CHAPTER 2 THE INTERSECTION PAIRINGS L, 𝔏, AND ^{#}𝔏

A. Pairs of Index Pairs Admissible for the Intersection Pairing.

2.1 Definition. (A) Let Δ: M → M × M denote the diagonal embedding and suppose (X, A) and (Y, B) are topological pairs in M. Then the pair of pairs ((X, A), (Y, B)) satisfies the off-diagonal condition if, and only if,

(X × B ∪ A × Y) ∩ Δ(M) = ∅ ,

or equivalently, if, and only if,

(2.1)  X ∩ B = ∅ = A ∩ Y .

Alternatively, (X, A) and (Y, B) could be said to lie in general position.

(B) Let S be an isolated invariant set of a flow in M, let (P_{1}, P_{0}) be a nested index pair for S with respect to forward time and let (P ∗_{1}, P ∗_{0}) be a nested index pair for S with respect to reverse time. Then the pair of index pairs P := ((P_{1}, P_{0}), (P ∗_{1}, P ∗_{0})) is admissible for the intersection pairing on the Conley indices of S (hereinafter abbreviated to AIP for S) if, and only if,

(1) the pair of pairs P satisfies the off-diagonal condition,

(2) the quotient maps

p^{+}: (P_{1}, P_{0}) → (P_{1}/P_{0}, [P_{0}]) and p^{−}: (P ∗_{1}, P ∗_{0}) → (P ∗_{1}/P ∗_{0}, [P ∗_{0}])

induce isomorphisms on singular homology, coefficients in any R-module G. Proposition 4.4 below states the existence of index pairs that are AIP for S.

From condition (2), it follows that the canonical homomorphisms

p̄_{∗}^{+}: H_{∗}(P_{1}, P_{0}; G) → H̃_{∗}C (S; G) and p̄_{∗}^{−}: H_{∗}(P ∗_{1}, P ∗_{0}; G′) → H̃_{∗}C∗(S; G′)

induced by the quotient maps p^{+} and p^{−} are isomorphisms. It then follows that also p̄_{∗}^{+} ⊗ p̄_{∗}^{−} is an isomorphism because the tensor product of two R-module isomorphisms is again an isomorphism—e.g., see [Sp, Lemmas 5.1.4–5].

Remarks analogous to those of the previous paragraph hold if instead of singular homology any functor F on the homotopy category of compact Hausdorff pairs to the category of left R-modules, R a ring with unit 1 acting like the identity on every R-module, is considered. That is, whenever F(p^{+}) is an isomorphism, \overline{F(p^{+})} is an R-module isomorphism of F(P_{1}, P_{0}) onto F̃C (S) in the covariant case and of F̃C (S) onto F(P_{1}, P_{0}) in the contravariant with analogous properties holding for F(p^{−}) relative to the modules F(P ∗_{1}, P ∗_{0}) and F̃C∗(S); also \overline{F(p^{+})} ⊗ \overline{F(p^{−})} will be an isomorphism.

(C) For S as in (B), let AIP(S) be the small category whose set of objects consists of pairs of index pairs that are AIP for S. A morphism in this category from ((P_{1}, P_{0}), (P ∗_{1}, P ∗_{0})) to ((Q_{1}, Q_{0}), (Q∗_{1}, Q∗_{0})) is an ordered pair of inclusion maps (i, j):

i: (P_{1}, P_{0}) ↪ (Q_{1}, Q_{0}) and j: (P ∗_{1}, P ∗_{0}) ↪ (Q∗_{1}, Q∗_{0}) .

The following notation will be employed. If P ∈ ob(AIP(S)), then P is the pair of index pairs given by having P = ((P_{1}, P_{0}), (P ∗_{1}, P ∗_{0})). Also, if P_{j} ∈ ob(AIP(S)), j varying through some index set, then P_{j} = ((P_{1 j}, P_{0 j}), (P ∗_{1 j}, P ∗_{0 j})). If P and Q are objects of AIP(S) and if there exists a morphism in AIP(S) from P to Q, this fact is denoted symbolically by writing P ↪ Q or Q ↩ P as is convenient.

B. The Euclidean Case: the Homology Intersection Number Pairing L. In this section let S be an isolated invariant set of a flow in R^{m}. All homology modules occurring in this section are assumed to have coefficients in R and the coefficient module is therefore suppressed from the notation for simplicity. The definition of the homology intersection number pairing on the Conley indices of S is based on the following notion of intersection numbers in R^{m} taken from [D].

2.2 Definition. Let ((X, A), (Y, B)) be a pair of topological pairs in R^{m} satisfying the off-diagonal condition, set Δ_{m} := Δ(R^{m}), let

i: (X × Y, (X × B) ∪ (A × Y)) ↪ (R^{m} × R^{m}, R^{m} × R^{m} \ Δ_{m})

be inclusion, and define

d: (R^{m} × R^{m}, R^{m} × R^{m} \ Δ_{m}) → (R^{m}, R^{m} \ {0})

by

d(u, v) := u − v.

Then for α ∈ H_{m−k}(X, A) and γ ∈ H_{k}(Y, B), the homology intersection number of the pair (α, γ), denoted α # γ, is defined by

α # γ := (−1)^{k}d_{∗}i_{∗}(α × γ).

As the map on H_{m−k}(X, A) × H_{k}(Y, B) to H_{m}(R^{m}, R^{m} \ {0}) given by

(α, γ) ↦ α # γ

is bilinear, it defines a linear map on the module of elements of degree m in the tensor product of the homology modules:

\begin{matrix}[H_{∗}(X, A) ⊗ H_{∗}(Y, B)]_{m} & ⟶ & H_{m}(R^{m}, R^{m} \ \{0\}) ≃ R \\ α ⊗ γ & ⟼ & α # γ \end{matrix} .

Both the bilinear map on the Cartesian product and the linear map on the tensor product will be referred to as the homology intersection number pairing in R^{m}, and in general throughout this work the word “pairing” is used as a gloss for the phrase “bilinear pairing” and refers either to a bilinear map on a Cartesian product or the corresponding linear map on the tensor product depending on context.

2.2.1 Remark. The homology intersection number pairing in R^{m} is invariant under inclusion maps. That is, if (X′, A′) ⊂ (X, A) and (Y′, B′) ⊂ (Y, B) are pairs in R^{m} with ((X, A), (Y, B)) satisfying the off-diagonal condition, then so too does ((X′, A′), (Y′, B′)), and if α and γ in the definition above are the images under the inclusion induced maps of α′ ∈ H_{m−k}(X′, A′) and γ′ ∈ H_{k}(Y′, B′), respectively, then α′ # γ′ = α # γ. This is immediate from the naturality of the homology cross-product with respect to inclusions.

The naturality with respect to inclusions of the homology intersection number pairing in R^{m} and its generalization in the manifold setting is crucial to our development of intersection pairings on Conley indices.

2.2.2 Remark. The homology intersection number pairing is graded commutative; i.e., γ # α = (−1)^{k(m−k)}α # γ with α and γ as in Definition 2.2. For a proof see [D, Proposition VII.4.12].

2.2.3 Remark. If X ∩ Y = ∅, then α # γ = 0 because the diagram of inclusions of pairs

(X × Y, X × B ∪ A × Y)  ^{i}  (R^{m} × R^{m}, R^{m} × R^{m} \ Δ_{m}) (R^{m} × R^{m} \ Δ_{m}, R^{m} × R^{m} \ Δ_{m})

commutes, whence i induces the zero homomorphism on homology since H_{∗}(Z, Z) = 0 for any non-empty space Z. Thus, if α # γ ≠ 0, then X ∩ Y ≠ ∅, and because of (2.1) in fact (X \ A) ∩ (Y \ B) ≠ ∅.

2.3 Definition. For each P ∈ ob(AIP(S)), for k = 0, …, m, define a pairing of R-modules

L_{P k}: H_{m−k}(P_{1}, P_{0}) ⊗ H_{k}(P ∗_{1}, P ∗_{0}) → H_{m}(R^{m}, R^{m} \ {0})

by assigning α # γ as given by Definition 2.2 to a generating element α ⊗ γ, i.e.,

L_{P k}(α ⊗ γ) := α # γ ,

and then extending by linearity. Because the pair of pairs P satisfies the off-diagonal condition, L_{P k} is well-defined. As

[H_{∗}(P_{1}, P_{0}) ⊗ H_{∗}(P ∗_{1}, P ∗_{0})]_{m} := ⨁_{k=0}^{m} H_{m−k}(P_{1}, P_{0}) ⊗ H_{k}(P ∗_{1}, P ∗_{0})

there is a pairing

L_{P}: [H_{∗}(P_{1}, P_{0}) ⊗ H_{∗}(P ∗_{1}, P ∗_{0})]_{m} → H_{m}(R^{m}, R^{m} \ {0})

defined by

L_{P} := ⨁_{k=0}^{m} L_{P k} .

Remembering that for each P ∈ AIP(S), p̄_{∗}^{+} ⊗ p̄_{∗}^{−} is an isomorphism, consider the family of pairings L̄_{P} := L_{P} ^{∘} (p̄_{∗}^{+} ⊗ p̄_{∗}^{−})^{−1}, P ∈ ob(AIP(S)), and for P, Q ∈ ob(AIP(S)), define P to be equivalent to Q if, and only if, L̄_{P} = L̄_{Q}. It is a consequence of Remark 2.2.1 (see the proof of Theorem 4.7 below) that if P ↪ Q is an inclusion morphism in AIP(S), then P is equivalent to Q. One can then show with some work (see the proofs of Lemma 4.6 and Theorem 4.7 below) that given P_{1}, P_{2} ∈ ob(AIP(S)) one can produce a string of inclusion morphisms in AIP(S) mediating between P_{1} and P_{2}—in general the arrows do not all point in the same direction; hence, there is no composite from P_{1} to P_{2} or vice versa. The observation from Remark 2.2.1 and transitivity of the relation then yield that the equivalence relation on ob(AIP(S)) has a unique equivalence class, whence the following is immediate.

2.4 Theorem. The family of pairings L_{P}, P ∈ ob(AIP(S)), determines a pairing

L: [H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S)]_{m} → H_{m}(R^{m}, R^{m} \ {0})

in such manner that for each P ∈ ob(AIP(S)),

L = L_{P} ^{∘} (p̄_{∗}^{+} ⊗ p̄_{∗}^{−})^{−1} .

2.5 Definition. Call L the homology intersection number pairing for the Conley indices of S, and for each integer k ∈ [0, m], for each element α⊗γ of the R-module H̃_{m−k}C (S) ⊗ H̃_{k}C∗(S), set

α # γ := L(α ⊗ γ) .

Call α # γ the homology intersection number of α ⊗ γ.

2.5.1 Remark. Theorem 2.4 asserts that L is completely determined by L_{P} for any choice of P ∈ ob(AIP(S)); i.e., there are unique representatives α ∈ H_{m−k}(P_{1}, P_{0}) and γ ∈ H_{k}(P ∗_{1}, P ∗_{0}) representing α and γ, respectively, and α # γ = α # γ. Consequently, computing L_{P} for one P computes L_{P} for all P given that one knows the maps on homology determined by the morphisms of C (S) and C∗(S).

There are many properties of L that follow from the analogous property for homology intersection numbers in R^{m} and Remark 2.5.1. The derivations of two simple ones of interest follow.

First, letting L∗ denote the pairing obtained from Theorem 2.4 by regarding S as an isolated invariant set of the reverse time flow associated to the given flow and defining τ_{∗}(γ ⊗ α) := (−1)^{k(m−k)}α ⊗ γ for α and γ as in Definition 2.5, we immediately obtain from Remark 2.5.1 and Remark 2.2.2 the following graded commutativity relationship between L and L∗.

2.6 Proposition. L∗ = Lτ_{∗}, i.e., γ # α = (−1)^{k(m−k)}α # γ.

Second, noting that for any topological pair (X, A), the homology class of a relative singular cycle c in X modulo A is the image under the inclusion induced homomorphism of the homology class of c regarded as a relative cycle in its support |c| modulo the support of its boundary chain |∂c|, we immediately obtain from Remarks 2.2.1, 2.2.3, and 2.5.1 the following result basic to the use of L in proving existence theorems for singularly perturbed two-point boundary value problems.

2.7 Proposition. Let α ∈ H̃_{m−k}C (S), γ ∈ H̃_{k}C∗(S), and suppose α # γ ≠ 0. Then for any P ∈ ob(AIP(S)), if c_{α} is a relative singular cycle in P_{1} modulo P_{0} representing α and c_{γ} is a relative singular cycle in P ∗_{1} modulo P ∗_{0} representing γ,

then (|c_{α}| \ P_{0}) ∩ (|c_{γ}| \ P ∗_{0}) ≠ ∅.

In the applications the most commonly and easily constructed index pairs are those that consist of an isolating block and either its exit or entrance set, the former (latter) yielding an index pair with respect to the forward (reverse) time flow. An obvious question therefore is: Given a block for S is it possible to express every homology intersection number of homology classes in the Conley indices of S directly from a pair of singular chains, one representing non-zero homology in the block modulo its exit set, the other in the block modulo its entrance set? The answer is a qualified yes with minor restrictions on how the representing chains sit inside the block. The precise statement is deferred until after the development of the intersection class pairing and the intersection number pairing for Conley indices in the setting of a continuous flow on a manifold because the result in the Euclidean setting (Corollary 6.3 below) is but a special case of more general results holding for the intersection class pairing that appear below as Theorem 6.1 and Corollary 6.2. As other properties of L of interest to us here are also subsumed by results obtainable for the more general intersection class pairing, we turn now to its development.

C. The Manifold Case: the Intersection Class and Number Pairings 𝔏 and ^{#}𝔏. The development of the intersection class pairing on Conley indices as carried out in the rest of this work requires specific notions of orientation, Poincaré duality, Čech homology and cohomology, and the definition of certain transfer maps on homology induced by the diagonal map Δ. An exposition of the required notions follows. Throughout, all manifolds are assumed second countable.

An R-orientation of the m-dimensional manifold X is a continuous global section O^{X} of the orientation bundle of X with coefficients in R with the property that O^{X}(u) generates H_{m}(X, X \{u}; R) ≃ R for each u ∈ X. For O^{X} an R-orientation of X, for each non-empty, compact K ⊂ X, there exists a unique class o_{K}^{X} ∈ H_{m}(X, X \ K; R), called the fundamental class along K, with the property that for each u ∈ K the image of o_{K}^{X} under the inclusion induced homomorphism to H_{m}(X, X \ {u}; R) is o_{u}^{X} := O^{X}(u). It follows that if K ⊂ K′ are compact subsets of X then o_{K′}^{X} ↦ o_{K}^{X} under the inclusion induced homomorphism. See [D, §VIII.2] for a full exposition.

Because the current setting includes only manifolds that are ENR’s, Čech homology and cohomology modules with coefficients in an R-module G will be identified with limits of the usual inverse and direct systems of singular homology and cohomology modules; specific notations and restrictions follow.

For X a topological space, for each K ⊂ X, let N (K) denote the family of neighborhoods of K directed by containment (i.e., U ⪯ U′ ⟺ U ⊃ U′), and for each pair (K, L) in X, let N (K, L) denote the family of neighborhood pairs of (K, L) directed by containment. When L = ∅ the set of neighborhood pairs of the form (U, ∅) is cofinal in N (K, ∅) and as used in arguments below it is therefore convenient to abuse notation and identify N (K, ∅) with N (K).

For (K, L) an arbitrary pair in a manifold X, its graded Čech homology module with coefficients in G, denoted Ȟ_{∗}(K, L; G), is identified with

Ȟ_{∗}(K, L; G) := \underleftarrow{lim} H_{∗}(U, V; G)

where the limit is over (U, V) ∈ N (K, L) directed by containment and where for (U′, V′) ⪯ (U, V) ∈ N (K, L), the bonding homomorphism i_{(U′,V′) ∗}^{(U,V)} is the inclusion induced homomorphism on singular homology. By the definition of an inverse limit, for each neighborhood pair (U, V) ∈ N (K, L), there is a natural projection homomorphism

π_{(U,V)}^{(K,L)}: Ȟ_{∗}(K, L; G) → H_{∗}(U, V; G) .

To simplify notation, a subscript (U, V) will often be appended to a Čech homology class to denote its image under this natural projection; i.e., define

ξ̌_{(U,V)} := π_{(U,V)}^{(K,L)}ξ̌ whenever ξ̌ ∈ Ȟ_{∗}(K, L; G) and (U, V) ∈ N (K, L).

Clearly, if (K, L) ⊂ (K′, L′) ⊂ X, then there is an inclusion induced homomorphism

ǐ_{(K′,L′)}^{(K,L)}: Ȟ_{∗}(K, L; G) → Ȟ_{∗}(K′, L′; G)

determined by the property that if (U′, V′) ∈ N (K′, L′), then π_{(U′,V′)}^{(K′,L′)} ^{∘} ǐ_{(K′,L′)}^{(K,L)} =

π_{(U′,V′)}^{(K,L)}. With these definitions Ȟ_{∗}( ·; G) is a functor on the category of pairs in X and inclusions of such pairs which suffices for our purposes.

For (K, L) a pair of locally compact subsets of a manifold X, its graded Čech cohomology module with coefficients in G, denoted by Ȟ(K, L; G), is identified with

Ȟ^{∗}(K, L; G) := \underrightarrow{lim} H^{∗}(U, V; G)

where the limit is over (U, V) ∈ N (K, L) and where for (U′, V′) ⪯ (U, V) the bonding map ρ_{(U′,V′)}^{(U,V)} is the inclusion induced restriction homomorphism on singular cohomology. For a pair of such locally compact pairs (K′, L′) ⊃ (K, L) there is a restriction homomorphism

ρ̌_{(K′,L′)}^{(K,L)}: Ȟ^{∗}(K′, L′; G) → Ȟ^{∗}(K, L; G)

obtained by passage to the limit over (U′, V′) ∈ N (K′, L′) ⊂ N (K, L) through the canonical homomorphisms H^{∗}(U′, V′; G) → Ȟ^{∗}(K, L; G). More generally, Ȟ( ·; G) can be extended to a contravariant functor on the category of pairs of locally compact subsets of a manifold and continuous maps between such pairs; see [D, §VIII.6]. The last assertion requires X to be an ENR; that and metrizability are the two reasons for assuming that all manifolds mentioned herein are second countable.

For (K, L) a closed pair in a manifold X, its graded Čech cohomology module with compact supports and coefficients in G is denoted by Ȟ_{c}^{∗}(K, L; G) and is defined by

Ȟ_{c}^{∗}(K, L; G) := \underrightarrow{lim} Ȟ^{∗}(K, ω; G)

where the limit is over the family, directed by containment, of locally compact subsets ω satisfying L ⊂ ω ⊂ K and ω is cobounded in K (i.e., cl (K \ ω) is compact) and where each bonding homomorphism is a restriction ρ̌_{(K′,L′)}^{(K,L)}. Čech cohomology with compact supports is a contravariant functor on the category of closed pairs in manifolds and continuous maps f: (K, L) → (K′, L′) between such pairs that are proper over K′ \ L′ (i.e, f^{−1}(C) is compact for each compact C ⊂ K′ \ L′); see [D,§VIII.6.26].

In [D,§VIII.7, pp. 291–297] relative Poincaré-Alexander duality isomorphisms are described for three successively more inclusive situations; all in the context of X an m-dimensional manifold with R-orientation O^{X}. The first two of these situations are (i) (K, L) is a compact pair in X and (ii) (K, L) is a closed pair in X with L cobounded in K. Both result in a duality isomorphism

Ȟ^{i}(K, L; G) ⟶_{≃}^{⌢o^{X}} H_{m−i}(X \ L, X \ K; G)

for any R-module G. The third situation, (iii) (K, L) is an arbitrary closed pair in X, results in a duality isomorphism

Ȟ_{c}^{i}(K, L; G) ⟶_{≃}^{⌢o^{X}} H_{m−i}(X \ L, X \ K; G)

and is the one of immediate interest to us although all three types of duality isomorphisms occur in Chapter 9. It is not difficult to show (a) that all three duality isomorphisms are the same for a compact pair (K, L) and (b) that the second and third are the same for (K, L) a closed pair and L cobounded in K modulo the natural identification Ȟ^{∗}(K, L; G) ≃ Ȟ_{c}^{∗}(K, L; G) that occurs in situations (a) and (b) via simple cofinality arguments.

Although it does not play a role in any of the arguments below, the reader may find it of interest that usage of the notation ⌢ o^{X} to denote the duality isomorphisms is particularly apt because where H_{q}^{c}(X, Ẋ; R) is, in the sense of [Sp,§6.3], the module of compatible 𝔄-families of homology classes (of dimension q with coefficients in R) for 𝔄 the family of compact subsets of X (= X \ Ẋ as X is boundaryless) the R-orientation O^{X} defines the compatible family o^{X} := (o_{K}^{X})_{K∈𝔄} which generates H_{m}^{c}(X, Ẋ; R) and because the method used to define the duality isomorphisms in each of situations (i)–(iii) is naturally extended to define in situations (i) and (ii) a bilinear pairing having the properties of a cap product

Ȟ^{i}(K, L; G) ⊗ H_{q}^{c}(X, Ẋ; R) ⟶^{⌢} H_{q−i}(X \ L, X \ K; G)

and similarly for situation (iii) where Ȟ_{c}^{i}(K, L; G) replaces Ȟ^{i}(K, L; G); i.e., with reference to the bilinear pairings just mentioned, the duality isomorphisms of situations (i)–(iii) are given by fixing the second argument in the appropriate pairing at o^{X} ∈ H_{m}^{c}(X, Ẋ; R).

It is convenient to use M_{Δ̸}^{(2)} to denote the category of pairs of topological pairs in M satisfying the off-diagonal condition with morphisms pairs of inclusions; thus AIP(S) becomes a full subcategory of M_{Δ̸}^{(2)}. Accordingly, the notation used to de-

note objects and morphisms of AIP(S) is extended to M_{Δ̸}^{(2)}; e.g, if X ∈ ob(M_{Δ̸}^{(2)}),

then X = ((X_{1}, X_{0}), (X∗_{1}, X∗_{0})). The notations M_{Δ̸ c}^{(2)} and M_{Δ̸ o}^{(2)} respectively denote

the full subcategories of M_{Δ̸}^{(2)} whose objects are respectively pairs of compact pairs

and pairs of open pairs. Also, for each X ∈ ob(M_{Δ̸}^{(2)}), let M_{Δ̸ c}^{(2)}(X) and M_{Δ̸ o}^{(2)}(X)

denote the full subcategories of M_{Δ̸ c}^{(2)} and M_{Δ̸ o}^{(2)} respectively with sets of objects de-

fined as follows: K ∈ ob(M_{Δ̸ c}^{(2)}(X) if, and only if, there exists an inclusion morphism

K ↪ X, and V ∈ ob(M_{Δ̸ o}^{(2)}(X) if, and only if, there exists an inclusion morphism X ↪ V.

Each of the categories defined in the previous paragraph can be regarded as a full subcategory of a corresponding category of pairs of pairs in M where objects are no longer required to satisfy the off-diagonal condition; the corresponding supercategory is denoted by dropping the subscript Δ / from the notation. For example, M^{(2)} denotes the category of arbitrary pairs of topological pairs in M and M_{o}^{(2)} its full subcategory of pairs of open pairs, etc.

For the rest of this section, fix an R-orientation O^{M} of M and give M × M the canonical product R-orientation: O^{M×M} := O^{M} × O^{M}. Then for any V ∈ ob(M_{o}^{(2)}), for i ∈ N, for coefficients taken in any R-module G but suppressed from the notation to aid in printing, there is a homology transfer homomorphism

Δ_{!}: H_{i}(V_{1} × V ∗_{1}, V_{0} × V ∗_{1} ∪ V_{1} × V ∗_{0}) → H_{i−m}(V_{1} ∩ V ∗_{1}, (V_{0} ∩ V ∗_{1}) ∪ (V_{1} ∩ V ∗_{0}))

(read Δ_{!} as Δ-shriek) defined by

Δ_{!} := (⌢ o^{M})^{∘} Δ̌_{c}^{∗} ^{∘} (⌢ o^{M×M})^{−1}

where

Ȟ_{c}^{2m−i}(M × M \ (V_{0} × V ∗_{1} ∪ V_{1} × V ∗_{0}), M × M \ V_{1} × V ∗_{1})

⟶_{≃}^{⌢o^{M×M}} H_{i}(V_{1} × V ∗_{1}, V_{0} × V ∗_{1} ∪ V_{1} × V ∗_{0})

and

Ȟ_{c}^{2m−i}(M \ ((V_{0} ∩ V ∗_{1}) ∪ (V_{1} ∩ V ∗_{0})), M \ (V_{1} ∩ V ∗_{1}))

⟶_{≃}^{⌢o^{M}} H_{i−m}(V_{1} ∩ V ∗_{1}, (V_{0} ∩ V ∗_{1}) ∪ (V_{1} ∩ V ∗_{0}))

are relative Poincaré-Alexander duality homomorphisms and where

Ȟ_{c}^{2m−i}(M × M \ (V_{0} × V ∗_{1} ∪ V_{1} × V ∗_{0}), M × M \ V_{1} × V ∗_{1})

⟶^{Δ̌_{c}^{∗}} Ȟ_{c}^{2m−i}(M \ ((V_{0} ∩ V ∗_{1}) ∪ (V_{1} ∩ V ∗_{0})), M \ (V_{1} ∩ V ∗_{1}))

is the homomorphism induced by the diagonal map Δ. Note that Δ is proper because the diagonal is closed and Δ is an embedding. The transfer Δ_{!} should be thought of as the map on homology induced by taking inverse images under Δ of open pairs in M × M. For the general definition of transfer homomorphisms on homology and cohomology see [D, §VIII.10].

As mentioned in the Introduction, the definition of the intersection class pairing on Conley indices is based on the intersection pairings defined in [D, §VIII.13] where the intersection pairing is first defined for arbitrary pairs of open pairs in M and then extended via limiting arguments to pairs of arbitrary topological pairs in M. However, in the current work, interest is focused almost entirely on intersection pairings for pairs of pairs satisfying the off-diagonal condition; all exceptions to this occur in the proof of Lemma 7.7 wherein the intersection pairing on Conley indices is computed for normally hyperbolic invariant sets of a smooth flow. Thus, to avoid repeated mention of cofinality arguments to reduce to the off-diagonal situation, it is technically convenient to construct intersection pairings for off-diagonal pairs of pairs entirely within the category M_{Δ̸}^{(2)}. For the sake of Lemma 7.7 and the reader’s convenience, however, the definition of the pairing for a general pair of open pairs is made below as the first step since the definition and notation is the same for the general and off-diagonal cases for pairs of open pairs. Then, after the extension of the definition to arbitrary off-diagonal pairs of pairs is made, the minor changes needed to make the extension to the general case are pointed out.

2.8 Definition. (A) For V ∈ ob(M_{o}^{(2)}) and G and G′ R-modules, define a pairing ⌢_{•}^{V} on the tensor product of H_{∗}(V_{1}, V_{0}; G) and H_{∗}(V ∗_{1}, V ∗_{0}; G′) by defining it first on generating elements and then extending by linearity where the formula on generating elements is (2.3)

\begin{matrix}H_{i}(V_{1}, V_{0}; G) & ⊗ H_{j}(V ∗_{1}, V ∗_{0}; G′) & → & H_{i+j−m}(V_{1} ∩ V ∗_{1}, V_{0} ∩ V ∗_{1} ∪ V_{1} ∩ V ∗_{0}; G ⊗ G′) \\ α & ⊗ γ & ↦ & α ⌢_{•}^{V} γ := (−1)^{m(m−j)}Δ_{!}(α × γ) \end{matrix} .

The pairing ⌢_{•}^{V} is natural with respect to inclusions of open pairs [D, Proposition VIII.13.4]; i.e., if W := ((W_{1}, W_{0}), (W ∗_{1}, W ∗_{0})) is another pair of open pairs in M and if there is an inclusion morphism V ↪ W, then where the vertical arrows are inclusion induced homomorphisms and with coefficients suppressed the following is a commutative diagram: (2.4)

H_{i}(V_{1}, V_{0}) ⊗ H_{j}(V ∗_{1}, V ∗_{0}) ^{⌢_{•}^{V}} H_{i+j−m}(V_{1} ∩ V ∗_{1}, (V_{0} ∩ V ∗_{1}) ∪ (V_{1} ∩ V ∗_{0})) H_{i}(W_{1}, W_{0}) ⊗ H_{j}(W ∗_{1}, W ∗_{0}) _{⌢_{•}_{W}} H_{i+j−m}(W_{1} ∩ W ∗_{1}, (W_{0} ∩ W ∗_{1}) ∪ (W_{1} ∩ W ∗_{0})) .

This naturality follows from the corresponding naturality of homology cross-prod-ucts and of transfer homomorphisms, and the naturality of the latter in turn follows from the naturality of the duality isomorphisms and the functorial properties of Čech cohomology with compact supports.

An alternative definition of the pairing defined by (2.3) can be given in terms of cup and cap products. Specifically, if x ∈ Ȟ_{c}^{m−i}(M \ V_{0}, M \ V_{1}) and y ∈ Ȟ_{c}^{m−j}(M \ V ∗_{0}, M \ V ∗_{1}) are Poincaré dual in M to α and γ respectively, then (suppressing certain inclusion induced maps)

(2.5a)  α ⌢_{•} γ = (x ⌣ y) ⌢ o^{M}

= x ⌢ (y ⌢ o^{M}) = x ⌢ γ . (2.5b)

Note that some interpretation is required to make sense of these formulae, especially for (2.5b). The interpretation and verification of these formulae is outlined in [D] on p. 337 and in Exercise 1 of §VIII.13.30 on p. 345 proved below as part of Proposition 9.18. Analogs of the formulae (2.5) in the context of the Conley index theory are obtained in Chapter 9 below and used to prove non-degeneracy of the intersection pairing on Conley indices; see Theorem 9.4 and its corollaries.

(B) The extension of (2.3) to an arbitrary pair of topological pairs satisfying the off-diagonal condition is made as follows. Let (i, j) ∈ N^{2} and let X ∈ M_{Δ̸}^{(2)}.

For each U ∈ N (X_{1} ∩ X∗_{1}), for each K ∈ M_{Δ̸ c}^{(2)}(X), define a pairing

𝔏_{U}^{K}: H_{i}(K_{1}, K_{0}; G) ⊗ H_{j}(K∗_{1}, K∗_{0}; G′) → H_{i+j−m}(U; G ⊗ G′)

as follows. Choose V ∈ ob(M_{Δ̸ o}^{(2)}(K)) satisfying V_{1} ∩ V ∗_{1} ⊂ U. Such V are easily seen to exist via a simple construction using normality and the closedness of the sets in the pairs of K as well as that of K_{1} ∩ K∗_{1}. Note too that ob(M_{Δ̸ o}^{(2)}(K))

is directed by containment. That is, for V, V′ ∈ ob(M_{Δ̸ o}^{(2)}(K)), define V ∩ V′ to

be the element of ob(M_{Δ̸ o}^{(2)}(K)) obtained by taking the componentwise intersection of the sets in the pairs forming V and V′, whence there are inclusion morphisms V ↩ V ∩ V′ and V′ ↩ V ∩ V′. Then 𝔏_{U}^{K} is that pairing making the diagram

(2.6)  H_{i}(K_{1}, K_{0}; G) ⊗ H_{j}(K_{1}^{∗}, K_{0}^{∗}; G′) ^{𝔏_{U}^{K}} H_{i+j−m}(U; G ⊗ G′) H_{i}(V_{1}, V_{0}; G) ⊗ H_{j}(V ∗_{1}, V ∗_{0}; G′) _{⌢_{•}_{V}} H_{i+j−m}(V_{1} ∩ V ∗_{1}; G ⊗ G′)

commutative where the vertical arrows are inclusion induced and the bottom horizontal arrow is given by (2.3). This definition of 𝔏_{U}^{K} does not depend on the particular choice of V ∈ ob(M_{Δ̸ o}^{(2)}(K)) satisfying V_{1} ∩ V ∗_{1} ⊂ U that is made as

follows easily from our observation that ob(M_{Δ̸ o}^{(2)}(K)) is directed by containment and from the naturality expressed by (2.4). It follows from this independence as to the choice of V that if U ⊃ U′ ∈ N (X_{1} ∩ X∗_{1}), then

(2.7)  i_{U}^{U′} ^{∘} 𝔏_{U′}^{K} = 𝔏_{U}^{K} .

It is immediate from (2.7) that we can pass to an inverse limit over N (X_{1} ∩ X∗_{1}) directed by containment and thereby obtain a pairing of R-modules

(2.8) 𝔏_{X (i,j)}^{K}: H_{i}(K_{1}, K_{0}; G) ⊗ H_{j}(K∗_{1}, K∗_{0}; G′) ⟶ Ȟ_{i+j−m}(X_{1} ∩ X∗_{1}; G ⊗ G′) .

Note that if in fact X ∈ ob(M_{Δ̸ o}^{(2)}), then the singleton {X_{1} ∩ X∗_{1}} is cofinal in N (X_{1} ∩ X∗_{1}), whence the Čech and singular homology modules of X_{1} ∩ X∗_{1} are naturally isomorphic. Hence, 𝔏_{X}^{K} as given by (2.8) is naturally identified with 𝔏_{X_{1}∩X∗_{1}}^{K} as defined by diagram (2.6) with U := X_{1} ∩ X∗_{1} and V := X.

Next, observe that ob(M_{Δ̸ c}^{(2)}(X)) can be regarded as a directed set where K_{1} ⪯ K_{2} if, and only if, K_{1} ↪ K_{2} because where K_{1} ∪K_{2} is defined to be the pair of pairs obtained by taking the union of corresponding components in K_{1} and K_{2}, K_{1} ∪ K_{2} satisfies the off-diagonal condition because X does.

Thus, noting that if

j_{L}^{K} ≡ (j_{(L_{1},L_{0})}^{(K_{1},K_{0})}, j_{(L∗_{1},L∗_{0})}^{(K∗_{1},K∗_{0})}): K ↪ L

is a morphism in M_{Δ̸ c}^{(2)}(X) then

(2.9)  𝔏_{X (i,j)}^{K} = 𝔏_{X (i,j)}^{L} ^{∘} (j_{(L_{1},L_{0}) ∗}^{(K_{1},K_{0})} ⊗ j_{(L∗_{1},L∗_{0}) ∗}^{(K∗_{1},K∗_{0})}) ,

we find that we can pass to the direct limit over M_{Δ̸ c}^{(2)}(X) and thereby obtain a pairing

𝔏_{X (i,j)}: H_{i}(X_{1}, X_{0}; G) ⊗ H_{j}(X∗_{1}, X∗_{0}; G′) → Ȟ_{i+j−m}(X_{1} ∩ X∗_{1}; G ⊗ G′)

because direct limits commute with tensor products and because the singular homology of any pair is the direct limit of the singular homology of the compact pairs it contains. Further, should it be the case that X ∈ ob(M_{Δ̸ c}^{(2)}), then the pairing just defined coincides with 𝔏_{X (i,j)}^{X} because the singleton {X} would in this case be

cofinal in ob(M_{Δ̸ c}^{(2)}(X)). Similarly, should it be the case that X ∈ ob(M_{Δ̸ o}^{(2)}), then 𝔏_{X (i,j)} coincides with the pairing ⌢_{•}^{X} defined in the manner of (2.3).

The intersection class pairing on X ∈ ob(M_{Δ̸}^{(2)}) is the R-module homomorphism

𝔏_{X}: H_{∗}(X_{1}, X_{0}; G) ⊗ H_{∗}(X∗_{1}, X∗_{0}; G′) → Ȟ_{∗}(X_{1} ∩ X∗_{1}; G ⊗ G′)

defined by 𝔏_{X} := ⨁ 𝔏_{X (i,j)} where the sum is over all pairs of non-negative integers.

Also, for α ∈ H_{∗}(X_{1}, X_{0}; G) and γ ∈ H_{∗}(X∗_{1}, X∗_{0}; G′), set

α ⌢_{•}^{X} γ := 𝔏_{X}(α ⊗ γ)

and call α ⌢_{•}^{X} γ the intersection class of α ⊗ γ.

(C) To construct an analogous intersection class pairing 𝔏_{X} for X ∈ ob(M^{(2)}), i.e., X fails to satisfy the off-diagonal condition hereinafter referred to as the general case, the following minor changes need to be made. First, for any Y ∈ ob(M^{(2)}) set

Y^{∩} := (Y_{1} ∩ Y ∗_{1}, (Y_{0} ∩ Y ∗_{1}) ∪ (Y_{1} ∩ Y ∗_{0})) .

Then 𝔏_{U}^{K} is now defined for K ∈ ob(M_{c}^{(2)}(X) and U ≡ (U_{1}, U_{0}) ∈ N (X^{∩}) via the diagram obtained from diagram (2.6) by replacing V_{1} ∩ V ∗_{1} with V^{∩} where V ∈ ob(M_{o}^{(2)}(X) and satisfies V^{∩} ↪ U such V existing by an argument similar to that used in the off-diagonal case. Then equality (2.7) holds in the context of the general case and passage to the inverse limit yields a pairing

𝔏_{X (i,j)}^{K}: H_{i}(K_{1}, K_{0}; G) ⊗ H_{j}(K∗_{1}, K∗_{0}; G′) → Ȟ_{i+j−m}(X^{∩}; G ⊗ G′)

so that (2.9) remains satisfied in the general case. Therefore, passage to the direct limit over M_{c}^{(2)}(X) through the pairings 𝔏_{X (i,j)}^{K} yields a pairing

𝔏_{X (i,j)}: H_{i}(X_{1}, X_{0}; G) ⊗ H_{j}(X∗_{1}, X∗_{0}; G′) → Ȟ_{i+j−m}(X^{∩}; G ⊗ G′) ,

and as before 𝔏_{X} := ⨁ 𝔏_{X (i,j)}.

(D) From (2.8), for each X ∈ M_{Δ̸}^{(2)}, the restriction of 𝔏_{X} to the module of elements of degree m, viz., [H_{∗}(X_{1}, X_{0}) ⊗ H_{∗}(X∗_{1}, X∗_{0})]_{m} with all coefficients taken in R, has image in the module Ȟ_{0}(X_{1} ∩ X∗_{1}; R) allowing us to generalize to the manifold setting the notion of intersection number pairing given in Definition 2.1 for the Euclidean setting as follows.

For X ∈ M_{Δ̸}^{(2)}, define the intersection number pairing on X to be the pairing

#𝔏_{X}: [H_{∗}(X_{1}, X_{0}) ⊗ H_{∗}(X∗_{1}, X∗_{0})]_{m} → R

given on generators by

#𝔏_{X}(α ⊗ γ) := ⟨1, (𝔏_{X}(α ⊗ γ))_{M}⟩

and extended by linearity. Here ⟨ · , · ⟩ denotes the Kronecker pairing of singular cohomology and homology and 1 denotes the unit cohomology class in H^{0}(M), i.e., the class of the usual augmentation (i.e., the Kronecker Index) on singular chains in M to R so that ^{#}𝔏_{X}(α ⊗ γ) gives a signed count (modulo the characteristic of R) of the number of points in the 0-chain (𝔏_{X}(α ⊗ γ))_{M} assuming |α| + |γ| = m. As a matter of convenience, we also set

#(α ⌢_{•}^{X} γ) := ^{#}𝔏_{X}(α ⊗ γ)

and call it the intersection number of α⊗γ. Too, the superscript X in the notation ⌢_{•}^{X} will often be dropped when clear from context.

(E) To see that the notion of intersection number just defined is indeed a generalization to the manifold setting of definition 2.1, let o_{m} ∈ H_{m}(R^{m}, R^{m} \ {0}) be the generator specified by a given orientation of R^{m}, let ν^{m} ∈ H^{m}(R^{m}, R^{m} \ {0}) be the element dual to o_{m}, i.e., ⟨ν^{m}, o_{m}⟩ = 1, let X ∈ ob((R^{m})_{Δ̸}^{(2)}), and define

#L_{X}: [H_{∗}(X_{1}, X_{0}) ⊗ H_{∗}(X∗_{1}, X∗_{0})]_{m} → R

by setting ^{#}L_{X}(α ⊗ γ) := (−1)^{|α||γ|}⟨ν^{m}, α # γ⟩ on generators and extending by linearity. Thus

(2.10)  ^{#}L_{X}(α ⊗ γ)o_{m} = (−1)^{|α||γ|}α # γ

because the isomorphism H_{m}(R^{m}, R^{m} \ {0}) ≃ R specified by sending o_{m} to 1 ∈ R is given by the map

(2.11)  ξ ↦ ⟨ν^{m}, ξ⟩ = ⟨1 ⌣ ν^{m}, ξ⟩ = ⟨1, ν^{m} ⌢ ξ⟩ .

On the other hand with α and γ as in the last paragraph,

(2.12)  ν^{m} ⌢ (γ # α) = (α ⌢_{•}^{X} γ)_{R^{m}} .

as an immediate consequence of [D, Proposition VIII.13.29].

The equalities (2.10)–(2.12) together with Remark 2.2.2 immediately yield the following result which states, in essence, that the two methods available for defining intersection numbers in the Euclidean case are the same.

2.9 Proposition. Let X ∈ ob((R^{m})^{(2)}). Then ^{#}L_{X} = ^{#}𝔏_{X}; that is, for α ∈

H_{m−k}(X_{1}, X_{0}) and γ ∈ H_{k}(X∗_{1}, X∗_{0}),

ν_{m} ⌢ (γ # α) = (α ⌢_{•} γ)_{R^{m}} .

Hence

(γ # α) = ^{#}(α ⌢_{•} γ)o_{m} .

The construction of the intersection pairing on the homology modules of the Conley indices of an isolated invariant set as carried out by the proof of Theorem 2.11 below has as key element the following statement of naturality following from the double limit construction used in defining 𝔏_{X}; see §4.10 for the proof.

2.10 Lemma. The families of pairings {𝔏_{X} : X ∈ ob(M_{Δ̸}^{(2)})} and {^{#}𝔏_{X} : X ∈

ob(M_{Δ̸}^{(2)})} are natural relative to the morphisms of M_{Δ̸}^{(2)}. The family of pairings {𝔏_{X} : X ∈ ob(M^{(2)}} is natural relative to the morphisms of M^{(2)}.

For the remainder of this section, fix S to be an isolated invariant set of some flow in the manifold M.

Let N (S; AIP) consist of those neighborhoods in N (S) expressible in the form P_{1} ∩ P ∗_{1} for some P ∈ ob(AIP(S)). Recall too that for P ∈ ob(AIP(S)) and coefficient R-modules G and G′, the tensor product of the canonical maps

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

is an isomorphism.

2.11 Theorem. N (S; AIP) is cofinal in N (S), and if P, Q ∈ ob(AIP(S)) with

P_{1} ∩ P ∗_{1} ⊃ Q_{1} ∩ Q∗_{1}, then

(2.13)  ǐ_{P_{1}∩P ∗_{1}}^{Q_{1}∩Q∗_{1}} ^{∘} 𝔏_{Q} ^{∘} (q̄_{∗}^{+} ⊗ q̄_{∗}^{−})^{−1} = 𝔏_{P} ^{∘} (p̄_{∗}^{+} ⊗ p̄_{∗}^{−})^{−1} ,

whence passage to the inverse limit yields a pairing of degree −m

𝔏: H̃_{∗}C (S; G) ⊗ H̃_{∗}C∗(S; G′) → Ȟ_{∗}(S; G ⊗ G′)

upon identification of \underleftarrow{lim} Ȟ_{∗}(P_{1} ∩ P ∗_{1}) and Ȟ_{∗}(S).

Remark 2.11.1. Note that for each V ∈ N (S), if P ∈ ob(AIP(S)) and P_{1}∩P ∗_{1} ⊂ V, then where π_{V}^{S} and π_{V}^{P_{1}∩P ∗_{1}} are the projections from the Čech homology of the superscript to the singular homology of the subscript,

π_{V}^{S} ^{∘} 𝔏 = π_{V}^{P_{1}∩P ∗_{1}} ^{∘} 𝔏_{P} ^{∘} (p̄_{∗}^{+} ⊗ p̄^{−})^{−1}

and that this relation completely characterizes 𝔏 by (2.13) and the universality properties of inverse limits. In particular, the right-hand side is independent of the choice of P. The proof of the theorem consists in showing this via an application of the method used to prove Theorem 2.4 which requires the naturality expressed by Lemma 2.10. See Proposition 4.4, Theorem 4.7, Corollary 4.8, and §4.11 for the proof.

2.12 Definition. (A) Call 𝔏 the intersection class pairing for the Conley indices of S, and for (i, j) ∈ N^{2} and element α ⊗ γ ∈ H̃_{i}C (S; G) ⊗ H̃_{j}C∗(S; G′), set

α ⌢_{•} γ := 𝔏(α ⊗ γ)

and call the Čech homology class α ⌢_{•} γ the intersection class of α ⊗ γ.

If it is necessary to distinguish between intersection class pairings for different flows, the flow or an index corresponding to the flow is appended to 𝔏 as a superscript. For simplicity if ψ∗ is the reverse time flow associated to a flow ψ understood from context, set 𝔏∗ := 𝔏^{ψ∗}, and for γ ∈ H̃_{j}C∗(S) and α ∈ H̃_{i}C (S),

set γ ⌢_{•} α := 𝔏∗(γ ⊗ α).

As is the case for the homology intersection number pairing L, there is a simple graded commutativity relationship between 𝔏 and 𝔏∗. Namely with α and γ as above,

γ ⌢_{•} α = (−1)^{(m−i)(m−j)}t_{∗}(α ⌢_{•} γ)

where t_{∗} is the change in coefficient isomorphism induced by G ⊗ G′ ≃ G′ ⊗ G. This identity follows easily from the definition of 𝔏, from equality (2.5a) above, and from the well-known graded commutativity of cup products. Details are left to the reader.

(B) In analogy with Definition 2.8(D), for (i, j), α, and γ as in (A) but assuming that i+j = m and G = G′ = R and making the identification R⊗R ≃ R, define the intersection number of α ⊗ γ, denoted by ^{#}𝔏(α ⊗ γ) or, more often, by ^{#}(α ⌢_{•} γ), via the formula

#(α ⌢_{•} γ) := ⟨1, (α ⌢_{•} γ)_{M}⟩ .

Extend ^{#}𝔏 by linearity to a pairing

#𝔏: [H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S)]_{m} → R

called the intersection number pairing on the singular homology of the Conley indices of S, coefficients in R.

(C) In analogy with Definition 2.8(E), when M := R^{m} and α and γ are as in (B), set ^{#}L(α ⊗ γ) := (−1)^{|α||γ|}⟨ν^{m}, α # γ⟩ and extend ^{#}L by linearity to an R-module homomorphism

#L: [H̃_{∗}C (S; R) ⊗ H̃_{∗}C∗(S; R)]_{m} → R .

If P ∈ AIP(S) and α ∈ H_{m−k}(P_{1}, P_{0}; R) represents α and γ ∈ H_{k}(P ∗_{1}, P ∗_{0}; R)

represents γ, then (α ⌢_{•}^{P} γ)_{M} = (α ⌢_{•} γ)_{M} as can be read off from Theorem 2.11 and the fact that π_{M}^{S} = π_{M}^{P_{1}∩P ∗_{1}} ^{∘}ǐ_{P_{1}∩P ∗_{1}}^{S}. Also, when M := R^{m} Remark 2.5.1 states that α # γ = α # γ . Thus from Propositions 2.6 and 2.9 we get the following.

2.13 Proposition. For M = R^{m}, ^{#}L = ^{#}𝔏; i.e.,

ν_{m} ⌢ (−1)^{k(m−k)}(α # γ) = (α ⌢_{•} γ)_{R^{m}} .

Hence,

(−1)^{k(m−k)}α # γ = ^{#}(α ⌢_{•} γ)o_{m} .