CHAPTER 1 BASIC NOTATION AND BACKGROUND DEFINITIONS

A. Basic Notation. The symbols N, Z, R, R^{+}, and R^{−} will respectively denote the natural numbers (0 included), the integers, the reals, the non-negative reals, and the non-positive reals. For a, b ∈ R, ]a, b[ and [a, b] respectively denote the open and closed intervals from a to b; similarly, ]a, b] and [a, b[ denote halfopen intervals open respectively on the left and right. Throughout, M denotes a second countable C^{r} manifold without boundary (0 ≤ r ≤ ∞) of dimension m. The second countability assumption ensures that M is a Euclidean neighborhood retract (hereinafter abbreviated to ENR) and that M is metrizable. Further, M is assumed oriented over the PID R.

A C^{k} flow ( 0 ≤ k ≤ r) in M is a C^{k} map on an open 𝔊 in M × R into M, generally denoted by (u, t) ↦ u · t, with the following properties: (1) 𝔊 ⊃ M × {0} and u · 0 = u for each u ∈ M, (2) (u · t_{1}) · t_{2} = u · (t_{1} + t_{2}) whenever (u, t_{1}), (u · t_{1}, t_{2}), and (u, t_{1} + t_{2}) ∈ 𝔊, (3) for each u ∈ M, the trajectory of u, i.e., the map t ↦ u · t defined for all t satisfying (u, t) ∈ 𝔊, has closed, connected graph in the topology inherited from R × M. It is immediate from the openness of 𝔊 and properties (3) and (1) that the domain of each trajectory is an open interval containing 0. Note that any locally Lipschitz vectorfield on a C^{1} manifold generates a C^{0} flow in the manifold assuming integral curves of the vectorfield are taken with maximally extended domain. Also, any continuous Hamiltonian vectorfield on a two-dimensional symplectic manifold generates a continuous flow in M.

Assume given a flow in M. The orbit through u ∈ M is the image of the trajectory of u and is complete if the trajectory has domain R; else it is non-complete. The positive (resp. negative) semi-orbit of u is the image of the trajectory restricted to the non-negative (resp. non-positive) reals in its domain and is complete if the restricted trajectory has domain R^{+} (resp. R^{−}); else it is non-complete. Because each trajectory has an open interval as domain, no non-complete semi-orbit can lie in a compact set as a simple consequence of property (3) of a flow. A subset of M is called invariant (resp. positively invariant, resp. negatively invariant) if, and only if, it is the union, the empty union not excluded, of complete orbits (resp. complete positive semi-orbits, resp. complete negative semi-orbits). Thus, the union of invariant sets is invariant; hence, each N ⊂ M contains a unique maximal invariant set relative to the partial order induced by inclusion on the invariant sets contained in N. It is a simple consequence of property (3) of a flow that the closure of a relatively compact, positively (resp. negatively) invariant set (relative compactness is not needed if 𝔊 = M × R) is again positively (resp. negatively) invariant. It follows that the maximal invariant set in each compact N ⊂ M is itself compact. Call a compact N ⊂ M an isolating neighborhood relative to the given flow if the maximal invariant set it contains lies in its interior, int_{M}(N). An invariant set S of the flow is an isolated invariant set if it is the maximal invariant set in some isolating neighborhood. For those Y contained in some compact, positively (resp. negatively) invariant subset of M, the maximal invariant subset of cl (Y · R^{+}) (resp. cl (Y · R^{−})) is denoted by ω(Y) (resp. ω∗(Y)) and coincides with the usual omega (resp. alpha) limit set of Y. In particular, for those u ∈ M having a complete positive semi-orbit and having cl (u · R^{+}) compact, ω(u) is a well-defined, non-empty, compact, connected subset of M; similarly for ω∗(u) for u having a complete negative semi-orbit. For S an isolated invariant set and U a relatively compact neighborhood of S, the notations W^{u}(S; U) and W^{s}(S; U) respectively denote the local unstable set and the local stable set of S within U relative to the given flow. Explicitly, u ∈ W^{s}(S; U) if, and only if, u · R^{+} ⊂ U and ω(u) ⊂ S; replacement of R^{+} with R^{−} and of ω(u) with ω∗(u) yields W^{u}(S; U). Also, set A^{±}(U) := {u ∈ U : u · R^{±} ⊂ U}. In general, W^{u}(S; U) is a proper subset of A^{−}(U), but the two coincide if S is the maximal invariant set in U.

To facilitate the investigation of the variance of the intersection pairings under continuation of the flow and isolated invariant set, the following conventions and notations are adopted in discussing parameterized families of flows. Let Λ be a locally compact Hausdorff space and assume given a C^{k} flow φ_{λ} in M with domain 𝔊_{λ} for each λ ∈ Λ. To conform to the previously introduced notation denoting a flow as a right R-action, set

u ·^{λ}t := φ_{λ}(u, t) for (u, t) ∈ 𝔊_{λ} , λ ∈ Λ .

The family of flows {φ_{λ}}_{λ∈Λ} is a continuous family of C^{k} flows in M if, and only if, the set 𝔊_{Λ} that is the union over λ ∈ Λ of the sets 𝔊_{λ} × {λ} is open in M × R × Λ and the map φ: 𝔊_{Λ} → M defined by φ(u, t, λ) := u ·^{λ}t is continuous. By an abuse of language, given 𝔊_{Λ} open in M × R × Λ, a map φ: 𝔊_{Λ} → M is referred to as a continuous family of C^{k} flows in M whenever φ is continuous and its restriction to 𝔊_{λ} × {λ} := 𝔊_{Λ} ∩ M × R × {λ} defines a C^{k} flow φ_{λ}: 𝔊_{λ} → M for each λ ∈ Λ. Note that if M is at least C^{1} and λ ↦ X_{λ} is a continuous map of Λ into the space of locally Lipschitz vectorfields on M with any topology at least as fine as the compact-open topology, then the family of vectorfields {X_{λ}}_{λ∈Λ} generates a C^{0} family of flows in M.

For φ: 𝔊_{Λ} → M a continuous family of flows in M, the following glosses are used in referring to isolated invariant sets and isolating neighborhoods of a particular φ_{λ}. An invariant set of φ_{λ} is called a λ-invariant set, and if S is an isolated λinvariant set, then S is said to be λ-isolated. Also, if N is an isolating neighborhood relative to φ_{λ}, then N is called a λ-isolating neighborhood, and if S is λ-isolated with λ-isolating neighborhood N, then N is said to λ-isolate S.

1.1 Definition. (A) The space of isolated invariant sets associated to the continuous family of flows φ: 𝔊_{Λ} → M, denoted S(φ), is the set of ordered pairs (S, λ) where λ ∈ Λ and S is λ-isolated topologized as follows. For each compact N ⊂ M, let Λ(N) be the set of λ ∈ Λ for which N is a λ-isolating neighborhood, and for λ ∈ Λ(N), let S(N, λ) denote the isolated λ-invariant set λ-isolated by N. For each compact N ⊂ M, define σ_{N}: Λ(N) → S(φ) by σ_{N}(λ) := (S(N, λ), λ). The topology on S(φ) is declared to be the finest topology consistent with each σ_{N}: Λ(N) → S(φ) being continuous where N ranges over compact subsets of M and Λ(N) inherits its topology from Λ. In fact, Λ(N) is open in Λ for each compact N ⊂ M. The collection of sets σ_{N}(U) where U ranges over open subsets of Λ(N) and N ranges over compact subsets of M is a basis for this topology. Also, the natural projection π_{φ}: S(φ) → Λ is a surjective local homeomorphism with σ_{N} a local inverse on Λ(N). Details of the above can be found in [RAII], [C], or [Sl].

(B) Throughout this work, whenever g: Y → S(φ) is a continuous map, the components of g are denoted by S_{g} and λ_{g}, i.e., g(y) =: (S_{g}(y), λ_{g}(y)) where S_{g}(y) is an isolated invariant set of the flow φ_{λ_{g}(y)} for each y ∈ Y.

B. The Conley Index. The Conley index will be described below as a certain type of category called a connected simple system. The following notation is used in describing categories. For any category K, its class of objects is denoted below by ob(K) and for X, Y ∈ ob(K), the set of morphisms from X to Y is denoted by K(X, Y). Recall that a category K is a small category if ob(K) is a set.

1.2 Definition. A small category K is a connected simple system if K(X, Y) is a singleton for every X, Y ∈ ob(K). The unique element of K(X, Y) will be denoted by h_{K}^{X Y}. Consequently, morphisms in K satisfy the following identities: for X, X′, X′′ ∈ ob(K),

(1.1)  h_{K}^{X X} = 1_{X} ,  h_{K}^{X′ X′′} ^{∘} h_{K}^{X X′} = h_{K}^{X X′′} ,  h_{K}^{X′ X} = (h_{K}^{X X′})^{−1} .

The middle and left identities hold because K is a category and because K(X, Y) = {h_{K}^{X Y} }; the identity on the right holds as a simple consequence of the other two and uniqueness of inverses. Consequently, any two objects of K are equivalent.

For K a connected simple system, the facts that ob(K) is a set, that for every X, X′ ∈ ob(K), K(X, X′) = {h_{K}^{X X′}}, and that the identities (1.1) hold are collectively referred to as the connected simple system properties of K.

As shown in [K3], for any category H, there is a corresponding category, denoted CSS(H), of connected simple systems that are subcategories of H. The notion of morphism in CSS(H) is reviewed below in Chapter 5 and is of particular interest to us in the case H := T^{∗′}, the homotopy category of topological spaces with basepoint, because a Conley index is by definition an object of CSS(T^{∗′}) and because this notion of morphism undergirds the proof of invariance of intersection numbers under continuation. Other substitutions for H of interest to us in this context are G_{R}M and ∂_{R}M′, the former the category of graded left R-modules, the latter the homotopy category of chain complexes over R.

1.3 Definition. Let S be an isolated invariant set of a C^{k} flow ψ in M, set u · t := ψ(u, t), and let O^{+}(u) denote the positive semi-orbit of u. A description of the Conley index of S relative to the flow ψ follows.

(A) Compact subsets N_{1} and N_{0} of M form an index pair ⟨N_{1}, N_{0}⟩ for S relative to ψ if, and only if, the following three conditions are satisfied:

(1) S ⊂ int_{M}(N_{1} \ N_{0}) and cl (N_{1} \ N_{0}) is an isolating neighborhood of S;

(2) if u ∈ N_{0}, t > 0, and u · [0, t] ⊂ N_{1}, then u · [0, t] ⊂ N_{0};

(3) if u ∈ N_{1} and O^{+}(u) ⊄ N_{1}, then there exists t ≥ 0 so that u · [0, t] ⊂ N_{1} and u · t ∈ N_{0}.

These three properties are called respectively the isolating, the relative positive invariance, and the exit properties of an index pair.

If an index pair ⟨N_{1}, N_{0}⟩ in fact forms a topological pair, i.e., if N_{0} ⊂ N_{1}, then call it a nested index pair and use the notation (N_{1}, N_{0}) rather than ⟨N_{1}, N_{0}⟩ in writing the index pair. Note that if ⟨N_{1}, N_{0}⟩ is an index pair for S, then (N_{1}, N_{1} ∩ N_{0}) is a nested index pair for S, and (N_{1}, ∩N_{0}) is used as a shorthand for the latter. The notion of non-nested index pair is needed to prove invariance of intersection numbers under continuation in the non-smooth case, i.e., when either the ambient manifold is only C^{0} or when some flow in the continuation is only absolutely continuous but not C^{1}. Readers interested only in the smooth case (r ≥ k ≥ 1) can assume that all index pairs are nested. Existence of index pairs is proven in [C] and [Sl].

(B) The index space associated to an index pair ⟨N_{1}, N_{0}⟩ is the quotient space N_{1}/N_{0}; i.e., when N_{1} ∩ N_{0} is non-empty, N_{1}/N_{0} is the pointed space obtained from N_{1} by collapsing N_{1}∩N_{0} to a point denoted by [N_{0}] and taken as the basepoint, and when N_{1} ∩N_{0} is empty, with {∗} any one point space disjoint from M, N_{1}/N_{0} is the disjoint union N_{1} ∪ {∗} with basepoint ∗ which for convenience will also be denoted [N_{0}]. In both cases [u] denotes the image of a point u ∈ N_{1} under the quotient map N_{1} → N_{1}/N_{0}. Too, note that (N_{1}, ∩N_{0}) and ⟨N_{1}, N_{0}⟩ have the same index space. Unless stated explicitly to the contrary, an index space is always regarded as a topological space with basepoint, i.e., as a pair with the space as first member and the basepoint as second; however, the notation is usually abbreviated to just writing the space.

(C) The Conley index of S relative to the flow ψ, denoted by C (S) if ψ is clear from context but by C (S, ψ) if necessary to resolve possible ambiguity, is the connected simple system whose set of objects is the set of index spaces for S relative to ψ and for two such index spaces X := N_{1}/N_{0} and X′ := N′_{1}/N′_{0}, the unique morphism from X to X′ is the homotopy class of the map F^{t}: X → X′ defined for t ≥ 0 sufficiently large by

F^{t}([u]) := F ([u], t) := \{ \begin{matrix}[u · 3t] & if u · [0, 2t] ⊂ N_{1} \ N_{0} \\ & and u · [t, 3t] ⊂ N′_{1} \ N′_{0}; \\ [N′_{0}] & otherwise. \end{matrix}

Here, the phrase “t ≥ 0 sufficiently large” means that there exists T ≥ 0, called a common squeeze time for ⟨N_{1}, N_{0}⟩ and ⟨N′_{1}, N′_{0}⟩, so that for t ≥ T,

(1) u · [−t, t] ⊂ N_{1} \ N_{0} implies u ∈ N′_{1} \ N′_{0};

(2) u · [−t, t] ⊂ N′_{1} \ N′_{0} implies u ∈ N_{1} \ N_{0}. This definition of the unique morphism in C (S) from N_{1}/N_{0} to N′_{1}/N′_{0} and the proof of continuity of F for t ≥ 0 sufficiently large is due to Salamon [Sl]. See [K5] for a proof that C (S) with this definition of its morphisms is a connected simple system and is the same one obtained using the original definition of Conley [C] as modified by the author in [K1].

For simplicity, when the flow is clear from context abuse notation and write h_{S}^{X X′} for the unique morphism in C (S) from X to X′ rather than h_{C(S,ψ)}^{X X′} as specified by the general notational scheme for connected simple systems introduced in Definition 1.2.

(D) The Conley homotopy index of an isolated invariant set S is the common pointed homotopy type of the index spaces in C (S) and is denoted by [C (S)].

(E) Each flow ψ in M gives rise to a new flow ψ∗ in M defined by ψ∗(u, t) := ψ(u, −t) and called the reverse time flow associated to ψ. Note that ψ∗ has the same isolated invariant sets as ψ. However, an index pair relative to ψ∗ is not an index pair relative to ψ and conversely; hence, C (S, ψ) and C (S, ψ∗) are distinct connected simple systems.

Usually, the flow ψ under consideration will be clear from context without being explicitly named (e.g., a subscript on the invariant set could specify the flow) and will be viewed as defining the preferred time direction. In such instances, call ψ the forward time flow, set C (S) := C (S, ψ) and call it the forward time Conley index of S, and set C∗(S) := C (S, ψ∗) and call it the reverse time Conley index of S.

(F) Whenever a single upper-case Latin letter is used in denoting both members of a nested index pair in forward (respectively, reverse) time, the corresponding lower case Latin letter with a plus (respectively, minus) sign as superscript is used to denote the quotient map from the pair to its index space. For example, given (P_{1}, P_{0}) and (P ∗_{1}, P ∗_{0}), nested index pairs in forward and reverse time respectively, the quotient maps are denoted by

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

The Conley index as described above grew out of Conley’s earlier notion of homotopy index for an isolated invariant set based on the existence of isolating blocks for an isolated invariant set. Several definitions of isolating block have been made and their existence proven in various contexts, but they are all special cases of the following notion suggested by R. McGehee.

1.4 Definition. (A) For N an isolating neighborhood relative to a given flow, let τ_{N} denote the time until first exit map on N into [0, ∞] defined by

τ_{N}(u) := sup{t ≥ 0 : u · [0, t] ⊂ N} ,

and taken relative to the reverse time flow, let the time until first exit map on N be denoted by τ∗_{N} and call it the time since last entrance map on N. Call (τ_{N})^{−1}(0) the exit set of N and (τ∗_{N})^{−1}(0) the entrance set of N.

(B) Call an isolating neighborhood B an isolating block if, and only if, τ_{B} and τ∗_{B} are continuous maps. The exit and entrance sets of an isolating block B will be denoted here by B^{+} and B^{−} respectively. The existence of an isolating block interior to a given isolating neighborhood has been proved in various contexts and with varying choices of structure given to the boundary; see [C-E], [Ch], [WY]. Also see Proposition 4.4 below.

N.B. The usage here of a plus (minus) sign as superscript to denote the exit (entrance) set is the opposite of the historical usage introduced by Conley and Easton in [C-E], but coincides with the usage in [Sm].

(C) Continuity of τ_{B} and τ∗_{B} implies that the exit and entrance sets of a block are closed. Conversely, if an isolating neighborhood has closed exit and entrance sets it is a block as follows from [Conley, Theorem II.2.3]. It is trivial to verify that (B, B^{+}) and (B, B^{−}) are nested index pairs with respect to the forward and reverse time flows respectively. In keeping with the notation introduced above, b^{±}: (B, B^{±}) → (B/B^{±}, [B^{±}]) denotes the quotient map from the index pair to its index space. It is a simple consequence of the continuity of τ_{B} and τ∗_{B} that B^{+} and B^{−} are strong deformation retracts of closed neighborhoods in B, whence it follows [Sp, Theorem 4.8.9] that the quotient maps b^{+} and b^{−} induce isomorphisms on homology. This property is shown to generalize to certain classes of index pairs in Chapter 4.

1.4.1 Remark. One advantage of the definition of isolating block given above is that it is a triviality that the intersection of two isolating blocks for an isolated invariant set is again an isolating block for the invariant set: if B_{1} and B_{2} are the blocks, then certainly B_{1} ∩ B_{2} is an isolating neighborhood and τ_{B_{1}∩B_{2}} = min{τ_{B_{1}}, τ_{B_{2}}} so that τ_{B_{1}∩B_{2}} is continuous; similarly τ∗_{B_{1}∩B_{2}} is continuous.

C. Homology and Cohomology of Conley Indices. The definitions in this subsection yield the graded (co)homology modules of a Conley index.

1.5 Definition. (A) Let X be one of the following categories: _{R}M, the category of left R-modules, G_{R}M, the category of graded left R-modules, or ∂_{R}M′, the category of chain complexes over R and chain homotopy classes of chain maps. Also, let F be a functor on the homotopy category of topological pairs with values in X, let F̃ be the corresponding reduced functor on T^{∗′}, and let K be a connected simple system that is a subcategory of T^{∗′}.

Then ob(K) is a directed set with respect to the quasi-order (i.e., reflexive, transitive relation) ⪯ defined by X ⪯ X′ for every X, X′ ∈ ob(K). Hence, for F̃ covariant (respectively, contravariant), application of F̃ to the objects and morphisms of K yields a direct (respectively, inverse) system of objects and bonding isomorphisms in X indexed over the directed set ob(K), viz.,

(1.2)  {{F̃(X)}_{X∈ob (K)}, {F̃(h_{K}^{X X′})}_{X⪯X′∈ob (K)}} .

For F̃ covariant (respectively, contravariant), define F̃(K) to be the direct (respectively, inverse) limit of the system in (1.2). Also, for each X ∈ ob(K), let ι^{X}: F̃(X) → F̃(K) (respectively, π_{X}: F̃(K) → F̃(X)) denote the canonical homomorphism.

Key to much of what follows is that ι^{X} (respectively, π_{X}) is an X -isomorphism as follows from the connected simple system properties of K, in particular, the identities (1.1). In this regard it is helpful to note that for F̃ covariant the direct limit F̃(K) can be realized as the set of equivalence classes of the equivalence relation ∼ defined on the union of the F̃(X), X ∈ ob(K) as follows: for α ∈ F̃(X), α′ ∈ F̃(X′), α ∼ α′ if, and only if, F̃(h_{K}^{X X′})α = α′. Similarly, for F̃ contravariant, it is helpful to note that the inverse limit F̃(K) can be realized as those elements (ϕ_{X}) in the direct product ∏{F̃(X) : X ∈ ob(K)} satisfying F̃(h_{K}^{X X′})ϕ_{X′} = ϕ_{X}.

Should X = ∂_{R}M′, then in the case F is a covariant functor into ∂_{R}M′, the boundary operator ∂^{K} for F̃(K) is for any choice of X ∈ ob(K) unambiguously defined for each integer k by

∂_{k}^{K} := ι_{k−1}^{X} ^{∘} ∂_{k}^{X} ^{∘} (ι_{k}^{X})^{−1} .

No ambiguity arises from the choice of X because the collection in (1.2) is a direct system of chain complexes and chain homotopy equivalences over R. An analogous definition can be made in the contravariant case.

In general, elements of F̃(K) will be denoted with lower case, bold face Greek letters while correspondents in any F̃(X), X ∈ ob(K), will be represented by the corresponding lower case, normal weight Greek letter.

(B) Let S be an isolated invariant set of a C^{k} flow in M, let F be a covariant (respectively, contravariant) functor of the type considered in part (A), and apply the construction of (A) to the connected simple systems C (S) and C∗(S). The resulting direct (respectively, inverse) limit graded R-modules are, for simplicity, denoted by F̃C (S) and F̃C∗(S) respectively. If (P_{1}, P_{0}) and (P ∗_{1}, P ∗_{0}) are nested index pairs for S, the former in forward time, the latter in reverse, then if F is covariant (respectively, contravariant) set \overline{F(p^{±})} = ι^{X} ^{∘} F(p^{±}) (respectively, set \overline{F(p^{±})} := F (p^{±})^{∘} π_{X}) where X := P_{1}/P_{0} if the plus sign is taken, but X := P ∗_{1}/P ∗_{0} if the minus sign is taken. Call \overline{F(p^{±})} the canonical homomorphism. When F is covariant (respectively contravariant), \overline{F(p^{±})} may also be called the canonical injection (projection). Note that \overline{F(p^{±})} will be an X -isomorphism if F(p^{±}) is.

With F := H_{∗}( ·; G), the singular homology functor with coefficients in the Rmodule G, the construction of part (A) applied to C (S) and C∗(S) yields, respectively, the graded R-modules H̃_{∗}C (S; G) and H̃_{∗}C∗(S; G) with the former referred to as the graded singular homology module (coefficients in G) of the forward time Conley index of S and the latter referred to as the graded singular homology module (coefficients in G) of the reverse time Conley index of S.

If instead one takes the singular cohomology functor F = H^{∗}( ·; G), one obtains the graded singular cohomology modules (coefficients in G) of the forward and reverse time Conley indices of S as inverse limits denoted by H̃^{∗}C (S; G) and H̃^{∗}C∗(S; G), respectively. Obviously, the homology and cohomology modules of any Conley index obtained using any other homology or cohomology theory can be similarly constructed and will be referred to as needed. Here and throughout this work, the phrases “homology theory” and “cohomology theory” are used in the broad sense to include extraordinary theories, i.e., ones that fail to satisfy the dimension axiom although the other Eilenberg-Steenrod axioms are satisfied.

The following notation and glosses are used when F = H_{∗}( ·; G) and (P_{1}, P_{0}) is a nested index pair for S in forward time. For simplicity, p̄_{∗}^{+}: H_{∗}(P_{1}, P_{0}; G) → H̃_{∗}C (S; G) is used to denote the canonical homomorphism. Also, α ∈ H_{∗}(P_{1}, P_{0}; G) is said to represent α ∈ H̃_{∗}C (S; G) if, and only if, p̄_{∗}^{+}α = α. Similarly, a singular chain c that is a relative cycle in P_{1} modulo P_{0} with homology class α is said to represent α. Analogous notation is used in the reverse time case.

(C) Let us note that the singular homology and cohomology modules of a Conley index as just defined are in fact isomorphic to the homology and cohomology modules of a chain complex in ∂_{R}M′. In fact, for any connected simple system K that is a subcategory of T^{∗′}, apply the construction described in part (A) with F̃ = Δ̃_{∗}, the reduced singular chain functor with coefficients in R. The direct limit chain complex, Δ̃_{∗}(K), is called the singular chain complex of K with coefficients in R. The homology and cohomology of the chain complex Δ̃_{∗}(K) ⊗ G are isomorphic to H̃_{∗}(K; G) and H̃^{∗}(K; G), respectively, as follows from the observation made in (A) that the canonical homomorphisms to the direct and inverse limits are isomorphisms. In the case of homology, this also follows because the direct limit of the homology modules of the constituent chain complexes in a direct system of chain complexes is isomorphic to the homology of the direct limit of the chain complexes; see [Sp, Theorem 4.1.7].

D. Sign Conventions for Products in Homology and Cohomology. The statement and verification of the evaluations of the intersection pairing in Chapter 7 and of the product formulae in §8.D depend upon the choice of sign convention in the definition of the exterior cohomology product. Here we adopt the sign conventions of [D]. Thus, with all coefficients taken in R, if y_{i} ∈ H^{∗}(X_{i}, A_{i}) and η_{i} ∈ H_{∗}(X_{i}, A_{i}), i = 1, 2, then define the exterior cohomology product so that the following duality statement relating the exterior products on cohomology and homology holds:

⟨y_{1} × y_{2}, η_{1} × η_{2}⟩ = (−1)^{|η_{1}||y_{2}|}⟨y_{1}, η_{1}⟩⟨y_{2}, η_{2}⟩ .

Other authors define the exterior cohomology product so that in the corresponding duality statement there is no alternating sign; e.g., see [Sp]. The choice of sign convention affects the sign that appears in the standard formulae relating exterior products and cup and cap products. Specifically, for i = 1, 2, if y_{i} ∈ H^{∗}(X_{i}, A_{i}) and η_{i} ∈ H_{∗}(X_{i}, A_{i} ∪ B_{i}) and appropriate excision conditions are met, then in H_{∗}((X_{1}, B_{1}) × (X_{2}, B_{2})) as a consequence of using the sign conventions of [D]

(1.3)  (y_{1} × y_{2}) ⌢ (η_{1} × η_{2}) = (−1)^{|y_{2}||η_{1}|}(y_{1} ⌢ η_{1}) × (y_{2} ⌢ η_{2})

whereas if the sign conventions of [Sp] were used the exponent on minus one would be |y_{1}|(|η_{2}| − |y_{2}|).

Mnemonically, for the exterior, cup, and cap products, when two adjacent elements in an expression are interchanged, then the sign of the expression changes by a factor of (−1) raised to the product of the degrees of the interchanged elements.