CHAPTER 4 CONSTRUCTION OF BILINEAR

PAIRINGS ON CONLEY INDICES

In this section S denotes an isolated invariant set of a flow in M. The main goal is to give proofs of Theorems 2.4 and 2.11. The latter yields existence of the intersection class pairing 𝔏 on H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S) into Ȟ_{∗}(S); the former yields existence of the homology intersection number pairing L on [H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S)]_{m} in the special case M := R^{m}. It is convenient in proving these results to obtain Theorem 2.4 as a simple corollary of a generalization, namely Theorem 4.7, showing that whenever there exists a functor on AIP(S) that assigns to each admissible pair of nested index pairs a bilinear pairing on the product of R-modules functorially produced from the index pairs in the admissible pair, then there is an associated induced pairing on the tensor product of the R-modules functorially produced in the manner of Definition 1.5(B) from the forward and reverse time Conley indices. In Theorem 4.7 all pairings take values in the same R-module, but the result can be extended to allow the pairings assigned to admissible pairs to take values in an inverse (direct) system of R-modules over the set of neighborhoods of the isolated invariant set directed by containment; however, the induced pairing then has its values in the inverse (direct) limit of the inverse (direct) system. This generalization is Corollary 4.8, and Theorem 2.11 follows straightforwardly from it.

A. The Existence of Admissible Pairs of Index Pairs. The aim of the next several definitions and propositions is to describe the class of regular index pairs and the class of index pairs with thick exit set. The former plays an important role in the constructions that make up the proofs of Theorems 2.4 and 2.11, the latter in the proof of Theorem 3.2. In particular, we show that an index pair in either class has the property that the quotient map from the pair to its index space induces an isomorphism on homology for any homology theory. It is not known to the author as of this writing if there exists an index pair relative to some flow that does not have this property. However, as noted in Remark 4.1.1 below, a slight enlargement of the exit set of any index pair yields a regular index pair that necessarily has the property. The definitions of these classes and of some auxiliary constructions are made relative to the forward time flow, but of course can be given relative to any continuous flow in M; in particular for the reverse time flow.

4.1 Definition. (A) A nested index pair (L_{1}, L_{0}) for the forward time flow is regular if, and only if, the map τ_{(L_{1},L_{0})}: L_{1} → [0, ∞] is continuous where

τ_{(L_{1},L_{0})}(u) := \{ \begin{matrix}sup\{t ≥ 0 : u · [0, t] ∩ L_{0} = ∅\} & if u ∈ L_{1} \ L_{0}; \\ 0 & otherwise. \end{matrix}

The notion of regular index pair is introduced by Salamon in [Sl] and is the immediate generalization of the pair formed by an isolating block and its exit set.

(B) A nested index pair (L_{1}, L_{0}) for S has thick exit set if, and only if, there exists K_{0} ⊂ int_{L_{1}}(L_{0}) so that (L_{1}, K_{0}) is a nested index pair for S. The notion of an index pair with thick exit set was first defined in [K4] and is needed, for example, in the proof of invariance of the intersection number pairing on Conley indices under continuation of the isolated invariant set when the ambient manifold is at best C^{0}.

(C) To prove that the quotient map from an index pair with thick exit set to its index space induces isomorphisms on homology and for other important uses below, let us introduce auxiliary index pairs that can be constructed from a given pair. Accordingly, for t ≥ 0, for ⟨N_{1}, N_{0}⟩ an index pair for S (not necessarily nested) in the forward time direction, define

N_{1}^{t} := {u ∈ N_{1} : u · [−t, 0] ⊂ N_{1}} ,

and define

N_{0}^{−t} := N_{0} ∪ {u ∈ N_{1} : ∃s ∈ [0, t], u · [0, s] ⊂ N_{1} and u · s ∈ N_{0}} .

Then it follows, but not immediately, from [K1, Proposition 4.9] that given any index pair ⟨N_{1}, N_{0}⟩ for S, for any r, s ≥ 0, (N_{1}^{r}, ∩N_{0}^{−s}) is a nested index pair for S. The appeal to the result of [K1] just cited is not immediate because the notion of index pair used in the current work is much broader than that used in [K1]. Specifically, all index pairs in [K1] are taken relative to some isolating neighborhood; i.e., an index pair ⟨K_{1}, K_{0}⟩ is an index pair relative to an isolating neighborhood N if, and only if, each member of the index pair is positively invariant relative to N, i.e., u ∈ K_{i} and u · [0, t] ⊂ N imply u · [0, t] ⊂ K_{i} for i = 0, 1. Unfortunately there are a number of instances below where an appeal to some result of [K1] is made where a similar problem arises. Fortunately, it is always possible to reduce the situation to one in which the results of [K1] apply directly and always in the same manner. This reduction will now be carried out to prove that (N_{1}^{r}, ∩N_{0}^{−s}) is indeed a nested index pair for S for any r, s ≥ 0; however, all other such reductions will be left to the reader without further comment.

To proceed with the reduction, set N̄_{1} := cl (N_{1} \ N_{0}) and N̄_{0} := N_{0} ∩ N̄_{1}. It is easily verified that (N̄_{1}, N̄_{0}) is an index pair relative to the isolating neighborhood N̄_{1}, whence the results of [K1] apply. In particular, Proposition 4.9 of [K1] yields that (N̄_{1}^{r}, ∩N̄_{0}^{−s}) is a nested index pair for S for each r, s ≥ 0. Then, because N_{1}^{r} = N̄_{1}^{r} ∪ (N_{1}^{r} ∩ N_{0}) and because N_{0}^{−s} = N̄_{0}^{−s} ∪ N_{0}, it is easy to verify that (N_{1}^{r}, ∩N_{0}^{−s}) is a nested index pair for S.

Further, for s > 0 sufficiently large, (N_{1}^{r}, ∩N_{0}^{−s}) has thick exit set. For it follows from [K1, Proposition 4.9] that for each r ≥ 0 and for s > 0 sufficiently large,

(4.1)  N_{0} ∩ N_{1}^{r} ⊂ int_{N_{1}^{r}}(N_{0}^{−s} ∩ N_{1}^{r}) .

4.1.1 Remark. A key observation in the development of the intersection pairing is that given any nested index pair (N_{1}, N_{0}) for S and a neighborhood W of N_{0}, then there exists a compact N′_{0} so that W ⊃ N′_{0} ⊃ N_{0} and (N_{1}, N′_{0}) is a regular index pair for S. For it is proved in [Sl, Lemma 5.3 and Remark 5.4] that given any nested index pair (N_{1}, N_{0}) for S, there exists a continuous Lyapunov function g: N_{1} → [0, 1] so that where N_{ε} := g^{−1}([0, ε]), for 0 < ε < 1, (N_{1}, N_{ε}) is a regular index pair for S and N_{0} = g^{−1}(0) ⊂ N_{ε}. Then, with V := int(W), the intersection of the nested family of compact sets {N_{ε} \V : 0 < ε < 1} equals N_{0} \V = ∅, whence N_{ε} ⊂ V for all sufficiently small ε > 0.

4.1.2 Remark. A second important observation in the development of the intersection pairing on Conley indices is that both the class of regular index pairs and the class of index pairs with thick exit set are closed under the formation of the index pair (N_{1}^{r}, ∩N_{0}^{−s}) from the nested pair (N_{1}, N_{0}). Regularity propagates because τ_{(N_{1}^{r},∩N_{0}^{−s})} = max{0, −s + τ_{(N_{1},N_{0}) |N r 1}} and thick exit sets because K ⊂ int_{N_{1}}(N_{0})

implies K ∩ N_{1}^{r} ⊂ int_{N_{1}^{r}}(N_{1}^{r} ∩ N_{0}^{−s}).

4.2 Proposition. If an index pair is either regular or has thick exit set, then the quotient map from the index pair to its index space induces isomorphisms on homology for any homology theory.

Proof. Let us work for the sake of argument with a nested index pair (L_{1}, L_{0}) with respect to the forward time flow. Note that the quotient map l^{+}: (L_{1}, L_{0}) → (L_{1}/L_{0}, [L_{0}]) has factorization

(L_{1}, L_{0}) ↪^{e}(L_{1} ∪ CL_{0}, CL_{0}) ⟶^{k} (L_{1} ∪ CL_{0}/CL_{0}, [CL_{0}]) ≡ (L_{1}/L_{0}, [L_{0}])

where CL_{0} is the unreduced cone on L_{0} and L_{1} ∪ CL_{0} is the mapping cone of the inclusion L_{0} ⊂ L_{1}. Because the excision axiom implies e induces isomorphisms on homology, the proposition follows if k too induces isomorphisms, and as CL_{0} is contractible, the Five Lemma applied to k and the sequences of the pairs (L_{1} ∪ CL_{0}, CL_{0}) and (L_{1} ∪CL_{0}/CL_{0}, [CL_{0}]) shows it sufficient to prove that the collapse map

k′: L_{1} ∪ CL_{0} → L_{1} ∪ CL_{0}/CL_{0}

is a homotopy equivalence as a map of spaces without basepoint.

According to [K2, Lemma 3.1], k′ will be a homotopy equivalence if we can exhibit (i) a deformation D of L_{1} whose restriction to U × [0, 1] for some closed neighborhood U of L_{0} is a weak deformation retraction of U into L_{0} and (ii) a continuous function φ: L_{1}: → [0, 1] satisfying L_{0} ⊂ φ^{−1}(1) and L_{1} \ U ⊂ φ^{−1}(0). In the present context L_{1} is a normal space so that a Urysohn function φ satisfying (ii) can always be found. If (L_{1}, L_{0}) is a regular index pair, then choose any c > 0, take U := τ_{(L_{1},L_{0})}^{−1}([0, c]) and define D(u, s) := u · s min{c, τ_{(L_{1},L_{0})}(u)}. If, however, (L_{1}, L_{0}) is not regular but has thick exit set, then by Remark 4.1.1 choose K_{0} ⊂ int_{L_{1}}(L_{0}) with (L_{1}, K_{0}) an index pair which without loss of generality can be assumed regular; else choose K′_{0} with K_{0} ⊂ K′_{0} ⊂ int_{L_{1}}(L_{0}) and (L_{1}, K′_{0}) regular. As was done to get (4.1) above, choose t > 0 so that L_{0} ⊂ int_{L_{1}}(L_{0}^{−t} ∩ L_{1}), take U := L_{0}^{−t} ∩ L_{1}, set c := max{τ_{(L_{1},K_{0})}(u) : u ∈ U}, and define D(u, s) := u · s min{c, τ_{(L_{1},K_{0})}(u)}.

Note that when (L_{1}, L_{0}) is regular, D is actually a strong deformation retraction of U onto L_{0} and we could instead invoke [Sp, Theorem 4.8.9] to get l^{+} inducing isomorphisms on homology. □

4.3 Definition. When M is C^{r} (r ≥ 1) and the flow in M is generated by a continuous vectorfield X, call an isolating block B for S a C^{r} isolating block with corners if B is a topological manifold with boundary and there exist C^{r} submanifolds Σ^{+} and Σ^{−} of M which together with B have the following additional properties:

(1) Σ^{±} if non-void is of codimension one in M;

(2) X is transverse to Σ^{±};

(3) B^{±} is a codimension zero C^{r} submanifold with boundary of Σ^{±};

(4) ∂B = B^{+} ∪ B^{−} and B^{+} ∩ B^{−} = ∂B^{±}. The existence of isolating blocks with corners is proved in [WY]. The submanifold B^{+} ∩ B^{−} is the set of “corners” of B and is sometimes referred to as the tangency set of B since the orbit through a point u ∈ B^{+} ∩ B^{−} locally lies exterior to B intersecting it only in u. As just defined a block with corners need not be a C^{r} manifold with corners since it is not assumed that Σ^{+} and Σ^{−} can be simultaneously linearized at points of B^{+} ∩ B^{−}. Such simultaneous linearization can be carried out, however, if Σ^{+} and Σ^{−} are transverse in which case B will be a C^{r} manifold with corners.

It is convenient in what follows to use the notation AIP(regular) to denote the class of pairs of index pairs that are AIP for S wherein both pairs are regular index pairs. Similarly, the notation AIP(thick) is used below to denote the class of AIP pairs for S wherein both pairs have thick exit set.

4.4 Proposition. There exist AIP(regular) and AIP(thick) pairs of index pairs for S. If U is any neighborhood of S, there exists N AIP(regular) for S so that N_{1} ∩ N∗_{1} ⊂ U. If M is C^{r}, r ≥ 1, and the flow is generated by a vectorfield of class C^{s}, 0 ≤ s ≤ r − 1, then N_{1}, N∗_{1}, and N_{1} ∩ N∗_{1} can be chosen to be C^{r} isolating blocks with corners.

Proof. Let U be a neighborhood of S and choose K ⊂ U so that K isolates S. The construction of isolating blocks in [Ch] although carried out for a complete flow on a compact metric space works equally well in the present setting and yields a block B for S with B ⊂ int_{M}(K). For any c, d ≥ 0 define

B_{c d} := τ_{B}^{−1}([c, ∞]) ∩ τ_{B}^{∗ −1}([d, ∞]) .

Then B_{c d} is an isolating block for S and is called a squeeze of B. Fix d > 0 and for 0 ≤ c < d set

(P_{1 c}, P_{0 c}) := (B_{0 d}, τ_{B_{0 d}}^{−1}([0, c])) , (P ∗_{1 c}, P ∗_{0 c}) := (B_{d 0}, τ_{B_{d 0}}^{∗ −1}([0, c]))

and define P_{c} := ((P_{1 c}, P_{0 c}), (P ∗_{1 c}, P ∗_{0 c})). Note that B_{0 d} (respectively, B_{d 0}) is a squeeze of B along its entrance (respectively, exit) set. It is easily verified that if 0 ≤ c < d, then P_{c} is an AIP(regular) pair of index pairs for S and is also AIP(thick) when 0 < c < d because

P_{0 0} ⊂ int_{B_{0 d}}(P_{0 c}) and P ∗_{0 0} ⊂ int_{B_{d 0}}(P ∗_{0 c}) .

Set N := P_{0}. Then N is AIP(regular) for S and N_{1} ∩ N∗_{1} = B_{d,d} ⊂ U. The isolating block B as constructed in [Ch] has B^{+} and B^{−}disjoint, whence if d is chosen sufficiently small, N_{0} = B^{+} and N∗_{0} = B^{−}. Also as N_{1}, N∗_{1} and N_{1} ∩ N∗_{1} are squeezes of a block, they are themselves blocks.

When M is C^{r}, r ≥ 1, and the flow is generated by a continuous vectorfield X, then a result of W. Wilson and J. Yorke (see Theorem 2.1 and the Remark at the top of p. 114 of [WY]) shows that for some open neighborhood Ω of S with Ω ⊂ K, there exist continuous Lyapunov functions g_{+}, g_{−}: Ω → R^{+} (i.e., the Lie derivative Xg_{±} exists and is continuous on Ω) with the properties

(1) A^{±}(Ω) = g_{±}^{−1}(0);

(2) Xg_{+} > 0 on Ω \ A^{+}(Ω) and Xg_{−} < 0 on Ω \ A^{−}(Ω);

(3) g_{±} is C^{r} on Ω \ A^{±}(Ω). The prototypical examples of g_{+} and g_{−} arise from the system ẋ = x, ẏ = −y for x ∈ R^{n}, y ∈ R^{p} which has S = {0} as isolated invariant set: Take g_{+}(x, y) = ∥x∥^{2} and g_{−}(x, y) = ∥y∥^{2} where the norms are the Euclidean norms. The reader will probably find it helpful to keep this example in mind while reading the following construction of blocks.

The necessary C^{r} isolating blocks with corners can be constructed from g_{+} and g_{−} via a minor variation on the method of [WY, Theorem 2.4]. In particular, the blocks finally produced will be selected from the family of neighborhoods of S whose members have the form

C_{p q} := g_{+}^{−1}([0, q]) ∩ g_{−}^{−1}([0, p]) for p, q > 0.

To eventually obtain restrictions on p and q ensuring that C_{p q} is compact and isolates S, choose a relatively compact, open neighborhood W of S with cl_{M} (W) ⊂ Ω. Then, since g_{±} ≥ 0 , condition (1) implies that some member of the nested family of compact sets

{(g_{+} + g_{−})^{−1}([0, (i + 1)^{−1}]) ∩ K \ W }_{i∈N}

must be void. Therefore, for some d > 0, (g_{+} + g_{−})^{−1}([0, 2d]) ⊂ cl_{M} (W), whence (g_{+} + g_{−})^{−1}([0, 2d]) is compact and isolates S. It follows that C_{p q} is compact and isolates S whenever p, q ∈ ]0, d[. Further, for such p and q, note that the topological boundary of C_{p q} (i.e., its frontier relative to M) satisfies

∂_{M}(C_{p q}) = C_{p q} ∩ (g_{+}^{−1}(q) ∪ g_{−}^{−1}(p)) .

Observe too that condition (2) implies that

C_{p q}^{+} := (τ_{C_{p q}})^{−1}(0) = g_{+}^{−1}(q) ∩ C_{p q} and C_{p q}^{−} := (τ∗_{C_{p q}})^{−1}(0) = g_{−}^{−1}(p) ∩ C_{p q} ,

whence the exit and entrance sets of C_{p q} are closed relative to M. As observed in Definition 1.4(C) this implies that C_{p q} is a block for S.

Set Ω^{±} := Ω \ A^{±}(Ω), set Ω^{R} := Ω^{+} ∩ Ω^{−}, set Ω_{0} := Ω^{+} ∪ Ω^{−} = Ω \ S and observe that conditions (2) and (3) on g_{±} imply that each positive number is a regular value of g_{±}|Ω^{±}. Define

Σ_{ε}^{±} := (g_{±}|Ω^{±})^{−1}(ε) = g_{±}^{−1}(ε) for ε > 0.

For p, q ∈ ]0, d[, it follows that C_{p q} will be a C^{r} isolating block with corners for S if C_{p q}^{+} ∩ C_{p q}^{−} has the structure of a C^{r} submanifold in the topology inherited from M.

To obtain such structure, first, observe that conditions (2) and (3) on g_{±} imply that each c ∈ R is a regular value of (g_{+} − g_{−})|Ω^{R} since X(g_{+} − g_{−}) > 0 on Ω_{0} ⊃ Ω^{R}. Hence

h_{c} := ((g_{+} − g_{−})|Ω^{R})^{−1}(c)

is either void or a codimension one submanifold of Ω^{R}; hence of M.

The blocks N_{1} and N∗_{1} can now be defined as follows. Choose c ∈ ]0, d[; choose ε_{1} ∈ ]0, d − c[ so that ε_{1} is a regular value of g_{−}|h_{c}, g_{−}|h_{0}, and g_{+}|h_{−c}; set ε_{2} := c + ε_{1}. Set N_{1} := C_{ε_{1} ε_{2}}, set N∗_{1} := C_{ε_{2} ε_{1}}, note that N_{1} ∩ N∗_{1} = C_{ε_{1} ε_{1}}, and observe that

N_{1}^{+} ∩ N_{1}^{−} = (g_{−}|h_{c})^{−1}(ε_{1}) ,
N_{1}^{∗ +} ∩ N_{1}^{∗ −} = (g_{+}|h_{−c})^{−1}(ε_{1}) ,

(N_{1} ∩ N∗_{1})^{+} ∩ (N_{1} ∩ N∗_{1})^{−} = (g_{−}|h_{0})^{−1}(ε_{1}) .

Hence, by the choice of ε_{1} as regular value, each of these three intersections of exit and entrance set is a C^{r} manifold in M that is either void or of codimension two in M. Finally, observe that the pair of regular index pairs ((N_{1}, N_{1}^{+}), (N∗_{1}, N_{1}^{∗ −})) satisfies the off-diagonal condition since Σ_{ε_{2}}^{±} is disjoint from g_{±}^{−1}([0, ε_{1}]) by virtue of the inequality ε_{1} < ε_{2}. □

B. Functorially Produced Pairings on the Conley Indices. The next lemma proves the existence of several AIP(regular) pairs of index pairs that mediate between two such pairs via inclusion maps and is the basis for showing that a functor on AIP(S) with values that are bilinear pairings on the product of (co)homology modules of the index pairs in an admissible pair induces a pairing on the tensor product of (co)homology modules of the forward and reverse time Conley indices. Prior to the statement of the lemma we need the following.

4.5 Definition. (A) F ⊂ ob(AIP(S)) is a pre-cofinal family for S if, and only if, for each U ∈ N (S), there exists P ∈ F with P_{1} ∩ P ∗_{1} ⊂ U. Proposition 4.4 implies that there exist pre-cofinal families for S, and in particular, ob(AIP(S)) is one such family.

(B) A pre-cofinal family F for S is a saturated pre-cofinal family if, and only if, it satisfies the following condition:

(†) Q ∈ ob(AIP(S)) and Q_{1} ∩ Q∗_{1} ⊂ P_{1} ∩ P ∗_{1} for some P ∈ F implies Q ∈ F.

Evidently, if F is pre-cofinal for S, then the set of Q ∈ ob(AIP(S)) satisfying the premise of condition (†), viz., Q_{1} ∩ Q∗_{1} ⊂ P_{1} ∩ P ∗_{1} for some P ∈ F, is a saturated pre-cofinal family for S and is the smallest one containing F.

4.6 Lemma. Let F be a saturated pre-cofinal family for S. If, for j = 1, 2, N_{j} ∈ F is AIP(regular) for S and has the additional property that both N_{1 j} and N∗_{1 j} are isolating neighborhoods for S, then there exists L, P_{1}, P_{2}, Q_{1}, Q_{2} ∈ F all AIP(regular) for S and inclusion morphisms

(4.2)  L ↪ P_{j} ↩ Q_{j} ↪ N_{j}  (j = 1, 2) .

Also, if AIP(thick) is substituted for AIP(regular) in the previous sentence, the resulting statement is true.

In the proof of Lemma 4.6, local stable and unstable sets of S are always defined relative to the forward time flow, never the reverse time flow.

Proof of Lemma 4.6. For j = 1, 2, let N_{j} be an AIP(regular) pair of index pairs for S and assume N_{j} ∈ F. As noted in Remark 4.2.1, for any r, s ≥ 0, each index pair in the pair of auxiliary index pairs

(4.3)  ((N_{1 j}^{r}, ∩N_{0 j}^{−s}), (N_{1 j}^{∗ r}, ∩N_{0 j}^{∗ −s}))

is regular with respect to the forward or reverse time flow as appropriate.

Next, let us show that for each s > 0 there exists T (s) > 0 so that if r ≥ T (s), then the pair of index pairs given in (4.3) satisfies the off-diagonal condition; hence is AIP(regular) for S. Also this pair of index pairs will be in F because F is saturated and because for all r > 0, N_{1 j}^{r} ∩ N_{1 j}^{∗ r} ⊂ N_{1 j} ∩ N∗_{1 j}. Fix s > 0 and note that

(4.4)  W^{u}(S; N_{1 j}) ∩ N∗ −s_{0 j} ∩ N∗_{1 j} = ∅  (j = 1, 2) .

For if u is in the intersection, then for some t, 0 < t ≤ s and u · [−t, 0] ⊂ N∗_{1 j} and u · −t ∈ N∗_{0 j}, but also u · −t ∈ N_{1 j}, a contradiction as N_{1 j} and N∗_{0 j} are disjoint. The analogous argument with respect to the reverse time flow shows that

(4.5)  W^{s}(S; N∗_{1 j}) ∩ N_{0 j}^{−s} ∩ N_{1 j} = ∅  (j = 1, 2) .

Because N_{1 j} and N∗_{1 j} isolate S, the sets W^{u}(S; N_{1 j}) and W^{s}(S; N∗_{1 j}) are closed. (This need not be true if N_{1 j} or N∗_{1 j} does not isolate S; e.g., if S is the repellor of a repellor-attractor pair and N_{1 j} is also a neighborhood of the attractor, then W^{u}(S; N_{1 j}) will not be closed if their exist heteroclinic orbits in N_{1 j} from S to its complementary attractor—see [K2, Lemma 4.1].) By normality, (4.4) and (4.5) yield sets U_{j} and U∗_{j}, open in M, satisfying

(4.6)  W^{u}(S; N_{1 j}) ⊂ U_{j} , W^{s}(S; N∗_{1 j}) ⊂ U∗_{j} (j = 1, 2)

and

(4.7)  U_{j} ∩ N∗ −s_{0 j} ∩ N∗_{1 j} = ∅ , U∗_{j} ∩ N_{0 j}^{−s} ∩ N_{1 j} = ∅ (j = 1, 2) .

Then by (4.6) and [K1, Lemma 3.3(1)], there exists T (s) > 0 so that for r ≥ T (s),

N_{1 j}^{r} ⊂ U_{j} and N_{1 j}^{∗ r} ⊂ U∗_{j} (j = 1, 2) .

It then follows from (4.7) and Remark 4.1.2 on the regularity of auxiliary index pairs that the pair of index pairs in (4.3) is AIP(regular) for S.

The index pairs (L_{1}, L_{0}) and (L∗_{1}, L∗_{0}) will be constructed from index pairs (N_{1 3}, N_{0 3}) and (N∗_{1 3}, N∗_{0 3}) defined as follows:

\begin{matrix}N_{1 3} := N_{1 1} ∩ N_{1 2} , & N_{0 3} := N_{1 3} ∩ (N_{0 1} ∪ N_{0 2}) , \\ N∗_{1 3} := N∗_{1 1} ∩ N∗_{1 2} , & N∗_{0 3} := N∗_{1 3} ∩ (N∗_{0 1} ∪ N∗_{0 2}) . \end{matrix}

It follows from the proof of Proposition A1 in the appendix to [K3] that (N_{1 3}, N_{0 3}) and (N∗_{1 3}, N∗_{0 3}) are index pairs for S, the first with respect to the forward time flow, the second with respect to the reverse time flow. Further, both of these index pairs are regular. For example, τ_{(N_{1 3},N_{0 3})} equals min{τ_{(N_{1 j},N_{0 j})}|N_{1 3} : j = 1, 2}, hence is continuous; similarly τ_{(N∗_{1 3},N∗_{0 3})} is continuous.

We proceed to show that we can find s > 0 and r ≥ T (s) so that if P_{j} is the pair of index pairs given in (4.3) above (j = 1, 2) and if the pair of index pairs L is defined by the equalities

(4.8)  \begin{aligned}L_{1} & := N_{1 3}^{r} , \\ L∗_{1} & := N_{1 3}^{∗ r} , \end{aligned} \begin{aligned}L_{0} & := N_{1 3}^{r} ∩ N_{0 3} , \\ L∗_{0} & := N_{1 3}^{∗ r} ∩ N∗_{0 3} , \end{aligned}

then L and P_{j} are AIP(regular) for S and there exists an inclusion morphism L ↪ P_{j}. Because N_{1 3} ⊂ N_{1 j} (j = 1, 2) and because both isolate S, by [K1, Lemma 3.3(4)] there exist s′ > r′ > 0 so that

N_{1 3}^{r′} ∩ N_{0 3}^{−r′} ∩ N_{1 j} \ N_{0 j}^{−s′} = ∅ (j = 1, 2) ,

and since N_{0 3} ⊂ N_{0 3}^{−r′} and because for each r ≥ r′ and s ≥ s′

N_{1 3}^{r} ⊂ N_{1 3}^{r′} ⊂ N_{1 j} and N_{0 j}^{−s′} ⊂ N_{0 j}^{−s} (j = 1, 2) ,

it follows that for each r ≥ r′ and s ≥ s′,

(4.9)  N_{1 3}^{r} ∩ N_{0 3} ⊂ N_{0 j}^{−s} (j = 1, 2) .

Similarly there exist s′′ > r′′ > 0 so that for r ≥ r′′ and s ≥ s′′,

(4.10)  N_{1 3}^{∗ r} ∩ N∗_{0 3} ⊂ N∗ −s_{0 j} (j = 1, 2) .

Choose s ≥ max{s′, s′′} and choose r ≥ max{r′, r′′, T (s)}. Use these values of s and r to define (L_{1}, L_{0}) and (L∗_{1}, L∗_{0}) according to the prescription of (4.8), and define P_{j} (j = 1, 2) to be the pair of index pairs in (4.3) for these values of s and r. Then P_{j} is AIP(regular) for S by choice of r > T (s) and the index pairs in L satisfy the off-diagonal condition as follows straightforwardly from the definition of these index pairs because the pair of index pairs N_{j} satisfies this condition for j = 1, 2. Hence L is AIP(regular) for S, and for j = 1, 2, there exists an inclusion morphism L ↪ P_{j} as a consequence of (4.8), (4.9), and (4.10). Note that the existence of the inclusion morphism implies that L ∈ F because F is saturated and P_{j} ∈ F.

To finish, use the chosen value of r to define

Q_{j} := ((N_{1 j}^{r}, ∩N_{0 j}), (N_{1 j}^{∗ r}, ∩N∗_{0 j})) (j = 1, 2) .

It follows easily that Q_{j} so defined is AIP(regular) for S because N_{j} is and that the inclusion morphisms indicated in (4.2) exist. As with L, the existence of the inclusion morphisms implies that Q_{j} ∈ F, j = 1, 2.

It is straightforward to check that if the initial pairs are AIP(thick) rather than AIP(regular), the construction given above yields pairs that are AIP(thick). For example, if K_{j} ⊂ int_{N_{1 j}}(N_{0 j}), then K_{j} ⊂ V_{j} ∩ N_{1 j} ⊂ N_{0 j} for some V_{j} open in M (j = 1, 2). Hence, N_{1 3} ∩ (K_{1} ∪ K_{2}) ⊂ N_{1 3} ∩ (V_{1} ∪ V_{2}) ⊂ N_{0 3} from which it follows that (N_{1 3}, N_{0 3}) has thick exit set if (N_{1 j}, N_{0 j}) does for j = 1, 2. □

For the main result of this section, assume that F and G are covariant functors on the homotopy category of compact Hausdorff pairs to the category of left R-modules and that K is a left R-module.

4.7 Theorem. Let F be a saturated pre-cofinal family for S. For each P ∈ F assume given a pairing of R-modules

B_{P}: F(P_{1}, P_{0}) ⊗ G(P ∗_{1}, P ∗_{0}) → K

so that for any Q ∈ F, if there exist inclusions

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

then the diagram

(4.12)  \begin{matrix}F(Q_{1}, Q_{0}) ⊗ G(Q∗_{1}, Q∗_{0}) & ⟶^{B_{Q}} & K \\ ^{F(i) ⊗ G(j)}↓ & & ↓^{1_{K}} \\ F(P_{1}, P_{0}) ⊗ G(P ∗_{1}, P ∗_{0}) & ⟶^{B_{P}} & K \end{matrix}

commutes. Also assume that for each P ∈ F, 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}])

transform to R-module isomorphisms under F and G respectively. Then \overline{F(p^{+})} ⊗ \overline{G(p^{−})} is an isomorphism, and there exists a unique pairing of R-modules

B: FC (S) ⊗ GC∗(S) → K

satisfying

(4.13)  B = B_{P} ^{∘} (\overline{F(p^{+})} ⊗ \overline{G(p^{−})})^{−1}  for each P ∈ F.

4.7.1 Remark. If F and G were assumed to be contravariant rather than covariant, then the statement dual to that of Theorem 4.7 obtained by reversing the direction of the vertical arrows in diagram (4.12) and replacing statement (4.13) with the statement

(4.13∗)  B = B_{P} ^{∘} \overline{F(p^{+})} ⊗ \overline{G(p^{−})}  for each P ∈ F

is true and has a proof analogous to the one given below for Theorem 4.7; details are left to the reader.

4.7.2 Remark. If F = ob(AIP(S)), the hypotheses on the family of pairings B_{P}, P ∈ ob(AIP(S)), imply that this family defines a covariant functor on AIP(S) with values in the category of R-bilinear maps into K.

The proof of the theorem is delayed until after the proof of the following corollary whose proof is the reason for introducing the notion of saturated pre-cofinal families into the previous theorem rather than just stating and proving it for F = ob(AIP(S)).

For the corollary, let R, F and G be as for the theorem. Also, direct N (S) by containment and let { {K_{U}}_{U∈N (S)}, {i_{U}^{V} }_{U⪯V ∈N (S)}} be an inverse system of R-modules and homomorphisms.

4.8 Corollary. Let F be a saturated pre-cofinal family for S. For each P ∈ F assume given a pairing of R-modules

B_{P}: F(P_{1}, P_{0}) ⊗ G(P ∗_{1}, P ∗_{0}) → K_{P_{1}∩P ∗_{1}}

so that for any Q ∈ F, if there exist inclusions

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

then the diagram

(4.15)  \begin{matrix}F(Q_{1}, Q_{0}) ⊗ G(Q∗_{1}, Q∗_{0}) & ⟶^{B_{Q}} & K_{Q_{1}∩Q∗_{1}} \\ _{F(i) ⊗ G(j)}↓ & & ↓^{i_{P_{1}∩P∗_{1}}^{Q_{1}∩Q∗_{1}}} \\ F(P_{1}, P_{0}) ⊗ G(P ∗_{1}, P ∗_{0}) & ⟶^{B_{P}} & K_{P_{1}∩P ∗_{1}} \end{matrix}

commutes. Also assume that for each P ∈ F, 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}])

transform to R-module isomorphisms under F and G, respectively. For each V ∈ N (S), choose P ∈ F with P_{1} ∩ P ∗_{1} ⊂ V and define B_{V}^{S}: F̃C (S) ⊗ G̃C∗(S) → K_{V} by

(4.16)  B_{V}^{S} := i_{V}^{P_{1}∩P ∗_{1}} ^{∘} B_{P} ^{∘} (\overline{F(p^{+})} ⊗ \overline{G(p^{−})})^{−1} .

Then the definition of B_{V}^{S} is independent of the choice of P ∈ F satisfying P_{1}∩P ∗_{1} ⊂ V, and

(4.17)  i_{U}^{V} ^{∘} B_{V}^{S} = B_{U}^{S}  for U ⊃ V ∈ N (S),

whence passage to the inverse limit yields a pairing

B: FC (S) ⊗ GC∗(S) → \underleftarrow{lim} K_{U} .

4.8.1 Remark. As for the theorem, there is a version of the corollary for F and G contravariant where the inverse system of R-modules and its limit is replaced by a direct system and its limit, where the vertical arrows of (4.15) are reversed, where B_{V}^{S} in (4.16) is replaced by

(4.16∗)  B_{S}^{V} := i_{V}^{P_{1}∩P ∗_{1}} ^{∘} B_{P} ^{∘} (\overline{F(p^{+})} ⊗ \overline{G(p^{−})}) ,

and where (4.17) is replaced by

(4.17∗)  i_{U}^{V} ^{∘} B_{S}^{U} = B_{S}^{V} for U ⊃ V ∈ N (S) .

Proof of Corollary 4.8. For each V ∈ N (S), set F_{V} := {P ∈ F : P_{1} ∩ P ∗_{1} ⊂ V } and note that F_{V} is a saturated pre-cofinal family for S because F is. For each

P ∈ F_{V}, define B_{V}^{P} := i_{V}^{P_{1}∩P ∗_{1}} ^{∘} B_{P}, a pairing on F(P_{1}, P_{0}) ⊗ G(P ∗_{1}, P ∗_{0}) to K_{V}.

Application of Theorem 4.7 to the family of pairings B_{V}^{P}, P ∈ F_{V}, yields a pairing B_{V}^{S} for which the equality of (4.16) holds for each P ∈ F_{V}; i.e., the definition of B_{V}^{S} as given by (4.16) is independent of the choice of P ∈ F_{V} as desired. Thus, if U ⊃ V ∈ N (S), by choosing P ∈ F satisfying P_{1} ∩ P ∗_{1} ⊂ V to define B_{U}^{S}, we see immediately from the properties of an inverse system that (4.17) holds, whence the family of transformations B_{V}^{S}, V ∈ N (S), pass to the inverse limit yielding B := \underleftarrow{lim} B_{V}^{S}. □

Proof of Theorem 4.7. Let P ∈ F and let p^{+} and p^{−} be the quotient maps as given in the statement of the theorem. Because F(p^{+}) and G(p^{−}) are by hypothesis isomorphisms, so too (e.g., see [Sp, Lemmas 4.1.4–5]) is \overline{F(p^{+})}⊗\overline{G(p^{−})} where \overline{F(p^{+})} and \overline{G(p^{−})} are defined as in Definition 4.1(A). Define

\overline{B}_{P} := B_{P}^{∘}(\overline{F(p^{+})} ⊗ \overline{G(p^{−})} )^{−1} for P ∈ F

and define an equivalence relation on F by P ∼^{B} Q if, and only if, \overline{B}_{P} = \overline{B}_{Q}.

To complete the proof it suffices to show that there is exactly one equivalence class for this equivalence relation, for we can then define B := \overline{B}_{P} for any P ∈ F yet have (4.13) hold. Obviously, (4.13) implies the uniqueness of B.

Basic to our proof is the observation that if P, Q ∈ F and there exists an inclusion morphism Q ↪ P, i.e., P and Q satisfy the inclusions (4.11), then P ∼^{B} Q. We call this observation “invariance of B under inclusions.” To prove it, first observe that the statement P ∼^{B} Q is equivalent to the commutativity of the diagram

(4.18)

F̃C (S) ⊗ G̃C∗(S) ⟵^{\overline{F(p^{+})} ⊗ \overline{G(p^{−})}} F(P_{1}, P_{0}) ⊗ G(P ∗_{1}, P ∗_{0}) ⟶^{B_{P}} K
1_{F̃C(S) ⊗ G̃C∗(S)}↑  ^{F(i) ⊗ G(j)} ↑  ↑^{1_{K}}
F̃C (S) ⊗ G̃C∗(S) ⟵^{\overline{F(q^{+})} ⊗ \overline{G(q^{−})}} F(Q_{1}, Q_{0}) ⊗ G(Q∗_{1}, Q∗_{0}) ⟶^{B_{Q}} K .

The right-hand rectangle is just diagram (4.12) which is commutative by hypothesis if P, Q ∈ F and satisfy the inclusions (4.11). The left-hand rectangle commutes when the inclusions (4.11) are satisfied because it expands (after rotating it counterclockwise 90^{∘}) to the diagram

(4.19)

F(P_{1}, P_{0}) ⊗ G(P ∗_{1}, P ∗_{0}) ⟵^{F(i) ⊗ G(j)} F(Q_{1}, Q_{0}) ⊗ G(Q∗_{1}, Q∗_{0})
F(p^{+}) ⊗ G(p^{−})↓  ↓^{F(q^{+}) ⊗ G(q^{−})}
F̃(P_{1}/P_{0}) ⊗ G̃(P ∗_{1}/P ∗_{0}) ⟵^{F^{}˜ (ī) ⊗ G^{}˜ (j̄)}  F̃(Q_{1}/Q_{0}) ⊗ G̃(Q∗_{1}/Q∗_{0})
↓  ↓
F̃C (S) ⊗ G̃C∗(S) ⟵^{1_{F̃C(S) ⊗ G̃C∗(S)}}  F̃C (S) ⊗ G̃C∗(S)

where ī and j̄are the maps on quotients induced by the inclusions i and j and where the unlabeled vertical arrows are the tensor product of canonical maps sending an element to its equivalence class: the bottom rectangle commutes by definition of the equivalence relations defining F̃C (S) and G̃C∗(S); the top rectangle commutes because it is the transform by the functor F ⊗ G of a commutative diagram of pairs of compact Hausdorff pairs and pairs of continuous maps between such pairs of pairs.

Claim I. For each Q ∈ F, there exists P ∈ F an AIP(regular) pair of index pairs for S and an inclusion morphism Q ↪ P; hence each pair of index pairs in F that is AIP for S is equivalent to an AIP(regular) pair of index pairs for S in F.

Proof of Claim I. Let Q ∈ F; write Q = ((Q_{1}, Q_{0}), (Q∗_{1}, Q∗_{0})). Choose an open neighborhood U of Q_{0} that is disjoint from Q∗_{1} and choose an open neighborhood U∗ of Q∗_{0} that is disjoint from Q_{1}. This choice of neighborhoods is possible because Q ∈ M_{Δ̸ c}^{(2)} and M is normal. As follows from Remark 4.1.1, we can find compact sets Q′_{0} and Q′∗_{0} satisfying the inclusions

Q_{0} ⊂ Q′_{0} ⊂ U and Q∗_{0} ⊂ Q′ ∗_{0} ⊂ U∗

so that (Q_{1}, Q′_{0}) is a regular index pair for S in forward time and (Q∗_{1}, Q′ ∗_{0}) is a regular index pair for S in reverse time. Set P := ((Q_{1}, Q′_{0}), (Q∗_{1}, Q′ ∗_{0})). By the choice of U and U∗ it follows that P is AIP(regular) for S, and by construction, P_{1} ∩ P ∗_{1} = Q_{1} ∩ Q∗_{1}, whence P ∈ F because F is saturated. Also by construction, there exists an inclusion morphism Q ↪ P. Hence by the invariance of B under inclusions Q ∼^{B} P. □

Claim II. Any two AIP(regular) pairs of index pairs for S, both elements of F, are ∼^{B}-equivalent.

Proof of Claim II. In proving the claim we need only consider AIP(regular) pairs of index pairs for S wherein the first compact set of each index pair in the pair of index pairs is an isolating neighborhood for S. For suppose N is AIP(regular) for S. If say N_{1} does not isolate S, then define

N′_{1} := cl (N_{1} \ N_{0}) and N′_{0} := N_{0} ∩ N′_{1} .

Then (N′_{1}, N′_{0}) is a regular index pair since τ_{(N′_{1},N′_{0})} is the restriction to N′_{1} of τ_{(N_{1},N_{0})} and also N′_{1} isolates S. Thus ((N′_{1}, N′_{0}), (N∗_{1}, N∗_{0})) is AIP(regular) for S, and by construction (N′_{1}, N′_{0}) ⊂ (N_{1}, N_{0}). Because F is saturated and N ∈ F, the inclusion implies that ((N′_{1}, N′_{0}), (N∗_{1}, N∗_{0})) ∈ F, whence ((N_{1}, N_{0}), (N∗_{1}, N∗_{0})) is ∼^{B}- equivalent to ((N′_{1}, N′_{0}), (N∗_{1}, N∗_{0})) by invariance of B under inclusions. Similarly we can assume that N∗_{1} isolates S.

Thus assume given N_{1}, N_{2} ∈ F both AIP(regular) for S and with the property that N_{1 j} and N∗_{1 j} are isolating neighborhoods of S for j = 1, 2. From the transitivity and symmetry of the equivalence relation ∼^{B}, it is then a simple consequence of Lemma 4.6 and invariance of B under inclusions that N_{1} ∼^{B} N_{2}. □

By Claims I and II there is exactly one ∼^{B}-equivalence class. □

C. The Proofs of Theorems 2.4 and 2.11. For the proof of Theorem 2.4 let S be an isolated invariant set of a flow in R^{m}. All homology modules occurring in the proof are taken with coefficients in R.

4.9 Proof of Theorem 2.4. Let P, Q ∈ ob(AIP(S)) and assume there exists an inclusion morphism (i, j): Q ↪ P. It follows that there is a commutative diagram of inclusion maps

(Q_{1}, Q_{0}) × (Q∗_{1}, Q∗_{0})  (P_{1}, P_{0}) × (P ∗_{1}, P ∗_{0}) (R^{m} × R^{m}, R^{m} × R^{m} \ Δ)

whence naturality of homology cross-products and Definition 2.2 imply that

(4.20)  \begin{matrix}[H_{∗}(Q_{1}, Q_{0}) ⊗ H_{∗}(Q∗_{1}, Q∗_{0})]_{m} & \underrightarrow{L_{Q}} & H_{m}(R^{m}, R^{m} \ \{0\}) \\ ^{i_{∗}⊗j_{∗}}↓ & & ↓^{1_{∗}} \\ [H_{∗}(P_{1}, P_{0}) ⊗ H_{∗}(P ∗_{1}, P ∗_{0})]_{m} & \underrightarrow{L_{P}} & H_{m}(R^{m}, R^{m} \ \{0\}) \end{matrix}

is a commutative diagram; i.e., intersection numbers as defined in Definition 2.2 are invariant under inclusion maps. As diagram (4.20) corresponds in the present context to diagram (4.12) of Theorem 4.7, the desired result is immediate from that theorem by applying it to each family of summands {L_{N k} : N ∈ ob(AIP(S))} for each integer k ∈ [0, m] thereby obtaining a pairing L_{k}: H̃_{m−k}C (S) ⊗ H̃_{k}C∗(S) → H_{m}(R^{m}, R^{m} \ {0}) and then setting L := ⨁_{k=0}^{m} L_{k}. □

For the proofs of Lemma 2.10 and Theorem 2.11 assume G and G′ are R-modules; for the proof of Theorem 2.11, S is an isolated invariant set of a flow in M.

4.10 Proof of Lemma 2.10. The naturality of the pairings 𝔏_{X}, X ∈ M_{Δ̸}^{(2)} will be

proved; the naturality of the pairings ^{#}𝔏_{X}, X ∈ M_{Δ̸}^{(2)} follows trivially. The proof of naturality of the pairings 𝔏_{X}, X ∈ M^{(2)} is entirely analogous to that for X ∈ M_{Δ̸}^{(2)}

and is omitted. Accordingly, let X, Y ∈ M_{Δ̸}^{(2)} with X ↪ Y. It must be shown that

(4.21)  \begin{matrix}H_{∗}(Y_{1}, Y_{0}; G) ⊗ H_{∗}(Y ∗_{1}, Y ∗_{0}; G′) & ⟶^{𝔏_{Y}} & Ȟ_{∗}(Y_{1} ∩ Y ∗_{1}; G ⊗ G′) \\ ↑ & & ↑ \\ H_{∗}(X_{1}, X_{0}; G) ⊗ H_{∗}(X∗_{1}, X∗_{0}; G′) & ⟶^{𝔏_{X}} & Ȟ_{∗}(X_{1} ∩ X∗_{1}; G ⊗ G′) \end{matrix}

is a commutative diagram where the vertical arrows are inclusion induced. It will suffice to show that (4.21) commutes in the special case X ∈ M_{Δ̸ c}^{(2)}(Y). For if so, it

follows for the general case that if K ∈ M_{Δ̸ c}^{(2)}(X) is chosen, then also K ∈ M_{Δ̸ c}^{(2)}(Y)

as a consequence of the inclusion j_{Y}^{X} = (j_{(Y_{1},Y_{0})}^{(X_{1},X_{0})}, j_{(Y ∗_{1},Y ∗_{0})}^{(X∗_{1},X∗_{0})}): X ↪ Y, whence the

diagram

(4.22) H_{∗}(Y_{1}, Y_{0}) ⊗ H_{∗}(Y ∗_{1}, Y ∗_{0})  Ȟ_{∗}(K_{1} ∩ K∗_{1})^{𝔏_{Y}}  Ȟ_{∗}(Y_{1} ∩ Y ∗_{1}) H_{∗}(K_{1}, K_{0}) ⊗ H_{∗}(K∗_{1}, K∗_{0}) _{𝔏_{K}} H_{∗}(X_{1}, X_{0}) ⊗ H_{∗}(X∗_{1}, X∗_{0})  _{𝔏_{X}}  Ȟ_{∗}(X_{1} ∩ X∗_{1})

is commutative where all the non-horizontal arrows are inclusion induced and where coefficients have been suppressed to facilitate the printing. The commutativity of (4.21) in the general case then follows element by element. That is, if α ⊗ γ ∈ H_{∗}(X_{1}, X_{0}; G)⊗H_{∗}(X∗_{1}, X∗_{0}; G′), then by taking (K_{1}, K_{0}) to equal (|c|, |∂c|) where c is a chain representing α and similarly defining (K∗_{1}, K∗_{0}) using a chain representing γ, it follows that α⊗γ is the image under the inclusion induced map of an element of H_{∗}(K_{1}, K_{0}; G)⊗H_{∗}(K∗_{1}, K∗_{0}; G′). Thus, the commutativity of diagram (4.22) yields that diagram (4.21) commutes on α ⊗γ, therefore, on each generating element, and is therefore commutative.

To establish commutativity of (4.21) in the special case X ∈ M_{Δ̸ c}^{(2)}(Y), it will suffice to show that for each neighborhood W ∈ N (Y_{1} ∩ Y ∗_{1}),

(4.23)  π_{W}^{Y} ^{∘} ǐ_{Y}^{X} ^{∘} 𝔏_{X} = π_{W}^{Y} ^{∘} 𝔏_{Y} ^{∘} (j_{(Y_{1},Y_{0})∗}^{(X_{1},X_{0})} ⊗ j_{(Y ∗_{1},Y ∗_{0})∗}^{(X∗_{1},X∗_{0})})

where π_{W}^{Y}: Ȟ_{∗}(Y_{1} ∩ Y ∗_{1}) → H_{∗}(W) is the canonical projection. To see that (4.23) holds, note that the left-hand side of (4.23) equals 𝔏_{W}^{X} by definition of 𝔏_{X}^{X} as the inverse limit over U ∈ N (X_{1} ∩ X∗_{1}) of the pairings 𝔏_{U}^{X} because π_{W}^{Y} ^{∘} ǐ_{Y}^{X} = π_{W}^{X} and because 𝔏_{X} = 𝔏_{X}^{X} for X ∈ M_{Δ̸ c}^{(2)} as observed in Definition 2.8(C) where the right-hand side of the last equality corresponds to the map in (2.8). On the other hand, by definition of 𝔏_{Y} as the direct limit over K ∈ M_{Δ̸ c}^{(2)}(Y) of the pairings 𝔏_{Y}^{K}, it follows that

𝔏_{Y} ^{∘} (j_{(Y_{1},Y_{0})∗}^{(X_{1},X_{0})} ⊗ j_{(Y ∗_{1},Y ∗_{0})∗}^{(X∗_{1},X∗_{0})}) = 𝔏_{Y}^{X} .

Hence, by definition of 𝔏_{Y}^{X} as the inverse limit over U ∈ N (Y_{1} ∩ Y ∗_{1}) of the pairings 𝔏_{U}^{X}, it follows that the right-hand side of (4.23) also equals 𝔏_{W}^{X}. □

4.11 The Proof of Theorem 2.11. That N (S; AIP) is cofinal in N (S) is a simple consequence of Proposition 4.4.

Set K_{U} := Ȟ_{i+j−m}(U; G ⊗ G′) for each U ∈ N (S) and let ι̌_{U}^{V}: K_{V} → K_{U} be the inclusion induced map for U ⊃ V ∈ N (S). Then

{{K_{U}}_{U∈N (S)}, {ι̌_{U}^{V} }_{U⪯V ∈N (S)}}

is an inverse system of R-modules and homomorphisms. Also set B_{P} := 𝔏_{P (i,j)} for each P ∈ ob(AIP(S)). By Lemma 2.10, the pairings B_{P} are natural relative to the inclusion morphisms of AIP(S). Thus, Corollary 4.8 applied to the family of maps B_{P} yields a pairing

B: H̃_{i}C (S; G) ⊗ H̃_{j}C∗(S; G′) → \underleftarrow{lim} Ȟ_{i+j−m}(U; G ⊗ G′)

where \underleftarrow{lim} Ȟ_{i+j−m}(U; G ⊗ G′) is taken over U ∈ N (S). Further, statement (4.17) of Corollary 4.8 applied in the current context to neighborhoods P_{1} ∩ P ∗_{1} ⊃ Q_{1} ∩ Q∗_{1} ∈ N (S; AIP) yields that statement (2.13) of Theorem 2.11 holds. Because N (S; AIP) is cofinal in N (S), it is immediate that passage to the limit in (2.13) over N (S; AIP) yields the same pairing B as above.

To finish, note that the family of natural transformations ψ_{U}: H_{∗}(U; G ⊗ G′) → Ȟ_{∗}(U; G ⊗ G′), U ∈ N (S), pass to the limit and yield a natural transformation ψ: Ȟ_{∗}(S) := \underleftarrow{lim} H_{∗}(U) → \underleftarrow{lim} Ȟ_{∗}(U) which is easily seen to be an isomorphism using the fact derived from the normality of M that if U ∈ N (S) then there exists W ∈ N (S) with U ∈ N (W). Thus, we can define 𝔏_{(i,j)} := ψ^{−1} ^{∘} B. □