CHAPTER 8 PRODUCTS OF INTERSECTION PAIRINGS

For a product of flows on manifolds without boundary, the intersection class pairing on the Conley indices of an isolated invariant set of the product flow is, up to sign, the homology cross-product of the intersection class pairings on the Conley indices of the factor isolated invariant sets when the homology modules of the Conley indices of the factor isolated invariant sets are torsion free relative to the coefficient ring. The analogous result for the intersection number pairing is obtained as a corollary.

Preparatory to these results, we will (1) establish that the Conley index of a product invariant set is up to equivalence of connected simple systems the smash product of the Conley indices of its factors, (2) show that the usual reduced homology cross product (i.e., the homology smash product) is defined for homology classes in the Conley indices of the factors and using (1) yields a class in the homology of the Conley index of the product invariant set, and (3) using (2), prove a Kunneth theorem expressing the graded homology module of the Conley index of a product isolated invariant set in terms of the graded homology modules of the Conley indices of the factor isolated invariant sets.

A. Preliminary Observations and Definitions. For the rest of Chapter 8, S will be an isolated invariant set of the flow on the manifold M and S′ will be an isolated invariant set of a flow on a manifold M′. The flow on M′ is assumed defined on an open set 𝔊′, M′ × {0} ⊂ 𝔊′ ⊂ M′ × R, and is given by (v, t) ↦ v ·′ t for (v, t) ∈ 𝔊′. Then there is a product flow on M × M′ given by ((u, v), t) ↦ (u, v) · t := (u · t, v ·′ t) defined on the open set 𝔊𝔊′ := {((u, v), t) : (u, t) ∈ 𝔊 and (v, t) ∈ 𝔊′ }. Also ⟨N_{1}, N_{0}⟩ and ⟨N′_{1}, N′_{0}⟩ are assumed to be index pairs for S and S′, respectively.

In the course of establishing (1) and (2) above, it is convenient to have available the following observation and the definitions immediately following.

8.1 Proposition. For X and Y categories, each functor F: X → Y induces a functor F_{CSS}: CSS(X) → CSS(Y) of the same variance as F.

Proof. To map an object of CSS(X) to one of CSS(Y), just apply F to the objects and morphisms of the connected simple system. Similarly, F applied to the diagrams in a map between connected simple systems in CSS(X) yields one in CSS(Y) with the variance of F_{CSS} clearly determined by that of F. □

8.2 Definition. (A) Let \underrightarrow{lim} denote the functor on CSS(G_{R}M) to G_{R}M obtained by applying Theorem 5.2 to the identity functor on G_{R}M. The notation is appropriate because where F is any reduced homology theory on T^{∗′}, if A ∈ ob(CSS(T^{∗′})), then F_{CSS}(A) ∈ CSS(G_{R}M) and obviously \underrightarrow{lim} F_{CSS}(A) = F (A) where F (A) is as defined in Definition 1.5(A). That is, \underrightarrow{lim} F_{CSS}(A) := \underrightarrow{lim} F (A) where the limit on the right is over A ∈ A.

(B) Let A, A′ ∈ ob(CSS(T^{∗′})). Then A × A′ is a connected simple system in CSS(T^{∗′} × T^{∗′}). Thus, where ∨ and ∧ denote the usual wedge sum and smash product functors on T^{∗′} × T^{∗′}, Proposition 8.1 yields connected simple systems A∨A′ and A∧A′ in CSS(T^{∗′}) called respectively the wedge sum and smash product of A and A′.

(C) For an R-module G, let H̃_{∗}( ·; G) be the functor on CSS(T^{∗′}) to CSS(G_{R}M) induced by H̃_{∗}( ·; G) according to the prescription of Proposition 8.1. Of particular interest below are the connected simple systems of graded R-modules H̃_{∗}C (S; G), H̃_{∗}(C (S)∧C (S′) ; G), and H̃_{∗}C (S × S′; G). The last of these is well-defined because S×S′ is an isolated invariant set of the product flow on M ×M′; see Proposition 8.3 below.

(D) Let G and G′ be R-modules and let H̃_{∗}C (S; G) ⊗ H̃_{∗}C (S′; G′) denote the small category each of whose objects is of the form H̃_{∗}(X; G)⊗H̃_{∗}(X′; G′) where X and X′ are index spaces for S and S′, respectively, and for X and Y index spaces for S and X′ and Y′ index spaces for S′, let h_{S ∗}^{X Y} ⊗ h_{S′ ∗}^{X′ Y′} be the unique morphism from H̃_{∗}(X; G)⊗H̃_{∗}(X′; G′) to H̃_{∗}(Y; G)⊗H̃_{∗}(Y′; G′) in H̃_{∗}C (S; G)⊗H̃_{∗}C (S′; G′). Then H̃_{∗}C (S; G)⊗H̃_{∗}C (S′; G′) is a connected simple system in CSS(G_{R}M) since the tensor product of two R-module isomorphisms is again an isomorphism—see [Sp, Lemmas 5.1.4 and 5.1.5]. Note that \underrightarrow{lim}(H̃_{∗}C (S; G) ⊗ H̃_{∗}C (S′; G′)) = H̃_{∗}C (S; G) ⊗

H̃_{∗}C (S′; G′) because direct limits commute with tensor products.

B. Conley Indices of Product Invariant Sets. Let us observe that if (X, Y) and (X′, Y′) are topological pairs, then there is a natural homeomorphism of pointed spaces

e: (X/Y ∧ X′/Y′, ∗) ≃ (\frac{X × X′}{Y × X′ ∪ X × Y′} , ∗)

that is an excision; i.e., removing the basepoint from both spaces yields the same set, viz., X \ Y × X \ Y′. Thus both spaces are one-point compactifications of the same set. This observation is used to establish the natural homeomorphism in the following proposition, a first step in showing (1), and is left as an easy exercise for the reader. Also see [C, §III.6].

8.3 Proposition. A subset of M × M′ that is the Cartesian product of two isolated invariant sets, one from each factor flow is an isolated invariant set of the product flow on M × M′. In particular, S × S′ is an isolated invariant set of the product flow and has ⟨N_{1} × N′_{1}, N_{0} × N′_{1} ∪ N_{1} × N′_{0}⟩ as index pair. The index space of this index pair is naturally homeomorphic to the smash product N_{1}/N_{0} ∧ N′_{1}/N′_{0} via an excision map. Further, this index pair is regular or has thick exit set if both ⟨N_{1}, N_{0}⟩ and ⟨N′_{1}, N′_{0}⟩ are regular or if both have thick exit set.

Thus in the context of the product flow on M × M′, it makes sense to define the product of the index pairs ⟨N_{1}, N_{0}⟩ and ⟨N′_{1}, N′_{0}⟩ by

⟨N_{1}, N_{0}⟩ × ⟨N′_{1}, N′_{0}⟩ := ⟨N_{1} × N′_{1}, N_{0} × N′_{1} ∪ N_{1} × N′_{0}⟩ .

Note that if both index pairs are nested, then this product coincides with the usual notion of product in the category of topological pairs. To simplify notation in what follows let us assume without loss of generality that both index pairs are nested; for if not replace ⟨N_{1}, N_{0}⟩ with (N_{1}, ∩N_{0}) and ⟨N′_{1}, N′_{0}⟩ with (N′_{1}, ∩N′_{0}).

It is immediate from Proposition 8.3 that the homotopy index (i.e., the homotopy type of any index space) of a product of isolated invariant sets is the smash product of the homotopy indices. This carries over to the full Conley index; i.e., the Conley index of the product is equivalent to the smash product of the indices. This last statement requires proof, albeit minimal, because the connected simple systems C (S) ∧ C (S′) and C (S × S′) are generally distinct; i.e., in general there are index pairs for S × S′ that are not the product of an index pair for S with one for S′. For example, the product flow on R^{2} given by (x, y) · t := (xe^{t}, ye^{−t}) has the origin as isolated invariant set, and any closed disk D_{r} of radius r centered at the origin is an isolating block for the origin with exit set D_{r}^{+} consisting of those points on the bounding circle of the disk admitting polar coordinates (r, θ) where θ lies either in the interval [−π/4, π/4] or in the interval [3π/4, 5π/4]. Clearly, the regular index pair (D_{r}, D_{r}^{+}) is not the product of two index pairs for the factor flows; hence, its index space is not homeomorphic to the smash product of any factor index spaces via an excision of the type described above.

8.4 Proposition. C (S) ∧ C (S′) is naturally equivalent to C (S × S′) relative to the morphisms of CSS(T^{∗′}), whence H̃_{∗}(C (S) ∧ C (S′) ; G) ≃ H̃_{∗}C (S × S′; G).

Proof. Define E: C (S) ∧ C (S′) → C (S × S′) a map of connected simple systems as follows. Set A := N_{1}/N_{0} ∧ N′_{1}/N′_{0}. Then with e the natural homeomorphism of Proposition 8.3, e(A) is an index space in C (S × S′); hence, the assignment E_{A}^{e(A)} := e defines a map of connected simple systems E: C (S)∧C (S′) → C (S × S′) via the observation used to get equality (5.2) above. This definition is independent of the choice of index spaces N_{1}/N_{0} and N′_{1}/N′_{0} because e is natural and makes E an equivalence in CSS(T^{∗′}) because e is a homeomorphism. Hence Theorem 5.2 applied to E yields the desired isomorphism on homology. □

8.4.1 Remark. In a similar manner, when S and S′ are disjoint isolated invariant sets of the same flow, one can define an equivalence C (S ∪ S′) ≃ C (S) ∨ C (S′) using any nested index pair for S ∪ S′ of the form (N_{1} ∪ N′_{1}, N_{0} ∪ N′_{0}) where (N_{1}, N_{0}) and (N′_{1}, N′_{0}) are nested index pairs for S and S′ respectively with N and N′ disjoint. Since H̃_{∗}(N_{1}/N_{0} ∨ N′_{1}/N′_{0}) ≃ H̃_{∗}(N_{1}/N_{0}) ⊕ H̃_{∗}(N′_{1}/N′_{0}) and as direct limits commute with direct sums, the factorization H̃_{∗} = \underrightarrow{lim}^{∘}H̃_{∗} then shows

H̃_{∗}C (S ∪ S′) ≃ H̃_{∗}C (S) ⊕ H̃_{∗}C (S′).

C. A Kunneth Theorem for Conley Indices. The usual exterior product for singular homology, denoted above by ×, induces an exterior product on reduced singular homology for pointed spaces, denoted below by ¯∧, as follows. For (X, x_{0}) and (Y, y_{0}) pointed spaces and G and G′ R-modules, define

(8.1a)  ¯∧: H̃_{∗}(X; G) ⊗ H̃_{∗}(Y; G′) → H̃_{∗}(X ∧ Y; G ⊗ G′)

as the composition

(8.1b) H̃_{∗}(X; G) ⊗ H̃_{∗}(Y; G′) ≡ H_{∗}(X, x′_{0}G) ⊗ H_{∗}(Y, y_{0}; G′) ⟶^{×} H_{∗}(X × Y, X ∨ Y; G ⊗ G′)

⟶^{Q_{∗}} H_{∗}(X ∧ Y, ∗; G ⊗ G′) ≡ H̃_{∗}(X ∧ Y; G ⊗ G′)

where Q: (X × Y, X ∨ Y) → (X ∧ Y, ∗) is the quotient map which is natural relative to morphisms f × g in T^{∗} or T^{∗′}. The product ¯∧ is natural; i.e., if f: (X, x_{0}) → (X′, x′_{0}) and g: (Y, y_{0}) → (y′, y′_{0}) are continuous maps of pointed spaces there is the commutative diagram (coefficients suppressed)

(8.2)  \begin{matrix}H̃_{∗}(X) ⊗ H̃_{∗}(Y) & \underrightarrow{¯∧} & H̃_{∗}(X ∧ Y) \\ ^{f_{∗} ⊗ g_{∗}}↓ & & ↓^{(f∧g)_{∗}} \\ H̃_{∗}(X′) ⊗ H̃_{∗}(Y′) & \underrightarrow{¯∧} & H̃_{∗}(X′ ∧ Y′) . \end{matrix}

The commutativity of this diagram is an immediate consequence of the naturality of the usual exterior product for singular homology, i.e., ×^{∘} f_{∗} ⊗ g_{∗} = (f × g)_{∗} ^{∘} ×, and the equality (f ∧ g) ^{∘} Q^{X×Y} = Q^{X′×Y′} ^{∘} (f × g) at the space level where Q^{X×Y}: (X × Y, X ∨ Y) → (X × Y, ∗) and Q^{X′×Y′}: (X′ × Y, X′ ∨ Y′) → (X′ ∧ Y′, ∗) are the quotient maps.

In general, the homomorphism Q_{∗} in (8.1b) need not be one-to-one so information can be lost in passing from the product × to the product ¯∧. However, when X and Y are index spaces this is not the case as a consequence of the following.

8.5 Proposition. For index pairs ⟨N_{1}, N_{0}⟩ and ⟨N′_{1}, N′_{0}⟩, the quotient map

Q: (N_{1}/N_{0} × N′_{1}/N′_{0}, N_{1}/N_{0} ∨ N′_{1}/N′_{0}) → (N_{1}/N_{0} ∧ N′_{1}/N′_{0}, ∗)

induces isomorphisms on homology for any homology theory.

Proof. Choose t > 0 large enough so that U := N_{0}^{−t} ∩ N_{1}/N_{0} and U′ := N_{0}^{′ −t} ∩ N′_{1}/N′_{0} are closed neighborhoods of [N_{0}] and [N′_{0}], respectively, in N_{1}/N_{0} and N′_{1}/N′_{0}, respectively. Let I := [0, 1] and define f^{t}: N_{1}/N_{0} × I → N_{1}/N_{0} by

f^{t}([u], s) := \{ \begin{matrix}[u · st] & if u · [0, st] ⊂ N_{1} \ N_{0}, \\ [N_{0}] & otherwise; \end{matrix}

and define f^{′ t}: N′_{1}/N′_{0} × I → N′_{1}/N′_{0} analogously. Then f^{t} and f^{′ t} are continuous basepoint preserving deformations; see [K1, Proposition 3.1] or [C, §III.4.1.B]. Further, the restrictions D := f^{t}|U × I and D′ := f^{′ t}|U′ × I contract U and U′ to their respective basepoints. Also, by normality, we can choose Urysohn functions ϕ: N_{1}/N_{0} → I and ϕ′: N′_{1}/N′_{0} → I so that ϕ([N_{0}]) = 1 = ϕ′([N′_{0}]) and ϕ(N_{1}/N_{0} \ U) = 0 = ϕ′(N′_{1}/N′_{0} \ U′). It follows from the Corollary to Lemma 3.1 of [K2] that the quotient map (of unpointed spaces)

N_{1}/N_{0} × N′_{1}/N′_{0} ∪ C(N_{1}/N_{0} ∨ N′_{1}/N′_{0}) → N_{1}/N_{0} ∧ N′_{1}/N′_{0}

is a homotopy equivalence. Hence, as in the proof of Proposition 4.2, the Five Lemma implies that the quotient map Q induces isomorphisms on homology. □

To simplify the printing of Proposition 8.6 and Theorem 8.7 below, coefficient modules are suppressed whenever they can be inferred from context. Also, to simplify the statement and proof of the proposition, set X := (N_{1}, N_{0}), set X′ := (N′_{1}, N′_{0}), set \overline{X × X′} := (N_{1} × N′_{1})/(N_{0} × N′_{1} ∪ N_{1} × N′_{0}) and let n^{×}: X × X′ → (\overline{X × X′}, ∗) be the quotient map.

8.6 Proposition. The product ¯∧ of (8.1) induces a product

(8.3)  ¯∧: H̃_{∗}C (S; G) ⊗ H̃_{∗}C (S′; G′) → H̃_{∗}C (S × S′; G ⊗ G′) .

Further, if α ∈ H_{∗}(N_{1}, N_{0}; G) represents α ∈ H̃_{∗}C (S; G) and α′ ∈ H_{∗}(N′_{1}, N′_{0}; G′) represents α′ ∈ H̃_{∗}C (S′; G′), then α × α′ represents α ¯∧ α′; i.e., n_{∗}^{×}(α × α′) ∈ H̃_{∗}(\overline{X × X′}; G ⊗ G′) has equivalence class α ¯∧ α′ ∈ H̃_{∗}C (S × S′; G ⊗ G′).

Proof. Let H̃_{∗}C (S), H̃C (S′), H̃C (S × S′), and H̃_{∗}(C (S) ∧ C (S′)) be the connected simple systems mentioned in Definition 8.2(C), and let H̃_{∗}C (S) ⊗ H̃_{∗}C (S′) be the connected simple system described in Definition 8.2(D).

Where ⟨M_{1}, M_{0}⟩ and ⟨M′_{1}, M′_{0}⟩ are also index pairs for S and S′, respectively, we have as a special case of diagram (8.2) the commutative diagram

(8.4)  \begin{matrix}H̃_{∗}(M_{1}/M_{0}) ⊗ H̃_{∗}(M′_{1}/M′_{0}) & \underrightarrow{¯∧} & H̃_{∗}(M_{1}/M_{0} ∧ M′_{1}/M′_{0}) \\ ^{h_{S ∗} ⊗ h_{S′ ∗}}↓ & & ↓^{(h_{S}∧h_{S′})_{∗}} \\ H̃_{∗}(N_{1}/N_{0}) ⊗ H̃_{∗}(N′_{1}/N′_{0}) & \underrightarrow{¯∧} & H̃_{∗}(N_{1}/N_{0} ∧ N′_{1}/N′_{0}) . \end{matrix}

The collection of such diagrams as M_{1}/M_{0} and N_{1}/N_{0} are varied over index spaces for S and as M′_{1}/M′_{0} and N′_{1}/N′_{0} are varied over index spaces for S′ defines a map of connected simple systems from H̃C (S) ⊗ H̃C (S′) to H̃_{∗}(C (S) ∧ C (S′)) whose image under the functor \underrightarrow{lim}: CSS(G_{R}M) → G_{R}M defined in Definition 8.2(A) is a pairing

˜¯∧: H̃_{∗}C (S; G) ⊗ H̃_{∗}C (S′; G′) → H̃_{∗}(C (S) ∧ C (S′) ; G ⊗ G′) .

Also, application of Proposition 8.1 to the equivalence E of Proposition 8.4, yields an equivalence E_{∗}: H̃_{∗}(C (S) ∧ C (S′)) ≃ H̃_{∗}C (S × S′). Thus, the definition E_{∗} := \underrightarrow{lim}(E_{∗}) yields an isomorphism of graded R-modules E_{∗}: H̃_{∗}(C (S) ∧ C (S′)) ≃

H̃_{∗}C (S × S′).

Define the product in (8.3) by ¯∧:= E_{∗} ^{∘} ˜ ¯∧.

The second assertion of the proposition is equivalent to the equality

α ¯∧ α′ = \overline{n^{×}}_{∗}α × α′

where utilizing the notation established in Definition 1.5(B), \overline{n^{×}}_{∗} := i\overline{X×X′} ; ∗n_{∗}^{×}. To prove it we need to present a commutative diagram of graded R-modules, and to help keep it of a reasonable size let us utilize the following notation. In addition to having X, X′, and \overline{X × X′} defined as in the paragraph immediately preceding the statement of the proposition, set X̄ := N_{1}/N_{0} and X̄′ = N′_{1}/N′_{0}. Also, set

n̄_{∗} := iX̄ ; ∗n_{∗} and n̄′_{∗} = iX̄′n′_{∗}, and let e: (X̄ ∧ X̄′, ∗) ≃ (\overline{X × X′}, ∗) be the natural

homeomorphism of Proposition 8.3. The diagram

H_{∗}(X) ⊗ H_{∗}(X′) ⟶^{×}  H_{∗}(X × X′)  ⟶^{n_{∗}^{×}} H̃_{∗}(\overline{X × X′})
n_{∗} ⊗ n′_{∗}↓  ↓^{(n×n′)_{∗}}  ↓^{e_{∗}^{−1}}
H̃_{∗}(X̄) ⊗ H̃_{∗}(X̄′) ⟶^{×} H_{∗}(X̄ × X̄′, X̄ ∨ X̄′) ⟶^{Q_{∗}} H̃_{∗}(X̄ ∧ X̄′)

is commutative because naturality of homology cross-products makes the left-hand rectangle commute and commutativity of the diagram of continuous maps underlying the right-hand rectangle makes it commute. Also commutative by definition of the maps of connected simple systems ˜ ¯∧ and E_{∗} is the diagram

H̃_{∗}(X̄) ⊗ H̃_{∗}(X̄′) \underrightarrow{¯∧} H̃_{∗}(X̄ ∧ X̄′) ⟶^{e_{∗}} H̃_{∗}(\overline{X × X′})
iX̄ ; ∗ ⊗ iX̄′ ; ∗ ↓  ↓_{i_{∗}^{X̄∧X̄′}}  ↓_{i\overline{X×X′} ; ∗}
H̃_{∗}C (S) ⊗ H̃_{∗}C (S′) \underrightarrow{¯∧^{}}˜H̃_{∗}(C (S) ∧ C (S′)) ⟶^{E_{∗}} H̃_{∗}C (S × S′) .

Note that the top left-hand horizontal arrow of this diagram equals the bottom row of the previous diagram. Diagram chasing then yields the desired result. □

In [Mc1], C.K McCord gives a Kunneth Theorem for Conley indices. The context in which his result is stated is more general than that being considered here as it deals with a relative version of the Conley index for a flow and closed subflow on a locally compact metric space. Although the result in the absolute case to be stated presently could be derived from the result in [Mc1] it is no more difficult to give a new proof which can if desired be adapted to the relative case although to give here the requisite definitions would take us too far afield. For the statement of the theorem we drop the assumption that we are dealing with flows on manifolds and instead assume that S and S′ are isolated invariant sets of continuous flows on some locally compact Hausdorff spaces.

8.7 Theorem (Kunneth Theorem for Conley Indices). For each integer k ≥ 0, there is a split short exact sequence

(8.5) 0 ⟶ [H̃_{∗}C (S; G) ⊗ H̃_{∗}C (S′; G′)]_{k}

\underrightarrow{¯∧} H̃_{k}C (S × S′; G ⊗ G′) ⟶
[H̃_{∗}C (S; G) ∗ H̃_{∗}C (S′; G′)]_{k−1} ⟶ 0

that is natural with respect to morphisms between Conley indices, but the splitting is not natural.

Proof. Let ⟨N_{1}, N_{0}⟩ and ⟨N′_{1}, N′_{0}⟩ be nested index pairs for S and S′ and without loss of generality assume N_{1} and N′_{1} are isolating neighborhoods of S and S′ respectively. For if not, replace N_{1} with N̄_{1} := cl (N_{1} \ N_{0}) and replace N_{0} with N̄_{0} := N_{0} ∩ N̄_{1}; treat ⟨N′_{1}, N′_{0}⟩ similarly. Set X := N_{1}/N_{0} × [N′_{0}] and X′ := [N_{0}] × N′_{1}/N′_{0}. Then observe that {N_{1}/N_{0} × N′_{1}/N′_{0}; X, X′} is an excisive triad. To see this, note that for any t > 0, X is a weak deformation retract of Y := N_{1}/N_{0} × N_{0}^{′ −t}/N′_{0} and X′ is a weak deformation retract of Y′ := N_{0}^{−t}/N_{0} ×N′_{1}/N′_{0} where the deformations are induced by the flows on the ambient spaces. The observation then follows, since for sufficiently large t > 0, the interiors of Y and Y′ relative to X ∪X′ cover X ∪X′ as a consequence of [K1, Proposition 2.9(2)].

Thus, application of the Kunneth Theorem for singular homology yields the short exact sequence

(8.6) 0 ⟶ [H̃_{∗}(N_{1}/N_{0}) ⊗ H̃_{∗}(N′_{1}/N′_{0})]_{k}

⟶^{×} H_{k}(N_{1}/N_{0} × N′_{1}/N′_{0}, N_{1}/N_{0} ∨ N′_{1}/N′_{0}) ⟶

[H̃_{∗}(N_{1}/N_{0}) ∗ H̃_{∗}(N′_{1}/N′_{0})]_{k−1} ⟶ 0 .

By Proposition 8.5, H̃_{k}(N_{1}/N_{0} ∧ N′_{1}/N′_{0}) can be substituted for the middle term in sequence (8.6), whence the exterior product × becomes the product ¯∧ of (8.1). Exactness is preserved under direct limits; hence by Theorem 5.2, Definition 8.2, and Propositions 8.3 and 8.4, it follows that the sequence (8.5) is exact where the arrow labeled by ¯∧ is the product of Proposition 8.6, and the other arrows are the direct limits of the corresponding arrows of the sequence (8.6), i.e., the arrows obtained by applying Theorem 5.2 to the singular homology functor.

Again from Theorem 5.2, the sequence (8.5) is natural relative to morphisms between Conley indices as a consequence of the naturality statement of the standard Kunneth Theorem; it is split, but not naturally, because sequence (8.6) is split nonnaturally. Specifically, because the Kunneth Theorem implies that the third arrow from the left in (8.6) has a right inverse (which is not natural), this right inverse defines a non-natural map r: [H̃_{∗}C (S) ∗ H̃_{∗}C (S′)]_{k−1} → H̃_{k}C (S × S′) of connected simple systems in G_{R}M. Thus, \underrightarrow{lim}(r) splits (8.5). □

D. Factor and Product Intersection Pairings. In this section, for i = 1, 2, assume M_{i} is a manifold of dimension m_{i}, oriented over R, and let S_{i} be an isolated invariant set of a flow in M_{i}. Thus S_{1} ×S_{2} is an isolated invariant set of the product flow in M_{1}×M_{2}. Also, M_{1}×M_{2} is given the product orientation over R and G_{1}, G′_{1}, G_{2}, and G′_{2} denote R-modules. The map on homology induced by the isomorphism of coefficient modules G_{1} ⊗ G_{2} ⊗ G′_{1} ⊗ G′_{2} ≃ G_{1} ⊗ G′_{1} ⊗ G_{2} ⊗ G′_{2} obtained by transposition of the second and third factors is denoted by t_{∗}.

The reader is reminded that the statement and proof of the principal result of this section, Theorem 8.10, depends upon the choice of sign convention made in §1.D. The theorem depends on the earlier results of this section and the following two lemmas. For the first, for i = 1, 2, let M_{i} be a manifold without boundary of dimension m_{i}, assume M_{i} is oriented over R, and assume M_{1} × M_{2} is given the product orientation.

8.8 Lemma. For i = 1, 2, let ξ_{i} ∈ H_{s_{i}}(V_{1 i}, V_{0 i}; G_{i}) and η_{i} ∈ H_{t_{i}}(V ∗_{1 i}, V ∗_{0 i}; G′_{i})

where (V_{1 i}, V_{0 i}) and (V ∗_{1 i}, V ∗_{0 i}) are open topological pairs in M_{i}. Set ℓ := s_{1} + s_{2} + t_{1} + t_{2} − (m_{1} + m_{2}) and set (X, Y) := (X_{1}, Y_{1}) × (X_{2}, Y_{2}) where

(X_{i}, Y_{i}) := (V_{1 i} ∩ V ∗_{1 i}, (V_{1 i} ∩ V ∗_{0 i}) ∪ (V_{0 i} ∩ V ∗_{1 i})) .

Then as homology classes in H_{ℓ}(X, Y; G_{1} ⊗ G_{2} ⊗ G′_{1} ⊗ G′_{2}),

(ξ_{1} × ξ_{2}) ⌢_{•} (η_{1} × η_{2}) = (−1)^{(m_{2}−s_{2})(m_{1}−t_{1})}t_{∗}((ξ_{1} ⌢_{•} η_{1}) × (ξ_{2} ⌢_{•} η_{2})) .

Also, if the pair of open pairs ((V_{1 i}, V_{0 i}), (V ∗_{1 i}, V ∗_{0 i})) satisfies the off-diagonal condition (i = 1, 2), then the pair of product pairs

((V_{1 1}, V_{0 1}) × (V_{1 2}, V_{0 2}), (V ∗_{1 1}, V ∗_{0 1}) × (V ∗_{1 2}, V ∗_{0 2}))

also satisfies the off-diagonal condition and ∅ = Y_{1} = Y_{2} = Y.

Proof. For i = 1, 2, let x_{i} be the Poincaré dual in M_{i} of ξ_{i}. From the sign convention as expressed by (1.3) it follows that (−1)^{m_{1}(m_{2}−s_{2})}x_{1}×x_{2} is the Poincaré dual of ξ_{1} ×ξ_{2} relative to the product orientation on M_{1} ×M_{2}. Hence (2.5) (also see Proposition 9.18 below) and (1.3) validate the following string of equalities which yields the desired result:

(ξ_{1} × ξ_{2}) ⌢_{•} (η_{1} × η_{2}) = (−1)^{m_{1}(m_{2}−s_{2})}(x_{1} × x_{2}) ⌢ (η_{1} × η_{2})

= (−1)^{m_{1}(m_{2}−s_{2})+(m_{2}−s_{2})t_{1}}t_{∗}((x_{1} ⌢ η_{1}) × (x_{2} ⌢ η_{2}))

= (−1)^{(m_{2}−s_{2})(m_{1}+t_{1})}t_{∗}((ξ_{1} ⌢_{•} η_{1}) × (ξ_{2} ⌢_{•} η_{2}))

= (−1)^{(m_{2}−s_{2})(m_{1}−t_{1})}t_{∗}((ξ_{1} ⌢_{•} η_{1}) × (ξ_{2} ⌢_{•} η_{2})) .

The easy verification that Y_{1} = Y_{2} = Y = ∅ and that the pair of product pairs satisfies the off-diagonal condition if ((V_{1 i}, V_{0 i}), (V ∗_{1 i}, V ∗_{0 i})) does for i = 1, 2 is left to the reader. □

8.9 Lemma. Let

(8.7)  A_{h_{1}}  _{h_{4}} ^{i_{1}}  A_{′}_{h′_{1}}  _{h′_{4}} D _{i_{2}}  D′ B _{h_{2}}  _{j_{1}} B′ _{h′_{2}} C  _{j_{2}}_{h_{3}}  C′_{h′_{3}}

be a diagram of R-module homomorphisms such that i_{2}^{∘}h_{4} = h′_{4}^{∘}i_{1}, j_{1}^{∘}h_{1} = h′_{1}^{∘}i_{1}, j_{2} ^{∘} h_{2} = h′_{2} ^{∘} j_{1}, and i_{2} ^{∘} h_{3} = h′_{3} ^{∘} j_{2}. Then for each a ∈ A and with a′ := i_{1}(a),

i_{2}(h_{4}(a) − h_{3} ^{∘} h_{2} ^{∘} h_{1}(a)) = h′_{4}(a′) − h′_{3} ^{∘} h′_{2} ^{∘} h′_{1}(a′) .

Proof. The proof is an easy diagram chase left to the reader. □

The reader is reminded that if ξ_{j} ∈ Ȟ_{∗}(S_{j}) then ξ_{1} × ξ_{2} ∈ Ȟ_{∗}(S_{1} × S_{2}) is defined componentwise from the homology cross-product of singular theory using the cofinal family of product neighborhoods of S_{1} × S_{2}. That is, if U_{j} is any neighborhood of S_{j} (j = 1, 2), then

(ξ_{1} × ξ_{2})_{U_{1}×U_{2}} := (ξ_{1})_{U_{1}} × (ξ_{2})_{U_{2}}

where the product on the right is of singular homology classes, and if V is any neighborhood of S_{1} × S_{2}, choose U_{j} a neighborhood of S_{j} so that U_{1} × U_{2} ⊂ V and define (ξ_{1} × ξ_{2})_{V} := i_{∗}(ξ_{1} × ξ_{2})_{U_{1}×U_{2}} where i_{∗}: H_{∗}(U_{1} × U_{2}) → H_{∗}(V) is the inclusion induced homomorphism. The definition of (ξ_{1} × ξ_{2})_{V} does not depend on the particular choice of U_{j}, j = 1, 2, by naturality of the singular homology cross-product and because for any space X, {{H_{∗}(U)}_{U∈N (X)}, {i_{∗}^{U′ U}}_{U≤U′∈N (X)}} is an inverse system where i_{∗}^{U′ U}: H_{∗}(U′) → H_{∗}(U) is inclusion induced. It also follows that the Čech homology cross-product is natural relative to inclusion induced homomorphisms.

8.10 Theorem. For i = 1, 2, let α_{i} ∈ H̃_{s_{i}}C (S_{i}; G_{i}), let γ_{i} ∈ H̃_{t_{i}}C∗(S_{i}; G′_{i}), and set ℓ := s_{1} + s_{2} + t_{1} + t_{2} − (m_{1} + m_{2}). Then in Ȟ_{ℓ}(S_{1} × S_{2}; G_{1} ⊗ G_{2} ⊗ G′_{1} ⊗ G′_{2}),

(α_{1} ¯∧ α_{2}) ⌢_{•} (γ_{1} ¯∧ γ_{2}) = (−1)^{(m_{2}−s_{2})(m_{1}−t_{1})}t_{∗}((α_{1} ⌢_{•} γ_{1}) × (α_{2} ⌢_{•} γ_{2})) .

Remark. If at least one of H̃_{∗}C (S) and H̃_{∗}C (S′) is a torsion free R-module, then H̃_{∗}C (S) ∗ H̃_{∗}C (S′), their torsion product, is the trivial graded R-module. Hence, by Theorem 8.7,

¯∧: H̃_{∗}C (S) ⊗ H̃_{∗}C (S′) ≃ H̃_{∗}C (S × S′) ,

and the analogous result holds for the reverse time indices. Thus, in the case when at least one of H̃_{∗}C (S) and H̃_{∗}C (S′) is torsion free and when also at least one of H̃_{∗}C∗(S), and H̃_{∗}C∗(S′) is torsion free, Theorem 8.10 yields that the intersection pairing for the product invariant set is, up to sign, the product of the intersection pairings for the factor isolated invariant sets.

Proof. Let 𝔏^{j} denote the intersection pairing for S_{j} (j = 1, 2), and let 𝔏^{×} denote the intersection pairing for S_{1} × S_{2}. Also, we use 1_{∗} to denote the identity isomorphism of various graded R-modules, and for any graded R-modules E_{∗} and F_{∗}, we generically use τ_{∗}: E_{∗} ⊗ F_{∗} → F_{∗} ⊗ E_{∗} to denote the graded transposition isomorphism given by τ_{∗}(e ⊗ f) = (−1)^{|e||f|}f ⊗ e. Then where s̄:= s_{1} + s_{2}, t̄:= t_{1} + t_{2}, and m̄ := m_{1} + m_{2} and where

A := H̃_{s_{1}}C (S_{1}) ⊗ H̃_{s_{2}}C (S_{2}) ⊗ H̃_{t_{1}}C∗(S_{1}) ⊗ H̃_{t_{2}}C∗(S_{2}) ,
Ā := H̃s̄C (S_{1} × S_{2}) ⊗ H̃t̄C∗(S_{1} × S_{2}) ,
B := H̃_{s_{1}}C (S_{1}) ⊗ H̃_{t_{1}}C∗(S_{1}) ⊗ H̃_{s_{2}}C (S_{2}) ⊗ H̃_{t_{2}}C∗(S_{2}) ,
C := Ȟ_{s_{1}+t_{1}−m_{1}}(S_{1}) ⊗ Ȟ_{s_{2}+t_{2}−m_{2}}(S_{2}) ,
D := Ȟ_{s̄+t̄−m̄}(S_{1} × S_{2}) ,

the statement of the theorem is equivalent to the commutativity of the following diagram:

(8.8)  A ^{¯∧⊗¯∧}_{1_{∗}⊗τ_{∗}⊗1_{∗}}  Ā  ^{𝔏^{×}}  D B _{(−1)^{(m_{2}−s_{2})(m_{1}−t_{1})}𝔏^{1}⊗𝔏^{2}}  _{C.}_{t_{∗}∘×}

The proof of commutativity of diagram (8.8) will be carried out by showing that the assumption of non-commutativity leads via Lemma 8.9 to a contradiction of Lemma 8.8. Accordingly, suppose diagram (8.8) does not commute and let h_{1}, h_{2}, and h_{3} denote respectively the left, bottom, and right arrows of diagram (8.8), and let h_{4} denote the composite of the top horizontal arrows in that diagram. Then for some a ∈ A,

d := h_{4}(a) − h_{3} ^{∘} h_{2} ^{∘} h_{1}(a) ≠ 0 ∈ Ȟ_{s̄+t̄−m̄}(S_{1} × S_{2}) .

By definition of Ȟ_{∗}(S_{1} × S_{2}), for some neighborhood U of S_{1} × S_{2}, the image of d under the canonical projection to H_{s̄+t̄−m̄}(U), denote it d_{U}, is not zero, whence for every U′ ∈ N (S_{1} × S_{2}), if U ⊃ U′, then the image of d under the canonical projection to H_{s̄+t̄−m̄}(U′), denote it d_{U′}, is also non-zero. In particular, we can choose P_{j} ∈ ob(AIP(S_{j})), j = 1, 2, so that U′ := (P_{1 1} × P_{1 2}) ∩ (P ∗_{1 1} × P ∗_{1 2}) is a neighborhood of S_{1} × S_{2} contained in U.

As noted in Definition 2.8(C), for j = 1, 2, there exists V_{j} ∈ M_{Δ̸ o}^{(2)}(P_{j}) satisfying

U′′ := (V_{1 1} × V_{1 2}) ∩ (V ∗_{1 1} × V ∗_{1 2}) ⊂ U .

It follows that the image of d under the inclusion induced homomorphism

Ȟ_{s̄+t̄−m̄}(S_{1} × S_{2}) ⟶^{i_{2}} Ȟ_{s̄+t̄−m̄}((P_{1 1} × P_{1 2}) ∩ (P ∗_{1 1} × P ∗_{1 2}))

is non-zero, i.e., i_{2}(d) ≠ 0. Also, the projection of i_{2}(d) into H_{s̄+t̄−m̄}(U′′) is nonzero; i.e., where i′_{2}: Ȟ_{∗}(U′) → H_{∗}(U′′) is the natural projection, i′_{2} ^{∘} i_{2}(d) ≠ 0.

Set

A′ := H_{s_{1}}(P_{1 1}, P_{0 1}) ⊗ H_{s_{2}}(P_{1 2}, P_{0 2}) ⊗ H_{t_{1}}(P ∗_{1 1}, P ∗_{0 1}) ⊗ H_{t_{2}}(P ∗_{1 2}, P ∗_{0 2}) ,
Ā′ := Hs̄((P_{1 1}, P_{0 1}) × (P_{1 2}, P_{0 2})) ⊗ Ht̄((P ∗_{1 1}, P ∗_{0 1}) × (P ∗_{1 2}, P ∗_{0 2})) ,
B′ := H_{s_{1}}(P_{1 1}, P_{0 1}) ⊗ H_{t_{1}}(P ∗_{1 1}, P ∗_{0 1}) ⊗ H_{s_{2}}(P_{1 2}, P_{0 2}) ⊗ H_{t_{2}}(P ∗_{1 2}, P ∗_{0 2}) ,
C′ := Ȟ_{s_{1}+t_{1}−m_{1}}(P_{1 1} ∩ P ∗_{1 1}) ⊗ Ȟ_{s_{2}+t_{2}−m_{2}}(P_{1 2} ∩ P ∗_{1 2}) ,
D′ := Ȟ((P_{1 1} × P_{1 2}) ∩ (P ∗_{1 1} × P ∗_{1 2})) ,

and in analogy with the labeling of the arrows of diagram (8.8), in the diagram of homomorphisms

(8.9)  A′  ^{×⊗×}_{1_{∗}⊗τ_{∗}⊗1_{∗}}  Ā′  ^{𝔏_{(s̄,t̄)}^{P_{1}×P_{2}}}  D′ B′ _{(−1)^{(m_{2}−s_{2})(m_{1}−t_{1})}𝔏_{(s_{1},t_{1})}^{P_{1}}⊗𝔏_{(s_{2},t_{2})}^{P_{2}}} C′_{t_{∗}∘×}

let h′_{1}, h′_{2}, and h′_{3} denote respectively the left, bottom, and right arrows and let h′_{4} denote the composite of the top horizontal arrows of the diagram.

Diagrams (8.8) and (8.9) correspond to the similarly labeled pair of opposite faces in diagram (8.7) in the statement of Lemma 8.9. Note that i_{2} has already been defined, and let the remaining arrows of diagram (8.7) be given as follows: i_{1} and j_{1} are the inverses of the tensor product of canonical maps from the homology of index pairs to the homology of a Conley index, explicitly,

i_{1} := (p̄_{∗}^{1 +} ⊗ p̄_{∗}^{2 +} ⊗ p̄_{∗}^{1 −} ⊗ p̄_{∗}^{2 −})^{−1} , j_{1} := (p̄_{∗}^{1 +} ⊗ p̄_{∗}^{1 −} ⊗ p̄_{∗}^{2 +} ⊗ p̄_{∗}^{2 −})^{−1} ,

and

Ȟ_{s_{1}+t_{1}−m_{1}}(S_{1}) ⊗ Ȟ_{s_{2}+t_{2}−m_{2}}(S_{2})

⟶^{j_{2}} Ȟ_{s_{1}+t_{1}−m_{1}}(P_{1 1} ∩ P ∗_{1 1}) ⊗ Ȟ_{s_{2}+t_{2}−m_{2}}(P_{1 2} ∩ P ∗_{1 2})

is the tensor product of the inclusion induced homomorphisms. Note that the commutativity hypotheses of Lemma 8.9 are satisfied as a consequence of Theorem 2.11, Proposition 8.6, and by the naturality of the transposition homomorphism τ_{∗} and of homology cross-products relative to inclusions. Thus where a′ := i_{1}(a), because i_{2}(d) ≠ 0, application of Lemma 8.9 yields that

0 ≠ d′ := h′_{4}(a′) − h′_{3} ^{∘} h′_{2} ^{∘} h′_{1}(a′) = i_{2}(d) .

Next, set

A′′ := H_{s_{1}}(V_{1 1}, V_{0 1}) ⊗ H_{s_{2}}(V_{1 2}, V_{0 2}) ⊗ H_{t_{1}}(V ∗_{1 1}, V ∗_{0 1}) ⊗ H_{t_{2}}(V ∗_{1 2}, V ∗_{0 2}) ,
Ā′′ := Hs̄((V_{1 1}, V_{0 1}) × (V_{1 2}, V_{0 2})) ⊗ Ht̄((V ∗_{1 1}, V ∗_{0 1}) × (V ∗_{1 2}, V ∗_{0 2})) ,
B′′ := H_{s_{1}}(V_{1 1}, V_{0 1}) ⊗ H_{t_{1}}(V ∗_{1 1}, V ∗_{0 1}) ⊗ H_{s_{2}}(V_{1 2}, V_{0 2}) ⊗ H_{t_{2}}(V ∗_{1 2}, V ∗_{0 2}) ,
C′′ := Ȟ_{s_{1}+t_{1}−m_{1}}(V_{1 1} ∩ V ∗_{1 1}) ⊗ Ȟ_{s_{2}+t_{2}−m_{2}}(V_{1 2} ∩ V ∗_{1 2}) ,
D′′ := Ȟ((V_{1 1} × V_{1 2}) ∩ (V ∗_{1 1} × V ∗_{1 2})) ,

and in analogy with the previous two cases, let h′′_{1}, h′′_{2}, and h′′_{3} denote respectively the left, bottom, and right arrows and let h′′_{4} denote the composite of the top horizontal arrows of the diagram of R-module homomorphisms

(8.10)  A′′  ^{×⊗×}_{1_{∗}⊗τ_{∗}⊗1_{∗}}  Ā′′  ^{𝔏_{P_{1}×P_{2} (s̄,t̄)}^{V_{1}×V_{2}}}  D′′ B′′ _{(−1)^{(m_{2}−s_{2})(m_{1}−t_{1})}𝔏_{P_{1} (s_{1},t_{1})}^{V_{1}}⊗𝔏_{P_{2} (s_{2},t_{2})}^{V_{2}}} C′′_{t_{∗}∘×}

where V_{1} × V_{2} ∈ M_{Δ̸ o}^{(2)}(P_{1} × P_{2}) is defined by

V_{1} × V_{2} := ((V_{1 1}, V_{0 1}) × (V_{1 2}, V_{0 2}), (V ∗_{1 1}, V ∗_{0 1}) × (V ∗_{1 2}, V ∗_{0 2})) .

Now take diagrams (8.9) and (8.10) as the pair of non-commuting faces in diagram (8.7) with A′ taking the role of A, A′′ taking the role of A′, etc., and with the inclusion induced i′_{2} defined above corresponding to i_{2}. Then the remaining arrows, also with a prime appended, are the inclusion induced homomorphisms, i.e., i′_{1}: A′ → A′′, j′_{1}: B′ → B′′, and j′_{2}: C′ → C′′ are the tensor product of inclusion induced homomorphisms between corresponding factors.

Again we can apply Lemma 8.9. The commutativity hypotheses are now satisfied as a consequence of Lemma 2.10 and the naturality of τ_{∗} and of homology crossproducts relative to inclusions. Also i′_{2}(d′) ≠ 0 by choice of P_{j} and V_{j}, j = 1, 2. This time Lemma 8.9 yields that where a′′ := i′_{1}(a′) = i′_{1} ^{∘} i_{1}(a)

0 ≠ d′′ := h′′_{4}(a′′) − h′′_{3} ^{∘} h′′_{2} ^{∘} h′′_{1}(a′′) ,

whence diagram (8.10) does not commute. However, this contradicts Lemma 8.8 which implies diagram (8.10) does commute for any choice of V_{j} ∈ M_{Δ̸ o}^{(2)}(P_{j}), for j = 1, 2. □

8.11 Corollary. For i = 1, …, n, let S_{i} be an isolated invariant set of a flow in an m_{i}-dimensional manifold M_{i} oriented over R, suppose α_{i} ∈ H̃_{s_{i}}C (S_{i}; R), γ_{i} ∈ H̃_{t_{i}}C∗(S_{i}; R), and s_{i} + t_{i} = m_{i} for i = 1, …, n. Also, give M_{1} × ⋯ × M_{n} the product orientation. Then where σ = ∑_{i=1}^{n−1} ∑_{j=i+1}^{n} s_{i}t_{j},

#((α_{1} ¯∧ ⋯ ¯∧ α_{n}) ⌢_{•} (γ_{1} ¯∧ ⋯ ¯∧ γ_{n})) = (−1)^{σ} ∏_{ℓ=1}^{n}^{#}(α_{ℓ} ⌢_{•} γ_{ℓ})

Proof. By induction it suffices to consider the case n = 2, but that case follows immediately from Theorem 8.10 and the definition of intersection number because for the exterior product on cohomology 1 × 1 = 1, i.e., because the value of the augmentation on the exterior product of two 0-simplexes is the product of the values of the augmentations on the factors. □