q \ V ∗_{1});

(12) Each of the pairs (P_{1}, P_{0}), (P ∗_{1}, P ∗_{0}), and (B, B^{±}) has isomorphic Čech and singular cohomology modules (any coefficients).

Proof. Let us show that we can reduce to the case where ∅ ≠ c_{B} := B^{+} ∩ B^{−}. For if c_{B} is void, there are four possibilities: (i) B^{+} = B^{−} = ∅, (ii) B^{−} = ∅, but B^{+} ≠ ∅, (iii) B^{+} = ∅, but B^{−} ≠ ∅, (iv) B^{+} and B^{−} are both non-void, but are disjoint.

If (i) holds then ∂B = ∅. Thus, B is open in M as well as being closed and compact; also, necessarily S = B. Hence, statements (1)–(12) of the lemma will obviously be satisfied (many vacuously) if V := P := ((B, B^{+}), (B, B^{−})), W := Q := B, and Q^{±} := ∅.

Cases (ii) and (iii) are handled analogously; we do only case (ii). If (ii) holds, note that ∂B = B^{+} = Σ^{+} so that by definition of 𝔅(S), the flow determines an open bicollar V_{0} of ∂B; say V_{0} = B^{+}·]−d, d[. Then the statements of the lemma are obviously satisfied if P := ((B, B^{+}), (τ_{B}^{−1}[2d, ∞], B^{−})), Q := W := V_{1} := B ∪ V_{0},

V ∗_{1} := W \ cl_{W} (V_{0}), Q^{−} := V ∗_{0} := ∅, and Q^{+} := V_{0}.

If (iv) holds, then B^{±} coincides with Σ^{±} and by hypothesis the flow determines a closed bicollar neighborhood N^{±} of B^{±}; say N^{±} = B^{±} · [−2d, 2d] for some d > 0. Set P_{1} := B \ B^{−} · [0, 2d[ and P_{0} := B^{+}. Similarly, set P ∗_{1} := B \ B^{+} ·]−2d, 0] and P ∗_{0} := B^{−}. Also, set V_{0} := B^{+} ·]−d, d[ and define V ∗_{0} analogously using B^{−}. Set Q := W := B ∪ V_{0} ∪ V ∗_{0} and define V_{1} and V ∗_{1} via the equalities of statement (2) of the lemma. Set Q^{+} := Q \ (P ∗_{1} ∪ B^{−} ·]−d, 0]) and define Q^{−} analogously using P_{1} and B^{+}. It is easily verified that these choices of P, V, Q, Q^{±}, and W fulfill the requirements of the lemma.

Thus, without loss of generality we assume henceforth that c_{B} ≠ ∅ and note that c_{B} = ∂B^{±}. First some preliminary constructions are needed after which P, W, V, Q and Q^{±} will be constructed in that order.

To begin, where Σ^{±} ⊃ B^{±} is the local section of the flow that exists by the assumption that B ∈ 𝔅(S), let h: c_{B} × [−1, 1] → Σ^{−} be a homeomorphism onto a closed bicollar neighborhood E^{−} of c_{B} in Σ^{−} so that where U^{−} := h(c_{B} × [0, 1[) and Ū^{−} := h(c_{B} × [0, 1]), U^{−} is an open and Ū^{−} a closed collar of ∂B^{−} in B^{−} with both disjoint from W^{s}(S; B). Next, with regard to the block B, in condition (2) of Definition 9.1 replace d by 2d sufficiently small so that the resulting statement holds true; call it condition (2′). Delete from B those maximal orbit segments in B with one endpoint in U^{−} and denote the set remaining by B̂, an isolating block for S as Ū^{−}∩W^{s}(S; B) = ∅. Now, without loss of generality, assume that (i) the flow embeds B̂^{+} ×[−3d, 3d] and B̂^{−} ×[−3d, 3d] onto disjoint closed sets whose intersections with B̂ are respectively τ −1 ; B̂([0, 3d]) and τ∗ −1 ; B̂([0, 3d]) and (ii) E^{−} · −d ⊂ M \ Σ^{+} · [0, d] and on E^{−} · −d the map τ_{Σ^{+}} is finite, positive, and continuous where

τ_{Σ^{+}}(u) := sup{t ≥ 0 : u · [0, t] ⊂ M \ Σ^{+}} for u ∈ E^{−} · −d.

For as B̂^{+} and B̂^{−} are disjoint, clearly Definition 9.1 implies that shrinking d as needed will yield (i). Also, Σ^{+} ·]0, 2d]∩B = ∅; else for some u ∈ Σ^{+} and s ∈ ]0, 2d], 0 < s̄ := s + τ_{B}(u · s) ≤ 4d and Σ^{+} intersects Σ^{+} · s̄ contradicting condition (2′). Then B^{+} · [0, d] and B^{−} · −d are disjoint closed sets; else

∅ ≠ B^{+} ·]0, 2d] ∩ B^{−} ⊂ Σ^{+} ·]0, 2d] ∩ B = ∅ .

Via normality, continuity of the flow, and compactness of B^{±}, it follows that (B^{−} ∪ E^{−}) · −d and cl (Σ^{+}) · [0, d] will be disjoint closed sets if the bicollar E^{−} is shrunk sufficiently and if Σ^{+} is replaced with a relatively compact, open relative to Σ^{+} neighborhood of B^{+}. The inclusion in (ii) then follows immediately. As (c_{B} ·−d)·d lies in the open Σ^{+} · ]−ε, ε[ for 0 < ε < 6d, it follows that shrinking the bicollar E^{−}, as needed, will ensure that τ_{Σ^{+}} is finite, positive, and continuous on E^{−} · −d.

Define P ∈ ob(AIP(S)) by setting

P_{1} := τ∗ −1 ; B̂[2d, ∞] ,  P_{0} := B̂^{+} ,
P ∗_{1} := τ −1 ; B̂[2d, ∞] ,  P ∗_{0} := B̂^{−} .

Note that P_{1} = B̂ \ B̂^{−} · [0, 2d[ and P ∗_{1} = B̂ \ B̂^{+} · ]−2d, 0], whence P_{1} and P ∗_{1} are squeezes of B̂, the former along the entrance set, the latter along the exit set. Thus, (P_{1}, P_{0}) is a (block, exit set)-pair and (P ∗_{1}, P ∗_{0}) a (block, entrance set)- pair. Therefore, both are regular index pairs and by construction P satisfies the off-diagonal condition; hence, P ∈ ob(AIP(S)).

Define W and V as follows. First, define Y^{−} ⊂ U^{−} by Y^{−} := h(∂B^{−} ×] \frac{1}{2}, 1[) and let Y be the union of the maximal orbit segments in B with one endpoint in Y^{−}. Next, set Y^{+} := Y ∩B^{+} and set Z^{±} := Y^{±}∪B̂^{±}. Then Z^{±} is an open neighborhood of B̂^{±} relative to Σ^{±}. It follows that Z^{±} ·]−d, d[ is an open neighborhood of B̂^{±} relative to M. Define

W := B̂ ∪ Y ∪ Z^{−} ·]−d, 0[ ∪ Z^{+} ·]0, d[ .

It follows easily that W is an open neighborhood of B̂ relative to M (and therefore of P_{1} and P ∗_{1}) and that W has compact closure. Next, define V ∈ M_{Δ̸ o}^{(2)}(P) by

V_{1} := W \ Z^{−} ·]−d, d] ,  V_{0} := Z^{+} ·]−d, d[ ,
V ∗_{1} := W \ Z^{+} ·]−d, d] ,  V ∗_{0} := Z^{−} ·]−d, d[ .

Define

Q := B ∪B^{+} ·[0, d[∪B^{−} ·]−d, 0]∪⋃{u·]−d, τ_{Σ^{+}}(u·−d)+d[ : u ∈ h(c_{B} ×]−1, 0]} .

To define Q^{+} and Q^{−}, first define

P̃_{1} := P_{1} ∪ P_{0} · [0, d[ ,  and  P̃ ∗_{1} := P ∗_{1} ∪ P ∗_{0} ·]−d, 0]

and then set

Q^{+} := Q \ P̃ ∗_{1}  and  Q^{−} := Q \ P̃_{1} .

It is clear from their construction that P ∈ ob(AIP(S)), that V ∈ M_{Δ̸ o}^{(2)}(P), and that W, Q, Q^{+}, and Q^{−} are relatively compact, open subsets of M such that W, Q^{±} ⊂ Q and statements (1)–(4) are satisfied. It remains to describe the requisite weak and strong deformation retractions of statements (5)–(11) and show the isomorphisms of statement (12).

A weak deformation retraction of W into V_{1} is given by

(u, s) ↦ \{ \begin{matrix}u · s min\{2d, τ_{B}(u)\} & if u ∈ (W ∩ B) ∪ Z^{−} ·]−d, 0], s ∈ [0, 1] \\ u & if u ∈ Z^{+} · [0, d[, s ∈ [0, 1] \end{matrix}

where τ_{B}(u · r) := τ_{B}(u) − r for u ∈ Z^{−} and r ∈ ]−d, 0]. A weak deformation retraction of W into V ∗_{1} is defined similarly.

Strong deformation retractions of V_{0} onto P_{0} and of W \ V_{1} onto P ∗_{0} will be defined presently; strong deformation retractions of V ∗_{0} onto P ∗_{0} and of W \ V ∗_{1} onto P_{0} can be defined similarly and are left for the reader. Because of the way Y^{±} is defined in terms of the collar neighborhood of ∂B^{−}, where Ȳ^{−} := h(∂B^{−}×] \frac{1}{2}, 1]), there is a strong deformation retraction D̂^{±} of Z^{±} onto B̂^{±} with the deformations D̂^{−} and D̂^{+} satisfying the relation

D̂^{−}(u, s) · τ_{B}(D̂^{−}(u, s)) = D̂^{+}(u · τ_{B}(u), s) for u ∈ Ȳ^{−}, s ∈ [0, 1].

Note too that W \V_{1} = Z^{−}·]−d, d]. Then define a strong deformation retraction D^{+} of V_{0} onto P_{0} and a strong deformation retraction D^{−} of W \ V_{1} onto P ∗_{0} by the formula

D^{±}(u · t, s) := D̂^{±}(u, s) · (1 − s)t

where for D^{+} the formula is defined for each u ∈ Z^{+}, t ∈ ]−d, d[, and s ∈ [0, 1], but for D^{−} the formula is for each u ∈ Z^{−}, t ∈ ]−d, d], and s ∈ [0, 1].

The definition of strong deformation retractions of V_{1} onto P_{1}, of V ∗_{1} onto P ∗_{1}, of W \ V_{0} onto P ∗_{1}, and of W \ V ∗_{0} onto P_{1} are more involved. Definitions of the deformation retractions of V_{1} onto P_{1} and of W \ V_{0} onto P ∗_{1} will be made. The others are defined analogously. The deformations will be given piecewise. Accordingly, define

V_{+} := Z^{+} · [0, d[ , V_{−} := Z^{−} ·]d, 2d] ,

and

Ȳ_{2d} := ⋃{u · [2d, τ_{B}(u)] : u ∈ Ȳ^{−}} .

Note that V_{+}, V_{−}, Ȳ_{2d}, and P_{1} are all subsets of V_{1}, each is closed relative to V_{1}, and the union of these four sets equals V_{1}. Also for each u ∈ Ȳ_{2d}, there are unique y ≡ y(u) ∈ Ȳ^{−} and r ≡ r(u) ∈ [0, 1], both varying continuously with u, so that u = y · (2d + rτ_{B}(y · 2d)), and for each u ∈ V_{−}, there are unique z ≡ z(u) ∈ Z^{−} and t ≡ t(u) ∈ ]d, 2d], both varying continuously with u, so that u = z · t. Then define

D(u, s) := \{ \begin{matrix}u & for u ∈ P_{1}, s ∈ [0, 1]; \\ D^{+}(u, s) & for u ∈ V_{+}, s ∈ [0, 1]; \\ D̂^{−}(y, s) · [2d + rτ_{B}(D̂^{−}(y, s) · 2d)] & for u ∈ Ȳ_{2d}, s ∈ [0, 1]; \\ D̂^{−}(z, s) · [(1 − s)t + 2sd] & for u ∈ V_{−}, s ∈ [0, 1]. \end{matrix}

Clearly D is continuous on the closed subsets of V_{1} × [0, 1] obtained by taking the Cartesian product of each of V_{±}, Ȳ_{2d}, and P_{1} with [0, 1], and it is easily verified that the piecewise definitions agree on any overlap. Thus D is continuous and by its definition is a strong deformation retraction of V_{1} onto P_{1}.

In an analogous manner one can define a strong deformation retraction D∗ of W \ V_{0} onto P ∗_{1} by a piecewise definition on the Cartesian product of sets V ∗_{±} and Ȳ ∗_{2d} with [0, 1] where V ∗_{±} and Ȳ ∗_{2d} are defined as follows:

V ∗_{−} := Z^{−} ·]−d, 0] , V ∗_{+} := Z^{+} · [−2d, −d] ,

and

Ȳ ∗_{2d} := ⋃ {u · [0, τ_{B}(u) − 2d] : u ∈ Ȳ^{−}} .

Then V ∗_{−}, V ∗_{+}, Ȳ ∗_{2d}, and P ∗_{1} are all closed relative to W \ V_{0} and their union equals W \ V_{0}. Also for each u ∈ Ȳ ∗_{2d}, there exist unique y ≡ y(u) ∈ Ȳ^{−} and r ≡ r(u) ∈ [0, 1] so that u = y · r(τ_{B}(y) − 2d), and for each u ∈ V ∗_{+}, there exist unique ū ∈ Z^{+} and t ∈ [−2d, −d] so that u = ū · t. Define D∗ by

D∗(u, s) := \{ \begin{matrix}u & for u ∈ P ∗_{1}, s ∈ [0, 1]; \\ D^{−}(u, s) & for u ∈ V ∗_{−}, s ∈ [0, 1]; \\ D̂^{−}(y, s) · r[τ_{B}(D̂^{−}(y, s)) − 2d] & for u ∈ Ȳ ∗_{2d}, s ∈ [0, 1]; \\ D̂^{+}(ū, s) · [(1 − s)t − 2sd] & for u ∈ V ∗_{+}, s ∈ [0, 1]. \end{matrix}

It is easily verified that D∗ is a strong deformation retraction of W \ V_{0} onto P ∗_{1}.

Presently, a weak deformation retraction of Q \ V_{0} into W \ V_{0} will be defined; a weak deformation retraction of Q \ V ∗_{0} into W \ V ∗_{0} can be defined analogously. The deformation will be defined in three stages. First, deform each maximal orbit segment in Q \ V_{0} containing a point of the form h(u, s) where u ∈ c_{B} and −1 < s < \frac{1}{2} onto the maximal orbit segment in Q containing the point h(u, \frac{1}{2}) by using the bicollar to define the deformation and scaling time along the orbit appropriately. All other points are left fixed during the first stage. Each maximal orbit segment in Q through a point of the form h(u, \frac{1}{2}) contains the subsegment h(u, \frac{1}{2})·[τ_{B}(h(u, \frac{1}{2}))−d, τ_{B}(h(u, \frac{1}{2}))+d[ lying in cl_{Q} (V_{0}), and the second stage of the deformation contracts each such orbit segment to the point h(u, \frac{1}{2})·τ_{B}(h(u, \frac{1}{2}))−d. All other points are left fixed during the second stage. Successive application of the two stages defined so far yields a strong deformation retraction of Q \ V_{0} onto cl_{Q} (W \ V_{0}). Above, a strong deformation retraction of W \ V_{0} onto P ∗_{1} has been given and will easily be extended by the reader to a strong deformation retraction of cl_{Q} (W \ V_{0}) onto P ∗_{1} that can then be regarded as a weak deformation retraction of cl_{Q} (W \ V_{0}) into V ∗_{1} and taken as the third stage. Successive application of the three stages yields a weak deformation retraction of Q \ V_{0} onto W \ V_{0}.

A strong deformation retraction of Q onto B can be described as follows. Over the time interval 0 ≤ s ≤ \frac{1}{2} use the collar of c_{B} to deform maximal orbit segments in Q that do not intersect B into an orbit segment that intersects c_{B} and note that orbit segments of the latter type lie in c_{B} ·]−d, d[ ⊂ B^{−} ·]−d, 0] ∪ B^{+} · [0, d[. Then over the time interval \frac{1}{2} ≤ s ≤ 1 use the flow to deform points of B^{−} ·]−d, 0] into B^{−} and points of B^{+} · [0, d[ into B^{+}. Throughout the deformation leave points of B fixed. This yields a strong deformation retraction of Q onto B.

A strong deformation retraction of Q^{+} onto B^{+} will be described; a strong deformation retraction of Q^{−} onto B^{−} can be described similarly. Observe that Q^{+} is the union of bounded, maximal orbit segments of two types: (1) those that do not intersect B and (2) those that intersect B^{+} in a unique point. As with Q, over the time interval 0 ≤ s ≤ \frac{1}{2} use the bicollar of c_{B} to define a deformation of maximal orbit segments of Q^{+} that do not intersect B into a maximal orbit segment contained in c_{B} · ]−d, d[ while leaving all other maximal orbit segments pointwise fixed. Then, over the time interval \frac{1}{2} ≤ s ≤ 1, use the flow to contract each maximal orbit segment intersecting B^{+} into its unique point of intersection. This yields a strong deformation retraction of Q^{+} onto B^{+}.

A weak deformation retraction of Q \ V_{1} into W \ V_{1} is described presently; an analogous definition yields a weak deformation retraction of Q \ V ∗_{1} into W \ V ∗_{1}. First, observe that Q\V_{1} ⊂ Q\P̃_{1} =: Q^{−} and that the strong deformation retraction of Q^{−} onto B^{−} restricts to a strong deformation retraction of Q \ V_{1} onto B^{−} that deforms W \ V_{1} within itself. Second, it is easy to use the collar to define a strong deformation retraction of B^{−} onto P ∗_{0} that deforms W ∩ B^{−} within itself. The two taken successively yield a strong deformation retraction of Q \ V_{1} onto P ∗_{0} that deforms W \ V_{1} within itself. Hence, as P ∗_{0} ⊂ W \ V_{1}, the deformations taken successively yield a weak deformation retraction of Q \ V_{1} into W \ V_{1}.

As (B, B^{±}) is a pair of compact manifolds with boundary, it is a pair of ENR’s whence its Čech and singular cohomology modules are isomorphic; see [D, Proposition VIII.6.12]. For the pair (P_{1}, P_{0}), one might be able to verify that it too is a pair of compact manifolds with boundary; however, even if that is not the case, note that the construction of (V_{1}, V_{0}) and the definition of the strong deformation retraction of V_{i} onto P_{i} (i = 0, 1) is readily modified to produce a cofinal family {U_{j}}_{j∈N} of open neighborhood pairs of (P_{1}, P_{0}) such that P_{i} is a strong deformation retract of U_{i j}, i = 0, 1, j ∈ N. Thus, H^{∗}(U_{i j}) ≃ H^{∗}(P_{i}), i = 0, 1, and the Five Lemma implies H^{∗}(U_{1 j}, U_{0 j}) ≃ H^{∗}(P_{1}, P_{0}), j ∈ N. Then

Ȟ^{∗}(P_{1}, P_{0}) = \underrightarrow{lim} H^{∗}(U_{1 j}, U_{0 j}) ≃ H^{∗}(P_{1}, P_{0}) .

A similar argument shows Ȟ^{∗}(P ∗_{1}, P ∗_{0}) ≃ H^{∗}(P ∗_{1}, P ∗_{0}). □

9.15 Theorem (cf. [Mc2]). If 𝔅(S) ≠ ∅, then the Čech and singular cohomology modules (any coefficients) of the Conley indices of S are naturally isomorphic and the R-module homomorphism D_{k}: H̃_{k}C (S) → H̃^{m−k}C∗(S) is an isomorphism.

Proof. For any P ∈ ob(AIP(S)), let h_{(P ∗_{1},P ∗_{0})}^{∗} denote the natural transformation

from the Čech to the singular cohomology of the pair (P ∗_{1}, P ∗_{0}) and define a homo-

morphism h̄_{(P ∗_{1},P ∗_{0})}^{∗}: ˜ Ȟ^{∗}C∗(S) → H̃^{∗}C∗(S) by h̄_{(P ∗_{1},P ∗_{0})}^{∗} := (p̄^{− ∗})^{−1} ^{∘} h_{(P ∗_{1},P ∗_{0})}^{∗} ^{∘} ¯ p̌^{− ∗}.

Similarly, define a homomorphism h̄_{(P_{1},P_{0})}^{∗}: ˜ Ȟ^{∗}C (S) → H̃^{∗}C (S). An argument analogous to that used to prove Proposition 9.12 shows that the definitions of h̄_{(P ∗_{1},P ∗_{0})}^{∗} and h̄_{(P_{1},P_{0})}^{∗} are independent of the choice of P ∈ ob(AIP(S)). In particular, Lemma 9.14 provides an AIP(regular) P so that both (P_{1}, P_{0}) and (P ∗_{1}, P ∗_{0}) have isomorphic Čech and singular cohomology. This and Proposition 4.2 yield that each factor in the composites defining h̄_{(P_{1},P_{0})}^{∗} and h̄_{(P ∗_{1},P ∗_{0})}^{∗} are isomorphisms. Thus, the

Čech and singular modules of C∗(S) are naturally isomorphic as are those of C (S). We may therefore regard D_{k} as a homomorphism into H̃^{m−k}C∗(S).

To see that D_{k} is an isomorphism, let P, V, W, and Q be as guaranteed by Lemma 9.14 and note that by definition of D̄_{P k} and Proposition 9.12 it will suffice to show that D_{P k} is an isomorphism. To do this, let us examine the diagram

(9.2)  H_{k}(P_{1}, P_{0}) _{≃} H_{k}(Q \ P ∗_{0}, Q \ P ∗_{1}) ^{(⌢ o^{Q})^{−1}}_{≃}  Ȟ^{m−k}(P ∗_{1}, P ∗_{0}) H_{k}(V_{1}, V_{0}) _{(⌢ o^{Q})^{−1}}_{≃}  Ȟ^{m−k}(Q \ V_{0}, Q \ V_{1})_{≃}

where the unlabeled arrows are inclusion induced. The arrows labeled by (⌢ o^{Q})^{−1} are the inverses of relative Poincaré-Alexander duality isomorphisms determined by the orientation of Q induced by that of M. Note that there is no need to use Čech cohomology with compact supports with regard to the duality isomorphism in the bottom row because cl_{M} (V_{1} \ V_{0}) is a compact subset of Q by Lemma 9.14. The rectangle in the diagram commutes as a consequence of the naturality of Čech cap products (hence of the duality isomorphisms) with respect to inclusions. That the triangle commutes is obvious.

Each of the arrows in the diagram is an isomorphism. First, the vertical arrow that is the right edge of the diagram is an isomorphism. For Ȟ^{∗}(Q \ V_{0}, Q \ V_{1}) ≃ Ȟ^{∗}(W \ V_{0}, W \ V_{1}) as a consequence of statement (8) of Lemma 9.14 and the Five Lemma, and Ȟ^{∗}(W \ V_{0}, W \ V_{1}) ≃ Ȟ^{∗}(P ∗_{1}, P ∗_{0}) as a consequence of statement (7) of Lemma 9.14 and the Five Lemma. Second, the oblique arrow in the triangle is an isomorphism as follows from statement (6) of Lemma 9.14 and the Five Lemma. A trivial diagram chase shows that the remaining two unlabeled arrows in the diagram are isomorphisms.

Then D_{P k} is an isomorphism. For Proposition 9.7 implies that in the top row the inclusion induced isomorphism into H_{k}(Q\P ∗_{0}, Q\P ∗_{1}) and the duality isomorphism (⌢ o^{Q})^{−1} can be replaced respectively by the inclusion induced homomorphism into H_{k}(M \P ∗_{0}, M \P ∗_{1}) (an isomorphism because it factors via an excision isomorphism through H_{k}(Q \ P ∗_{0}, Q \ P ∗_{1})) and the duality isomorphism (⌢ o^{M})^{−1}. □

9.16 Lemma. Let B ∈ 𝔅(S) and let P, V, W, Q, and Q^{±} be as guaranteed by Lemma 9.14; then ⌢ o_{B}: H^{m−k}(B, B^{−}; G) → H_{k}(B, B^{+}; G) is an isomorphism, the diagram (all coefficients are in G)

(9.3)  H_{k}(P_{1}, P_{0})_{i_{1}}  _{i_{9}} H_{k}(V_{1}, V_{0})_{(⌢o^{Q})^{−1}}  H_{k}(Q \ P ∗_{0}, Q \ P ∗_{1})_{(⌢o^{Q})^{−1}}  H_{k}(B, B^{+})_{(⌢o_{B})^{−1}} Ȟ^{m−k}(Q \ V_{0}, Q \ V_{1})  Ȟ^{m−k}(P ∗_{1}, P ∗_{0})  H^{m−k}(B, B^{−})_{i_{15}} Ȟ^{m−k}(Q, Q \ V_{1})

is commutative where the unlabeled arrows and those labeled by i_{1}, i_{9}, or i_{15} are inclusion induced, and each arrow in the diagram is an isomorphism. The analogous statement holds if the roles of (P_{1}, P_{0}) and (P ∗_{1}, P ∗_{0}), of (V_{1}, V_{0}) and (V ∗_{1}, V ∗_{0}), and of (B, B^{+}) and (B, B^{−}) are interchanged.

9.16.1 Remark. Note Lemma 9.14(12) ensures the existence of the inclusion induced homomorphism i_{15}.

9.16.2 Remark. An immediate consequence of Lemma 9.16 and the proof of Theorem 9.15 is that the duality isomorphism of Conley indices D_{k} is in the current context the inverse of the duality isomorphism given in [Mc2].

Proof of Lemma 9.16. That subdiagram of diagram (9.3) consisting of the triangle at the top left together with the rectangle directly below it commutes and has each of its arrows an isomorphism because it is congruent with diagram (9.2) from the proof of Theorem 9.15.

Each of the two triangles at the bottom of diagram (9.3) is obviously commutative, and each of their arrows is an isomorphism as follows. The oblique arrow in the triangle at the bottom right is an isomorphism as follows from statements (10) and (11) of Lemma 9.14 and the Five Lemma. The horizontal arrow of that triangle, labeled i_{15}, is an isomorphism because the inclusion inducing it is between regular index pairs. In such a situation the commutative rectangle of continuous maps consisting of the inclusion between regular index pairs at the top, the induced map on quotients at the bottom, and the collapse maps from the pairs to the quotients (as pointed spaces) on the sides is transformed by homology or cohomology to a commuting rectangle of module isomorphisms: the sides are isomorphisms by Proposition 4.2; the bottom because an inclusion induced map between index spaces in a Conley index has homotopy class the unique morphism in the index from the source to the target index space (see [K5, Proposition 3.1]) and is therefore a homotopy equivalence; hence, commutativity of the rectangle forces its top to be an isomorphism. Thus, commutativity of the triangle at the bottom right of diagram (9.3) implies its vertical arrow to be an isomorphism. Hence, two of the three arrows in the bottom left triangle are isomorphisms whence so is the third.

It remains to show ⌢ o_{B} an isomorphism and to show the five-arrow subdiagram at the top right commutative and to consist of isomorphisms. This follows by analyzing the diagram

(9.4) H_{∗}(Q \ P ∗_{0}, Q \ P ∗_{1})  H_{∗}(P_{1}, P_{0})^{i_{1}}_{i_{3}}  _{i_{5}}  _{i_{7}}  _{i_{9}} H_{∗}(W \ P ∗_{0}, W \ P ∗_{1})_{i_{2}}  _{i_{4}} H_{∗}(W, W \ P̃ ∗_{1}) _{i_{6}} H_{∗}(Q, Q^{+})  H_{∗}(B, B^{+})_{i_{8}} H^{∗}(W \ P ∗_{0}, V ∗_{0} \ P ∗_{0})_{⌢ o′′_{K_{2}}}  H^{∗}(W, V ∗_{0})_{i_{10}}  _{⌢ o′_{K_{2}}}_{i_{11}}  H^{∗}(Q, Q^{−})_{⌢ o′_{K_{1}}}_{i_{12}}_{i_{13}}  _{i_{14}} H^{∗}(B, B^{−})_{⌢ o_{B}}_{i_{15}} H^{∗}(P ∗_{1}, P ∗_{0})

which will turn out to be an expansion of the five-arrow subdiagram. In diagram (9.4), P̃ ∗_{1} := Q \ Q^{+}, each arrow labeled with an i_{j} for some integer j is inclusion induced, and the homology classes o′′_{K_{2}}, o′_{K_{2}}, and o′_{K_{1}} used in the cap product homomorphisms remain to be defined.

The classes o′′_{K_{2}}, o′_{K_{2}}, and o′_{K_{1}} are determined from the given orientation of M and yield well-defined cap product homomorphisms in diagram (9.4) as follows. Their are inclusions

P_{1} ∩ P ∗_{1} ⊂ K_{1} := Q \ (Q^{+} ∪ Q^{−}) ⊂ P_{1} ∩ P ∗_{1} ⊂ K_{2} := P ∗_{1} ∩ W \ V ∗_{0} ⊂ B^{∘}

as a consequence of Lemma 9.14: the two leftmost follow from statement (4), the next from the inclusion implicit in statement (6), and the rightmost from statement (3). Thus, K_{1} = P_{1} ∩ P ∗_{1} and K_{1} and K_{2} are compact subsets of B^{∘}; hence, also of W and Q. For i = 1, 2, let o_{K_{i}} ∈ H_{m}(M, M \ K_{i}; R) be the fundamental class determined by the given orientation O^{M} of M. Then define o′_{K_{1}} to be the image of o_{K_{1}} under the excision isomorphism

H_{m}(M, M \ K_{1}; R) ≃ H_{m}(Q, Q \ K_{1}; R) = H_{m}(Q, Q^{+} ∪ Q^{−}; R)

and define o′_{K_{2}} to be the image of o_{K_{2}} under the excision isomorphism

H_{m}(M, M \ K_{2}; R) ≃ H_{m}(W, W \ K_{2}; R) .

It is immediate that the cap product homomorphism ⌢ o′_{K_{1}} in diagram (9.4) is well-defined; in particular the excisiveness conditions are met because Q^{±} is open. Too, the homomorphism ⌢ o′_{K_{2}} in diagram (9.4) is well-defined because

W \ K_{2} = V ∗_{0} ∪ (W \ P ∗_{1}) = V ∗_{0} ∪ (W \ P̃ ∗_{1})

where equality holds on the right as a consequence of Lemma 9.14(4); again, the excisiveness conditions are met because V ∗_{0} and W \ P̃ ∗_{1} are open. To define o′′_{K_{2}} note that

W \ (K_{2} ∪ P ∗_{0}) = (V ∗_{0} \ P ∗_{0}) ∪ (W \ P ∗_{1}) ;

hence, define o′′_{K_{2}} to be the image of o′_{K_{2}} under the excision isomorphism

(9.5)  H_{m}(W, W \ K_{2}) ≃ H_{m}(W \ P ∗_{0}, (V ∗_{0} \ P ∗_{0}) ∪ (W \ P ∗_{1})) ;

it follows that in diagram (9.4) the cap product homomorphism ⌢ o′′_{K_{2}} is welldefined; once again the excisiveness conditions are met because V ∗_{0} \P ∗_{0} and W \P ∗_{1} are open.

Diagram (9.4) is commutative. Certainly each triangle of morphisms in the diagram is commutative since the arrows in each triangle are inclusion induced. It remains to show that the three rectangles across the middle part of the diagram are commutative. As o′′_{K_{2}} is the image of o′_{K_{2}} under the excision isomorphism (9.5), it is immediate that the leftmost of the three rectangles commutes by naturality of cap products with respect to continuous maps; e.g., see [D, VII.12.6] or [Sp, 5.6.16]. Similarly, the other two rectangles will be known to commute once it is shown that the inclusion induced homomorphisms

H_{m}(W, W \K_{2}; R) → H_{m}(Q, Q\K_{1}; R) and H_{m}(B, ∂B; R) → H_{m}(Q, Q\K_{1}; R)

respectively map o′_{K_{2}} to o′_{K_{1}} and o_{B} to o′_{K_{1}}.

That o′_{K_{1}} is the image of o′_{K_{2}} follows from the definition of o′_{K_{i}} (i = 1, 2) and the commutativity of the diagram of inclusion induced homomorphisms

H_{m}(W, W \ K_{2}; R) ⟶ H_{m}(Q, Q \ K_{1}; R)
≃↓  ↓^{≃}
H_{m}(M, M \ K_{2}; R) ⟶ H_{m}(M, M \ K_{1}; R)

because o_{K_{2}} is necessarily mapped to o_{K_{1}} by the bottom horizontal arrow since both are fundamental classes determined by the same orientation.

To prove that o′_{K_{1}} is the image of o_{B} proceed as follows. First, as K_{1} ⊂ B^{∘} := int(B), choose K_{3} compact satisfying K_{1} ⊂ K_{3} ⊂ B^{∘} and B \ K_{3} is a collar of ∂B. Next, consider the commutative diagram of inclusion induced homomorphisms

H_{m}(M, M \ K_{3})  H_{m}(B^{∘}, B^{∘} \ K_{3})_{≃}  _{≃} H_{m}(B, B \ K_{3})  H_{m}(B, ∂B)_{≃} H_{m}(M, M \ K_{1})  H_{m}(Q, Q \ K_{1})_{≃}

(coefficients are in R) and note that each arrow in the top row is an isomorphism: the leftmost by excision, the middle one because the inclusion of pairs inducing it is a weak deformation retract, the rightmost because ∂B is a strong deformation retract of the collar B \ K_{3}. Also, the bottom horizontal arrow is an excision isomorphism. Let o_{K_{3}} be the fundamental class along K_{3} determined by O^{M}. Then the image in H_{m}(B, ∂B) of o_{K_{3}} under the composite isomorphism that is the top row is by definition o_{B} and is independent of the particular choice of K_{3} as is easily seen. Hence, as o_{K_{1}} is necessarily the image of o_{K_{3}} along the leftmost oblique arrow, the commutativity of the diagram and the definition of o′_{K_{1}} yield that o′_{K_{1}} is the image of o_{B} under the inclusion induced homomorphism. Thus, diagram (9.4) is commutative as claimed.

Each of the arrows i_{1}, …, i_{9} in diagram (9.4) is an isomorphism. To prove the previous statement, diagram chasing shows it sufficient to prove that i_{1}, i_{2}, i_{6}, i_{8}, and i_{9} are isomorphisms. The arrow i_{1} is because it occurs as an arrow in diagram (9.2) in the proof of Theorem 9.15 where it is shown to be an isomorphism. The arrows i_{2} and i_{6} are excision isomorphisms. The arrows i_{9} and i_{8} are: i_{9} by the general argument used above to show i_{15} an isomorphism; i_{8} as a consequence of Lemma 9.14(10) and the Five-Lemma.

Each of the arrows i_{10}, …, i_{15} in diagram (9.4) is an isomorphism. That i_{15} is one is already known, and i_{14} is by Lemma 9.14(10) and the Five Lemma, whence i_{13} is too. Next, statements (5) and (6) of Lemma 9.14 imply that H^{∗}(P ∗_{1}) ≃ H^{∗}(V ∗_{1}) ≃ H^{∗}(W) and that H^{∗}(P ∗_{0}) ≃ H^{∗}(V_{0}); the Five Lemma therefore implies i_{11} is an isomorphism. Therefore i_{12} is because i_{11} and i_{13} are. Finally, i_{10} is by excision.

The arrows ⌢ o′′_{K_{2}}, ⌢ o′_{K_{2}}, ⌢ o′_{K_{1}}, and ⌢ o_{B} in diagram (9.4) are isomorphisms, and up to natural isomorphism of the Čech and singular cohomology modules of the pair (P ∗_{1}, P ∗_{0}), the composite homomorphism k := i_{2} ^{∘} (⌢ o′′_{K_{2}})^{∘} i_{10} ^{∘} i_{11}^{−1} equals the Poincaré-Alexander duality isomorphism

⌢ o^{Q}: Ȟ^{m−k}(P ∗_{1}, P ∗_{0}; G) → H_{k}(Q \ P ∗_{0}, Q \ P ∗_{1}; G) .

For by definition of the Čech cap product given in [D], ⌢ o^{Q} is obtained by passage to the direct limit over homomorphisms of the type i_{2} ^{∘} (⌢ o′′_{K_{2}})^{∘} i_{10} where the pair (W, V ∗_{0}) is replaced by an arbitrary open neighborhood pair of (P ∗_{1}, P ∗_{0}) under the proviso that o′′_{K_{2}} is the image of the fundamental class o_{P ∗_{1}} under a certain composite of inclusion induced homomorphisms to be described momentarily. With the proviso met, it follows that ⌢ o^{Q} = k up to the isomorphism Ȟ^{∗}(P ∗_{1}, P ∗_{0}) ≃ H^{∗}(P ∗_{1}, P ∗_{0}) described for the proof of Lemma 9.14(12) by virtue of i_{11} being an isomorphism. The proviso is that o′′_{K_{2}} must be the image of o_{P ∗_{1}} under the homomorphism that is the composite of the arrows in the top row of the commutative diagram of inclusion induced homomorphisms (all coefficients are taken in R and excision isomorphisms are marked by ≃)

H_{m}(M, M \ P ∗_{1}) H_{m}(M, M \ P ∗_{1} ∪ V ∗_{0}) H_{m}(W \ P ∗_{0}, W \ P ∗_{1} ∪ V ∗_{0} \ P ∗_{0})_{≃} _{≃} H_{m}(W, W \ P ∗_{1})_{≃}  H_{m}(W, W \ K_{2})_{≃}  ;

however, this follows immediately from the commutativity of the diagram and the definitions of o′_{K_{2}} and o′′_{K_{2}} since o_{K_{2}} is necessarily the image of o_{P ∗_{1}} under the inclusion induced map. Thus, as ⌢ o^{Q}, i_{11}, and i_{10} are isomorphisms so too is ⌢ o′′_{K_{2}}. Diagram chasing then shows ⌢ o′_{K_{2}}, ⌢ o′_{K_{1}}, and ⌢ o_{B} to be isomorphisms.

It is immediate from the analysis of the last paragraph that diagram (9.4) is an expanded form of the five-arrow subdiagram in the top right corner of diagram (9.3). Hence, the subdiagram is commutative and has all its arrows isomorphisms since these properties hold for diagram (9.4). □

The next lemma is a statement about the diagrams in Figures 4 and 5. To aid in the printing of those diagrams all coefficient modules have been suppressed, but should be as follows: In the diagram of Figure 4 and in the top row of the diagram of Figure 5, each left-hand factor of a tensor product has coefficients in an R-module G and each right-hand factor has coefficients in an R-module G′; the cohomology and homology modules in the bottom two rows of the diagram of Figure 5 have coefficients in G ⊗ G′.

9.17 Lemma. Let B ∈ 𝔅(S), S ≠ ∅, and let P, V, W, Q, Q^{+}, and Q^{−} be as guaranteed by Lemma 9.14. Then the diagrams in Figures 4 and 5 are commutative where all arrows not labeled as cup or cap products are inclusion induced. Further, all arrows in the diagram in Figure 4 are isomorphisms, and all the inclusion induced arrows in the diagram in Figure 5 are isomorphisms except possibly the two horizontal arrows in the rectangle at the bottom right corner of that diagram.

Proof. The diagram in Figure 4 is obtained by tensoring each lattice point of diagram (9.3) with the corresponding lattice point of the diagram obtained from diagram (9.3) by interchanging the roles of (P_{1}, P_{0}) and (P ∗_{1}, P ∗_{0}), of (V_{1}, V_{0}) and (V ∗_{1}, V ∗_{0}), and of (B, B^{+}) and (B, B^{−}) and similarly tensoring the arrows. Thus, clearly the tensor product of the two diagrams is commutative since each factor is, and each arrow in the tensor product is an isomorphism since each arrow in each of the factors is.

To prove that the diagram in Figure 5 is commutative, let us first examine the definitions of the vertical arrows in that diagram. Each arrow marked as a cup product, but in particular the leftmost one, is an abbreviation for Δ̌^{∘}×, the composition of the map induced by the diagonal map with the cohomology cross-product. Where the source and target are Čech modules, the cross-product and map induced

H(V_{1}, V_{0}) ⊗ H(V ∗_{1}, V ∗_{0})_{(⌢ o^{Q}⊗⌢ o^{Q})^{−1}}  H(P_{1}, P_{0}) ⊗ H(P ∗_{1}, P ∗_{0})  H(B, B^{+}) ⊗ H(B, B^{−})_{(⌢ o_{B} ⊗ ⌢ o_{B})^{−1}} H(Q \ P ∗_{0}, Q \ P ∗_{1}) ⊗ H(Q \ P_{0}, Q \ P_{1})_{(⌢ o^{Q} ⊗ ⌢ o^{Q})^{−1}} Ȟ(P ∗_{1}, P ∗_{0}) ⊗ Ȟ(P_{1}, P_{0}) Ȟ(Q \ V_{0}, Q \ V_{1}) ⊗ Ȟ(Q \ V ∗_{0}, Q \ V ∗_{1})  Ȟ(Q, Q \ V_{1}) ⊗ Ȟ(Q, Q \ V ∗_{1})  H^{∗}(B, B^{−}) ⊗ H^{∗}(B, B^{+})

Figure 4

Ȟ(Q \ V_{0}, Q \ V_{1}) ⊗ Ȟ(Q \ V ∗_{0}, Q \ V ∗_{1})_{⌣}  Ȟ(Q, Q \ V_{1}) ⊗ Ȟ(Q, Q \ V ∗_{1})_{⌣}  H^{∗}(B, B^{−}) ⊗ H^{∗}(B, B^{+})_{⌣} Ȟ(Q \ (V_{0} ∪ V ∗_{0}), Q \ ((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}))_{⌢ o^{Q}}  Ȟ(Q, Q \ (V_{1} ∩ V ∗_{1}))_{⌢ o^{Q}}  H^{∗}(B, ∂B)_{⌢ o_{B}} H((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}), V_{0} ∪ V ∗_{0})  H(V_{1} ∩ V ∗_{1})  H(B)

by the diagonal are those obtained from the singular theory by passing to the limit over neighborhood pairs, or alternatively, can be viewed as those obtained from the Alexander-Spanier theory. In either case, the functoriality of those constructions implies that the rectangle at the top left of the diagram in Figure 5 is commutative. Similarly, the rectangle at the top right is commutative using the fact that because B, ∂B, and B^{±} are ENR’s the Čech and singular cohomology of the pairs (B, B^{±}) and (B, ∂B) are naturally isomorphic, and this is also used in defining the horizontal arrows in that rectangle. The cap product homomorphisms at the left and center of the diagram are Čech cap products and are well-defined, i.e., do not require the use of cohomology with compact supports, since cl_{M} (V_{1} \ V_{0}), cl_{M} (V ∗_{1} \ V ∗_{0}), and cl_{M} (V_{1} ∩ V ∗_{1}) are compact subsets of Q. Thus, by the naturality of Čech cap products with respect to inclusions (see formula VIII.(7.11) on page 296 of [D]) the rectangle at the bottom left of the diagram is commutative.

The proof that the rectangle at the bottom right of the diagram in Figure 5 is commutative is more difficult. First, as V_{1} ∩ V ∗_{1} is an open subset of B^{∘} recall that there is an inclusion induced homomorphism Ȟ_{c}^{∗}(V_{1} ∩ V ∗_{1}) → Ȟ_{c}^{∗}(B^{∘}). Second, as (B^{∘}, B^{∘} \ K) is an open pair in M, it is cofinal in the set of all neighborhood pairs of itself contained in M. Thus, by definition its Čech and singular cohomology modules are equal. Consequently, where K_{3} compact is chosen as in the proof of Lemma 9.16, i.e., B \ K_{3} is an open collar of ∂B, the diagram of cap product and inclusion induced homomorphisms (all coefficients are in G ⊗ G′)

(9.6)

Ȟ(Q, Q\V_{1} ∩ V ∗_{1}) _{≃}_{⌢ o^{Q}} Ȟ_{c}(V_{1} ∩ V ∗_{1})_{⌢ o^{V1∩V ∗_{1}}}  Ȟ_{c}(B^{∘})_{⌢ o^{B∘}}  H^{∗}(B^{∘}, B^{∘}\K_{3})_{≃} _{⌢ o_{K3}^{B∘}}  H^{∗}(B, ∂B)_{≃}  _{⌢ o_{B}} H_{∗}(V_{1} ∩ V ∗_{1})  H_{∗}(B^{∘})  H_{∗}(B)

is a well-defined expansion of the rectangle in the lower right corner of the diagram in Figure 5. The class o_{K_{3}}^{B^{∘}} ∈ H_{m}(B^{∘}, B^{∘} \ K_{3}) is the image under the inverse of the excision isomorphism of the fundamental class along K_{3} determined by O^{M}. The three non-horizontal arrows to the left of ⌢ o_{K_{3}}^{B^{∘}} are Poincaré duality isomorphisms; ⌢ o_{B} of course is the Lefschetz duality isomorphism.

Diagram (9.6) is commutative: the subdiagram consisting of the curved arrow on top together with the horizontal arrows of the top row because each is inclusion induced; the trapezoid on the right by naturality of cap products in the singular theory; the triangle to the immediate left of the trapezoid because ⌢ o^{B^{∘}} is the direct limit of triangles of this type where K_{3} is replaced by an arbitrary compact K ⊂ B^{∘}. The commutativity of the rectangle follows immediately from Proposition 9.8. The commutativity of the triangle at the left of the diagram is an exercise in unwinding the definitions of the two duality isomorphisms that are its non-horizontal sides and utilizes that cl_{M} (V_{1} ∩ V ∗_{1}) is a compact subset of Q so

that Ȟ(Q, Q \ V_{1} ∩ V ∗_{1}) ≃ Ȟ_{c}(Q, Q \ V_{1} ∩ V ∗_{1}) ≃ Ȟ_{c}(V_{1} ∩ V ∗_{1}).

The sought for commutativity of the rectangle in the lower right corner of the diagram in Figure 5 follows from a simple diagram chase on diagram (9.6) once it is shown that the the two rightmost horizontal arrows in its top row are invertible. That both are isomorphisms follows from an elementary diagram chase over the inclusion induced diagram (again coefficients are in G ⊗ G′)

Ȟ_{c}(B^{∘})  H^{∗}(B^{∘}, B^{∘} \ K_{3})  H(B, ∂B) Ȟ(B, B \ K_{3})_{≃}  _{≃}  Ȟ(B, ∂B)_{≃}_{≃}

where the two vertical arrows were shown to be isomorphisms in the course of proving Lemma 9.16 and where the curved arrow on the bottom is the isomorphism Ȟ(B, ∂B) = Ȟ_{c}(B, ∂B) ≃ Ȟ_{c}(B^{∘}).

Finally, the left and right horizontal arrows in the top row of the diagram in Figure 5 are isomorphisms because they are the same respectively as the left and right horizontal arrows in the bottom of the diagram in Figure 4, and the two horizontal arrows in the rectangle at the bottom left of the diagram in Figure 5 are isomorphisms by excision. □

The last proposition needed for the proof of Theorem 9.4 gives precise interpretation to the formulae (2.5). It is inspired by [D, §VIII.13.30, Exercise 1, p. 345] which in the current context states that the composite homomorphism of (9.9) coincides with ⌢_{•}^{V}. In the course of the proof, the cup product of two Čech classes one of which has compact support must be factored through the map on cohomology induced by the diagonal embedding Δ. To carry out this factorization requires the introduction of a relative Čech cohomology module whose elements have compact support along the vertical, i.e., along the fiber of a bundle. As our sole interest is in product bundles, only the following situation need be considered. Let (K, L) be a closed pair in a manifold and define

Ȟ_{cv}^{∗}((K, L) × K) := \underrightarrow{lim} Ȟ^{∗}(K × K, L × K ∪ K × ω)

where ω lies in the family of subsets of K that are locally compact and cobounded in K.

9.18 Proposition. Let V ∈ M_{Δ̸ o}^{(2)}. Then where each unlabeled arrow is inclusion induced, each of the following composite homomorphisms equals 𝔏_{V} upon identification of the singular and Čech homology modules of the open set V_{1} ∩ V ∗_{1}:

(9.7) H_{i}(V_{1}, V_{0}; G) ⊗ H_{j}(V ∗_{1}, V ∗_{0}; G′) ⟶^{(⌢ o^{M})^{−1}⊗(⌢ o^{M})^{−1}}

Ȟ_{c}^{m−i}(M \ V_{0}, M \ V_{1}; G) ⊗ Ȟ_{c}^{m−j}(M \ V ∗_{0}, M \ V ∗_{1}G′) ⟶^{×}

Ȟ_{c}^{2m−i−j}((M \ V_{0}, M \ V_{1}) × (M \ V ∗_{0}, M \ V ∗_{1}); G ⊗ G′) ⟶^{Δ̌^{∗}}

Ȟ_{c}^{2m−i−j}(M \(V_{0} ∪ V ∗_{0}), M \((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}); G ⊗ G′) ⟶^{⌢ o^{M}}

H_{i+j−m}((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}, V_{0} ∪ V ∗_{0}; G ⊗ G′) ⟶^{(e_{1 ∗})^{−1}}

H_{i+j−m}(V_{1} ∩ V ∗_{1}; G ⊗ G′) ;

(9.8) H_{i}(V_{1}, V_{0}; G) ⊗ H_{j}(V ∗_{1}, V ∗_{0}; G′) ⟶^{(⌢ o^{M})^{−1}⊗(⌢ o^{M})^{−1}}

Ȟ_{c}^{m−i}(M \ V_{0}, M \ V_{1}; G) ⊗ Ȟ_{c}^{m−j}(M \ V ∗_{0}, M \ V ∗_{1}; G′) ⟶

Ȟ^{m−i}(V ∗_{1} \ V ∗_{0}, V ∗_{1} \ ((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0}); G) ⊗ Ȟ_{c}^{m−j}(V ∗_{1} \ V ∗_{0}; G′) ⟶^{⌣}

Ȟ_{c}^{2m−i−j}(V ∗_{1} \ V ∗_{0}, V ∗_{1} \ ((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0}); G ⊗ G′) ⟶^{⌢ o^{V ∗_{1}}}

H_{i+j−m}((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0}, V ∗_{0}; G ⊗ G′) ⟶^{(e_{2 ∗})^{−1}}

H_{i+j−m}(V_{1} ∩ V ∗_{1}; G ⊗ G′) ;

(9.9) H_{i}(V_{1}, V_{0}; G) ⊗ H_{j}(V ∗_{1}, V ∗_{0}; G′) ⟶^{(⌢ o^{M})^{−1}⊗1_{∗}}

Ȟ_{c}^{m−i}(M \ V_{0}, M \ V_{1}; G) ⊗ H_{j}(V ∗_{1}, V ∗_{0}; G′) ⟶

Ȟ^{m−i}(V ∗_{1} \ V ∗_{0}, V ∗_{1} \ ((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0}); G) ⊗ H_{j}(V ∗_{1}, V ∗_{0}; G′) ⟶^{⌢}

H_{i+j−m}((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0}, V ∗_{0}; G ⊗ G′) ⟶^{(e_{2 ∗})^{−1}}

H_{i+j−m}(V_{1} ∩ V ∗_{1}; G ⊗ G′) .

Proof. The proof is a diagram chase on the diagram obtained by putting together the two diagrams of Figures 6 and 7 as follows: the two vertical arrows on the right edge of the diagram in Figure 6 coincide with the top two vertical arrows in the left edge of the diagram in Figure 7 and the two horizontal arrows in the bottom edge of the diagram of Figure 6 coincide with the bottom two vertical arrows in the left edge of the diagram of Figure 7. All unlabeled arrows are inclusion induced; those on Čech cohomology with compact support induced by an inclusion of the form (K \ L, ∅) ⊂ (K, L) where (K, L) is a closed pair in a manifold are natural isomorphisms; see statement (6.24) on [D, p. 289]. Although all coefficient modules have been suppressed from the diagrams, they are easily determined from the statement of the proposition after reading the following paragraph.

The composition of the arrows of (9.9) yields the top row of the diagram in Figure 6 followed by the top row of the diagram in Figure 7, and ⌢_{•}^{V} is the left edge of the diagram in Figure 6 followed by the bottom edge of the diagram in Figure 7 followed by the identity homomorphism on H_{∗}(V_{1} ∩ V ∗_{1}) as represented by the right edge of the diagram in Figure 7 in order to carry out the chase. The composite homomorphisms of (9.7) and (9.8) are easily seen to lie “between” the two just given. Assuming the diagrams of Figures 6 and 8 are commutative, the reader can easily perform the necessary diagram chase to see that the composite homomorphisms of (9.7), (9.8), and (9.9) coincide with ⌢_{•}^{V}.

The diagram of Figure 6 is seen to commute as follows. Where e_{7}^{∗} and e_{8}^{∗} are the excision induced isomorphisms (see [D, §VIII.6.22], in particular the isomorphism labeled (6.24), or see [Sp, Lemma 6.6.11])

Ȟ_{c}(M×M\(V_{0}×V ∗_{1}∪V_{1}×V ∗_{0}), M×M\V_{1}×V ∗_{1}) ⟶^{e_{7}^{∗}} Ȟ_{c}(V_{1}×V ∗_{1}\(V_{0}×V ∗_{1}∪V_{1}×V ∗_{0}))

and

Ȟ_{c}((M \ V_{0}, M \ V_{1}) × (M \ V ∗_{0}, M \ V ∗_{1})) ⟶^{e_{8}^{∗}} Ȟ_{c}(V_{1} × V ∗_{1} \ (V_{0} × V ∗_{1} ∪ V_{1} × V ∗_{0})) ,

H_{i}(V_{1}, V_{0}) ⊗ H_{j}(V ∗_{1}, V ∗_{0})  I_{×}  _{(⌢ o^{M})^{−1} ⊗ (⌢ o^{M})^{−1}}^{(⌢ o^{M})^{−1}⊗1_{∗}}_{≃}  _{Ȟ_{c}(M \V_{0}, M \V_{1}) ⊗ H(V ∗_{1}, V ∗_{0})}  _{Ȟ(V ∗_{1}\V ∗_{0}, V ∗_{1} \((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0})) ⊗ H(V ∗_{1}, V ∗_{0})}_{H((V_{1}, V_{0}) × (V ∗_{1}, V ∗_{0}))}_{(−1)^{m(m−j)}(⌢ o^{M} ×o^{M})^{−1}}  _{Ȟ_{c}(M \V_{0}, M \V_{1}) ⊗ Ȟ_{c}(M \V ∗_{0}, M \V ∗_{1})}_{1^{∗} ⊗ ⌢ o^{M}}_{≃} _{×}  _{Ȟ(V ∗_{1}\V ∗_{0}, V ∗_{1}\((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0})) ⊗ Ȟ_{c}(V ∗_{1}\V ∗_{0})}_{≃} _{1^{∗} ⊗ ⌢ o^{V ∗_{1}}}_{×}_{Ȟ_{c}(M × M \(V_{0} × V ∗_{1} ∪ V_{1} × V ∗_{0}), M × M \V_{1} × V ∗_{1})} _{e_{4}^{∗}}_{≃} _{Ȟ_{c}((M \V_{0}, M \V_{1}) × (M \V ∗_{0}, M \V ∗_{1}))}  _{Ȟ_{cv}((V ∗_{1}\V ∗_{0}, V ∗_{1}\((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0})) × V ∗_{1}\V ∗_{0})}

Figure 6

Ȟ(V ∗_{1}\V ∗_{0}, V ∗_{1} \((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0})) ⊗ H(V ∗_{1}, V ∗_{0}) II ^{⌢}  _{H((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0}, V ∗_{0})}  ^{(e_{2 ∗})^{−1}}_{≃}  _{H(V_{1} ∩ V ∗_{1})}_{Ȟ(V ∗_{1}\V ∗_{0}, V ∗_{1} \((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0})) ⊗ Ȟ_{c}(V ∗_{1}\V ∗_{0})}_{1^{∗} ⊗ ⌢ o^{V ∗_{1}}}  _{≃}  _{⌣}_{×}  _{Ȟ_{c}(V ∗_{1}\V ∗_{0}, V ∗_{1} \((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0}))}_{≃} _{⌢ o^{V ∗_{1}}}_{Ȟ_{cv}((V ∗_{1}\V ∗_{0}, V ∗_{1}\((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0})) × V ∗_{1}\V ∗_{0})}  _{Δ̌_{cv}}  _{Ȟ_{c}(V ∗_{1}\V ∗_{0}, V ∗_{1}\((V_{1} ∩ V ∗_{1}) ∪ V ∗_{0}))}_{Ȟ_{c}((M \V_{0}, M \V_{1}) × (M \V ∗_{0}, M \V ∗_{1}))}  _{Δ̌_{c}}  _{Ȟ_{c}(M \(V_{0} ∪ V ∗_{0}), M \((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}))} _{⌢ o^{M}}_{≃}  _{H((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}, V_{0} ∪ V ∗_{0})}_{≃} _{(e_{1 ∗})^{−1}}_{e_{3 ∗}}_{Ȟ_{c}(M × M \(V_{0} × V ∗_{1} ∪ V_{1} × V ∗_{0}), M × M \V_{1} × V ∗_{1})}_{e_{4}^{∗}} _{≃}  _{Δ̌_{c}}  _{Ȟ_{c}(M, M \(V_{1} ∩ V ∗_{1}))}_{≃} _{e_{5}^{∗}}  _{⌢ o^{M}}_{≃}  _{H(V_{1} ∩ V ∗_{1})}_{≃} _{e_{1 ∗}}

the commutativity of the trapezoid labeled I is a consequence of Proposition 9.10 and the fact that e_{7}^{∗} = e_{8}^{∗} ^{∘} e_{4}^{∗}. That the rectangle at the top right corner of the diagram in Figure 6 commutes follows from the definition of the duality isomorphism that is the right-hand factor of its left edge. Specifically, that duality isomorphism is by definition the composite of the isomorphism Ȟ_{c}(M \V ∗_{0}, M \V ∗_{1}) ≃ Ȟ_{c}(V ∗_{1} \V ∗_{0}) which is the right-hand factor in the bottom arrow of the rectangle with the duality isomorphism that is the right-hand factor of the right edge of the rectangle. It is evident that the remaining rectangle and the triangle in the diagram of Figure 6 commute.

The commutativity of the diagram in Figure 7 is seen as follows. The commutativity of the rectangle labeled II is an immediate consequence of Proposition 9.11. The rectangle below rectangle II commutes because the cup product that is its top horizontal arrow and Δ̌_{cv} ^{∘} × are both obtainable by passage to the limit over homomorphisms of the form Δ^{∗} ^{∘} × on the tensor product of singular cohomology modules of open pairs. The rectangle at the bottom right corner of the diagram in Figure 7 and the triangle above it both commute as an immediate consequence of Proposition 9.9. The commutativity of the remaining two rectangles of the diagram and of the remaining triangle of excision isomorphisms is evident. □

9.19 The Proof of Theorem 9.4. The theorem follows from Lemma 9.17, Proposition 9.18, in particular, that 𝔏_{V} can be identified with the composite homomorphism of the sequence in (9.7), and a diagram chase upon the composite diagram obtained by putting together the diagram (9.10)

H(V_{1}, V_{0}; G) ⊗ H(V ∗_{1}, V ∗_{0}; G′)_{(⌢ o^{M} ⊗ ⌢ o^{M})^{−1}}  _{(⌢ o^{Q} ⊗ ⌢ o^{Q})^{−1}} Ȟ(M \V_{0}, M \V_{1}) ⊗ Ȟ(M \V ∗_{0}, M \V ∗_{1})_{⌣}  Ȟ(Q\V_{0}, Q\V_{1}) ⊗ Ȟ(Q\V ∗_{0}, Q\V ∗_{1})_{⌣} Ȟ(M \(V_{0} ∪ V ∗_{0}), M \((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}))_{⌢ o^{M}}  Ȟ(Q\(V_{0} ∪ V ∗_{0}), Q\((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}))_{⌢ o^{Q}} H((V_{1} ∩ V ∗_{1}) ∪ V_{0} ∪ V ∗_{0}), V_{0} ∪ V ∗_{0})

and the diagrams of Figures 4 and 5. Specifically, given B ∈ 𝔅(S), let P, V, Q, Q^{±}, and W be as guaranteed by Lemma 9.14, and join the diagrams of Figures 4 and 5 by identifying the bottom row of the former with the top row of the latter. Then combine this joined diagram with diagram (9.10) by identifying the left edge of the former with the right edge of the latter. Let us refer to the commutative diagram so obtained as the combined diagram. Note that in the sequence (9.7) each occurrence of Čech cohomology with compact support can be replaced by ordinary Čech cohomology because the sets V_{1} \ V_{0}, V ∗_{1} \ V ∗_{0}, and V_{1} ∩ V ∗_{1} have compact closure in M.

Then the composite homomorphism obtained by starting in the top row of the combined diagram at H(P_{1}, P_{0})⊗H(P ∗_{1}, P ∗_{0}) and traveling to the left along the top row, then down the outside left edge of the combined diagram, and then along the bottom row of the combined diagram to H(B) equals 𝔏_{B}^{P} := i_{B}^{V_{1}∩V ∗_{1}} ^{∘} 𝔏_{V}^{P} as follows from Proposition 9.18 using the composite homomorphism from (9.7). A fairly straightforward diagram chase, but one that makes full use of the fact that many of the arrows in the combined diagram have been shown to be isomorphisms, shows that the composite, call it C_{B}^{P}, obtained by starting at H(P_{1}, P_{0}) ⊗ H(P ∗_{1}, P ∗_{0}) and traveling to the right along the top row of the combined diagram and then down its outer right edge to H(B) equals 𝔏_{B}^{P}. Call the composite of the homomorphisms in the outer right edge of the combined diagram C_{B}^{B}. The equality 𝔏_{B}^{P} = C_{B}^{P} together with the commutativity of the diagram

H̃_{∗}C (S; G) ⊗ H̃_{∗}C∗(S; G′)_{(p̄_{∗}^{+}⊗p̄_{∗}^{−})^{−1}}  _{(b̄_{∗}^{+}⊗b̄_{∗}^{−})^{−1}} H_{∗}(P_{1}, P_{0}; G) ⊗ H_{∗}(P ∗_{1}, P ∗_{0}; G′)  _{≃}  H_{∗}(B, B^{+}; G) ⊗ H_{∗}(B, B^{−}; G′)

which follows from the definitions yields via the definition of 𝔏 that

π_{B}^{S} ^{∘} 𝔏 = C_{B}^{B} ^{∘} (b̄_{∗}^{+} ⊗ b̄_{∗}^{−})^{−1} .

Application to α ⊗ γ ∈ H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S) then yields

(α ⌢_{•} γ)_{B} = (x_{B} ⌣ y_{B}) ⌢ o_{B} = x_{B} ⌢ γ_{B}

as desired where recall that x_{B} and y_{B} have been defined in the paragraph preceding the statement of Theorem 9.4. □

D. The case of S ∩ ∂M ≠ ∅. As mentioned earlier McCord’s statement and proof of Poincaré duality of Conley indices is broader than that considered above. The sole goal of this section is to point out how his broader result allows one by analogy with Theorem 9.4 to associate two intersection pairings to an isolated invariant set S of a flow in a manifold with boundary M when

(9.11)  S ∩ ∂M ≠ ∅ .

Note that by invariance of domain, the flow in M restricts to a flow in ∂M. It is then easy to see that the intersection of an isolated invariant set in M with ∂M yields an isolated invariant set of the flow in ∂M. Suppose (9.11) holds and for convenience, set

S_{∂} := S ∩ ∂M .

In this context, one can see from simple examples that the homotopy Conley index of S is often trivial either in forward or backward time although the corresponding index of S_{∂} is non-trivial. In such situations, it is shown in [Mc1] that it is possible by dividing out what is happening in the boundary at the index space level to assign S a non-trivial relative Conley index. We refer the reader to [Mc1] for full generality, but here this index will only be described in terms of isolating blocks. For simplicity, assume M is C^{r} (r ≥ 1) and that the flow in M, hence also in ∂M, is generated by a continuous vectorfield. It is not difficult to modify the construction of isolating blocks in [WY] to obtain in each isolating neighborhood of S an isolating block B for S whose intersection with ∂M is an isolating block for S_{∂}. For convenience, set

B_{∂} := B ∩ ∂M .

In the modified construction B and B_{∂} will be compact, C^{0} manifolds with boundary. Also,

(9.12)  ∂B = B^{+} ∪ B^{−} ∪ B_{∂}

and B_{∂}^{±} = B^{±} ∩ B_{∂}. Accordingly, the definition of 𝔅(S) should be changed to reflect (9.12) whereas B_{∂} will be in 𝔅(S_{∂}) as defined in §9.A. However, for the purposes of the discussion here, we can define 𝔅(S) to consist of those isolating blocks B for S that are compact manifolds with boundary satisfying (9.12) with B^{±} and B_{∂} submanifolds with boundary of ∂B, either void or of codimension one in ∂B; also

∂B_{∂} = B_{∂}^{+} ∪ B_{∂}^{−}

and B_{∂}^{±} is a submanifold with boundary of ∂B_{∂}, either void or of codimension one.

Then as a connected simple system, the relative index of S in forward time, denote it C (S, S_{∂}), includes in its set of objects those quotient spaces of the form B/(B^{+} ∪B_{∂}). Similarly, the relative index of S in reverse time, denote it C∗(S, S_{∂}), includes in its set of objects those quotient spaces of the form B/(B^{−} ∪ B_{∂}). As in the absolute case one can assign homology modules to these relative indices which in keeping with the notation we have used in the absolute case we will denote by H̃_{∗}C (S, S_{∂}) and H̃_{∗}C∗(S, S_{∂}) where for simplicity coefficients are taken in R. Then note that there will be isomorphisms

H̃_{∗}C (S, S_{∂}) ≃ H_{∗}(B, B^{+} ∪ B_{∂})  and  H̃_{∗}C∗(S, S_{∂}) ≃ H_{∗}(B, B^{−} ∪ B_{∂})

for any block B ∈ 𝔅(S). Similar definitions and statements can be made for cohomology; their easy formulation is left to the reader. Then the full duality result of [Mc2] is the following:

9.20 Theorem. There are isomorphisms

H̃^{k}C (S, S_{∂}) ≃ H̃_{m−k}C∗(S) ,  H̃^{k}C∗(S) ≃ H̃_{m−k}C (S, S_{∂})

and

H̃^{k}C∗(S, S_{∂}) ≃ H̃_{m−k}C (S) ,  H̃^{k}C (S) ≃ H̃_{m−k}C∗(S, S_{∂})

that in each block B ∈ 𝔅(S) as above are given by cap product with the fundamental class o_{B} ∈ H_{m}(B, ∂B).

9.21 Definition. For B a block as above, for α ∈ H_{m−k}(B, B^{+}), let x_{α} ∈ H^{k}(B, B^{−} ∪ B_{∂}) be the unique element such that α = x_{α} ⌢ o_{B}. For such α and for γ ∈ H_{∗}(B, B^{−} ∪ B_{∂}), define a pairing

𝔏_{∂^{−} B}: H_{∗}(B, B^{+}) ⊗ H_{∗}(B, B^{−} ∪ B_{∂}) → H_{∗}(B)

by

𝔏_{∂^{−} B}(α ⊗ γ) = x_{α} ⌢ γ .

Similarly, one can define a pairing

𝔏_{∂^{+} B}: H_{∗}(B, B^{+} ∪ B_{∂}) ⊗ H_{∗}(B, B^{−}) → H_{∗}(B) .

Then define

𝔏̄_{∂^{−} B} := 𝔏_{∂^{−} B} ^{∘} (b̄^{+} ⊗ b̄^{−})^{−1}: H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S, S_{∂}) → H_{∗}(B)

and similarly define 𝔏̄_{∂^{+} B}. It follows from the proof of Theorem 9.20, in particular [Mc2, Lemma 2.8], that if B and B′ are two isolating blocks with B′ ⊂ B, then

i_{B}^{B′} ^{∘} 𝔏̄_{∂^{−} B′} = 𝔏̄_{∂^{−} B} .

Similarly,

i_{B}^{B′} ^{∘} 𝔏̄_{∂^{+} B′} = 𝔏̄_{∂^{+} B} .

As 𝔅(S) is cofinal in N (S) it follows that we can pass to the inverse limit to obtain pairings of degree −m

𝔏_{∂^{−}}: H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S, S_{∂}) → Ȟ_{∗}(S)

and

𝔏_{∂^{+}}: H̃_{∗}C (S, S_{∂}) ⊗ H̃_{∗}C∗(S) → Ȟ_{∗}(S) .

Simple examples suggest that only one of these two pairings will be non-trivial for S satisfying (9.11) and S = S_{∂}, e.g., hyperbolic rest points in the boundary, but the author has not investigated the matter further.

Also, it is certainly true that if one uses these pairings to define intersection number pairings in the same manner used to define^{#}𝔏, then the intersection number pairings so defined will be invariant under continuation and each of the full pairings has continuous lifts over a path of isolated invariant sets just as 𝔏 does. The proofs of these facts can either be obtained by suitably extending the notion of admissible index pairs with separate extensions required for each pairing and then mimicking the proofs for 𝔏, or at least for invariance of intersection numbers, by suitably adapting the proof in [Mc2] that the duality isomorphisms of Theorem 9.20 are invariant under continuation.

The cursory treatment here is not meant to imply a lack of importance of results that might be obtainable, but the present investigation draws to a close here. There are, however, numerous interesting differential equations modeling the dynamics of the relative population densities of n related entities where the natural phase space is the open n-simplex. Such systems often extend to the boundary of the simplex, but with singularities at the vertices, and the presence of critical points or more general isolated invariant sets in the (n − 1)-faces certainly affects the dynamics in the open simplex.