INTRODUCTION

A compact invariant set of a flow in a manifold is an isolated invariant set if it is the largest invariant set in some compact neighborhood of itself called an isolating neighborhood of the invariant set. The present work arose from the author’s study of two-point boundary value problems when in the course of proving some existence theorems it became necessary to ask how closely does the nature of intersections of local stable and unstable sets of an isolated invariant set and the behavior of such intersections under continuation along a path of flows resemble at the level of homology those found for the subclass of isolated invariant sets consisting of normally hyperbolic invariant submanifolds of flows generated by continuous vectorfields. The answer is that the resemblance is remarkably close with regard to intersection classes of degree zero; the resemblance is much weaker for intersection classes of positive degree. The definition of the intersection classes is based on the intersection product of singular homology classes as described in Dold’s book [D] and, requiring no assumption of additional smoothness, is naturally formulated in the context of continuous flows on a C^{0} manifold. However, the presence of a minimal amount of smoothness, viz., that the manifold is at least C^{1} and that the flow is generated by a continuous vectorfield, hereinafter referred to as the minimal smoothness conditions, does simplify the proofs of a few results as will be pointed out in the main body of the text and may be necessary to the proof of nonsingularity of the intersection number pairing on Conley indices developed here. A more detailed description of the contents of this work follows.

Let S be an isolated invariant set of a flow in M, an m-dimensional, second countable, topological manifold without boundary oriented over a PID R. For G any R-module, associated to S are the (reduced) singular homology modules with coefficients in G of its Conley indices in forward and reverse time, respectively denoted by H̃_{∗}C (S; G) and H̃_{∗}C∗(S; G). The graded module H̃_{∗}C (S; G) is a homological approximation to the reduced singular homology of the one-point compactification of any local unstable set of S with the approximation exact when S is a normally hyperbolic invariant submanifold of a smooth flow; the analogous remarks apply to H̃_{∗}C∗(S; G) and any local stable set of S. The precise definitions of these modules as well as other basic notions of the Conley index theory are reviewed in Chapter 1. To simplify the presentation, all homology modules henceforth mentioned in this introduction are taken with coefficients in R and the coefficient module is therefore suppressed from the notation. Thus, H_{∗}(M) denotes the singular homology of M and Ȟ_{∗}(S) denotes the Čech homology of S, both with coefficients in R.

Received by the editor August 30, 1990, and in revised form June 15, 1994.

The main objects of study in this work are defined in Chapter 2 and are a bilinear pairing of degree −m

𝔏: H̃_{∗}C (S) ⊗ H̃_{∗}C∗(S) → Ȟ_{∗}(S) ,

called the intersection class pairing on the Conley indices of S, and the closely related pairing

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

called the intersection number pairing on the Conley indices of S, obtained from 𝔏 by first restricting 𝔏 to the indicated submodule of elements of degree m, then composing the natural projection of Ȟ_{0}(S) into H_{0}(M) with the restricted 𝔏, and finally superposing on this composite the Kronecker index.

When M = R^{m} a third pairing

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

called the homology intersection number pairing on the Conley indices of S, is defined whose definition is independent of those of 𝔏 and ^{#}𝔏, but is related to ^{#}𝔏 by the identity

L = ±^{#}𝔏 · o_{m}

where o_{m} ∈ H_{m}(R^{m}, R^{m} \{0}) is the given orientation of R^{m}. (See Proposition 2.9 below for a resolution of the sign ambiguity; also see part (D) of Definition 2.8 immediately preceding it.) Given the identity, the obvious question is why bother with L at all. There are two reasons for doing so. First, it is much more difficult to define intersection numbers homologically in the general manifold setting than it is in the Euclidean setting and the definition of L is based on the simpler definition available in the Euclidean setting. Since those properties of 𝔏 most likely to find wide use in applications to the study of differential equations are in fact properties of ^{#}𝔏 necessarily shared with L, the author hopes that including the definition of L will make these results available to a wider audience. Second, that L is well-defined requires proof, and that proof when suitably abstracted has as corollary the result needed to show 𝔏 well-defined. Thus, in Chapter 2 first L is defined, then 𝔏, and finally ^{#}𝔏. However, the proofs of the theorems showing L and 𝔏 well-defined are delayed until Chapter 4.

The pairings 𝔏 and ^{#}𝔏 are of interest chiefly because of the properties they enjoy with respect to continuation of the flow and isolated invariant set. Stated precisely in Chapter 3, but with the proofs delayed until Chapter 5, these are, roughly, (1) the pairing 𝔏 defines continuous lifts of a path of flows and isolated invariant sets to a space of Čech homology classes associated to the isolated invariant sets of the family of flows under consideration; (2) the pairing ^{#}𝔏 is invariant along any path of flows and isolated invariant sets; i.e, the Kronecker index is constant along a lift by 𝔏 to a path of intersection classes in dimension zero. Here, a family of isolated invariant sets, each member determined by a flow along a continuous path of flows, defines a continuous path of isolated invariant sets if at each parameter value an isolating neighborhood of the invariant set at that parameter value is an isolating neighborhood of the invariant set at nearby parameter values. Continuity of the lift of such a path to the space of Čech classes means that at each parameter value if N is an isolating neighborhood of the invariant set at that parameter value, then the projection to H_{∗}(N) of the Čech class on the lift at that point is the common value of the projections to H_{∗}(N) of the Čech classes on the lift at parameter values sufficiently close to the original one. Statement (1) is weak, for as illustrated by Example 3.9 below, it is possible to have a non-zero intersection class of positive dimension continue to a zero intersection class in that dimension. This merely reflects the fact that continuity in the context of a space of isolated invariant sets associated to a continuous family of flows ensures upper, but not lower, semi-continuity of the isolated invariant set relative to the topology on compact subsets of M induced by the Hausdorff metric so that the homeomorphism type of isolated invariant sets along a path can vary drastically in dramatic contrast to what could happen if the path of invariant sets were confined to a path of normally hyperbolic invariant submanifolds where as a consequence of transversality theory the diffeomorphism type must remain constant. However, the continuity of the lifts determined by 𝔏 over a path of isolated invariant sets does provide more information about the isolated invariant sets on the path than is available with the Conley index alone. On the other hand, the invariance under continuation of the intersection number pairing ^{#}𝔏, a previously unsuspected algebraic manifestation of the invariance of the Conley index under continuation, is easily the most important property of 𝔏 proved here because it is the one most likely to have wide use in applications of the pairing to the study of differential equations. Specific application of these results to a problem arising in population genetics is discussed further on in this introduction.

The first part of Chapter 6 contains several technical results describing computation of 𝔏 and ^{#}𝔏 in terms of singular chains representing classes in H̃_{∗}C (S) and H̃_{∗}C∗(S) that will not be elaborated upon here. The second part of Chapter 6 shows that 𝔏 is invariant, i.e., transforms naturally, under orientation preserving, orbit preserving homeomorphisms. Such homeomorphisms are not assumed to preserve time parameterizations. This result extends to transformation by continuous, proper, orbit preserving surjections if the pairing 𝔏 is replaced by its analogue on the Čech homology of the Conley indices denoted by

𝔏̌: ˜ Ȟ_{∗}C (S) ⊗ ˜ Ȟ_{∗}C∗(S) → Ȟ_{∗}(S) .

In general, 𝔏 always factors through 𝔏̌ by the natural homomorphism of singular theory into Čech theory. When the minimal smoothness conditions are met the Čech and singular modules of the Conley indices of S are naturally identified, and modulo this identification 𝔏̌ and 𝔏 coincide.

To describe the computation of 𝔏 and ^{#}𝔏 on R-orientable, normally hyperbolic invariant submanifolds made in Chapter 7 (which in part relies on the technical results of Chapter 6) requires introducing the notion of an isolating block. For the purposes of this introduction, an isolating neighborhood B is an isolating block neighborhood (sometimes abbreviated to “block”) if its boundary is the union of two closed and possibly overlapping sets B^{+} and B^{−}, called respectively the exit and entrance sets of B, so that a point lies in B^{+} (respectively, B^{−}) if it exits B immediately in positive (respectively, negative) time under the action of the flow.

The flow is therefore externally “tangent” to a block at points in the intersection of its exit and entrance sets. When the minimal smoothness conditions are met, a result of Wilson and Yorke [WY, Theorem 2.1] shows that there exists a cofinal family of isolating block neighborhoods of any isolated invariant set with the property that each block in the family is a C^{0} manifold with boundary whose exit and entrance sets lie in codimension one, C^{1} submanifolds without boundary to which the vectorfield generating the flow is transverse. In the present context, isolating blocks are of interest because for any block B there are canonical isomorphisms

H_{∗}(B, B^{+}) ≃ H̃_{∗}C (S)  and  H_{∗}(B, B^{−}) ≃ H̃_{∗}C∗(S) .

A result of Pugh and Shub [PS, Theorem 2] shows that when S is a normally hyperbolic invariant submanifold of a smooth flow in M, the flow is topologically conjugate on a tubular neighborhood of S to the linearization of the flow restricted to the normal bundle. It is fairly clear that the technique of their proof which defines expanding-contracting product coordinates in a neighborhood of S and which is based in part on earlier work of Palis and Smale [Pa], [PaS] can be used to show that S admits a cofinal family of isolating blocks with each member block B homeomorphic to the total space of the Whitney sum of two disk bundles, one the unit disk bundle of the expanding subbundle of the normal bundle over S, the other the unit disk bundle of the contracting subbundle of the normal bundle over S with B^{+} corresponding to the Whitney sum of the unit sphere bundle of the expanding normal subbundle and the unit disk bundle of the contracting normal subbundle and with B^{−} described similarly by interchanging the roles of the two disk subbundles. Because [C (S)], the Conley homotopy index of S in forward time, must equal the homotopy type of the quotient B/B^{+}, it follows that [C (S)] equals the homotopy type of the Thom space of the expanding subbundle; similarly for the reverse time homotopy index [C∗(S)] and the contracting subbundle. It follows with the aid of the Thom isomorphism theorem that for S and the expanding and contracting normal subbundles R-orientable and where k equals either (i) the dimension of the unstable (stable) manifold of S or (ii) the rank of the expanding (contracting) subbundle of the normal bundle over S, a generator α of H̃_{k}C (S) (respectively, γ of H̃_{k}C∗(S)) can be geometrically realized: in case (i) as the local unstable (stable) manifold of S within B regarded as a relative cycle in B modulo its exit (entrance) set; in case (ii) as the local strong unstable (strong stable) manifold within B of any point in S regarded as a relative cycle in B modulo its exit (entrance) set. Hyperbolicity ensures that the intersection of such a representative for α with one for γ is a submanifold of S, and this intersection submanifold geometrically represents 𝔏(α ⊗ γ). When α represents a local unstable manifold of S and γ a local stable, it follows that 𝔏(α ⊗ γ) = ±o_{S} where o_{S} is the fundamental class of S; when α ⊗ γ has degree m (e.g., a strong local unstable manifold represents α and a local stable manifold represents γ) it follows by definition of the Kronecker index that^{#}𝔏(α⊗γ) gives an algebraic count of the number of points in the 0-dimensional intersection submanifold representing 𝔏(α ⊗ γ)—in the cases at hand, ^{#}𝔏(α ⊗ γ) = ±1. In all these cases, the ambiguous sign is determined by the choice of orientation classes for the expanding and contracting subbundles of the normal bundle. In particular, the generators described in cases (i) and (ii) can be characterized in terms of Thom classes of the expanding and contracting subbundles of the normal bundle of S, a fundamental class of S, and a cofinal family of isolating blocks of S with the requisite Whitney sum structure in which generators of the homology modules of the Conley indices can be geometrically realized as described above by cases (i) and (ii) compatibly over the cofinal family.

Below, although a partial computation of 𝔏 and ^{#}𝔏 as just described will be made for the general normally hyperbolic invariant submanifold modulo the detailed construction of the isolating block described above and assuming that S and the expanding and contracting subbundles of the normal bundle of S are R-orientable, a complete proof of the statements of the previous paragraph will be made only for S a hyperbolic critical point or a hyperbolic closed orbit. In these two cases, the partial computation is in fact complete due to dimensional considerations. Also, the description of [C (S)] in terms of a Thom space results, to the author’s knowledge, in the first complete computation of the Conley indices of a hyperbolic closed orbit: In [C] the correct values of the indices as pointed homotopy types were given for the two possibilities of the local unstable manifold being orientable or non-orientable, viz., S^{P +1} ∨ S^{P} in the orientable case and S^{P −1}∧ RP^{2} in the nonorientable when there are (counting multiplicities) P ≥ 1 Floquet multipliers of modulus greater than one; however, no computation is made in the non-orientable case, and the heuristic computation made in the orientable case assumes that the closed orbit is an isolated invariant set of a product flow, viz., the product of S^{1} regarded as the unique non-empty isolated invariant set of a uniform rotation on the circle and a hyperbolic critical point of a flow on a Euclidean space of suitable dimension, and therefore has a product block. In general this assumption is invalid: for example, if M is non-orientable, the periodic orbit might have an orientable unstable manifold but a non-orientable stable manifold precluding the possibility that a tubular neighborhood of the orbit is trivializable. However, existence of a product block is provable via Floquet normal form when the linearization of the Poincaré map at one (hence every) point on the closed orbit has a real square root.^{1} Thus, Conley’s computation is salvageable by lifting the vectorfield to a connected double cover of a tubular neighborhood of the closed orbit and then using Floquet normal form to trivialize the problem in the double cover. In particular, a product block for the doubled closed orbit having the Whitney sum structure described earlier is easily constructed which when “averaged” over the Z_{2}-action of the double cover yields an invariant block whose projection to the base tubular neighborhood is a block for the original closed orbit also having the desired Whitney sum structure. Further, the construction does not depend on the orientability of the invariant manifolds and makes it easy to compute the index when the unstable manifold is non-orientable.

The main result of Chapter 8 is that modulo torsion the intersection pairing on the Conley indices of an isolated invariant set of a product flow is the product of the intersection pairings on the Conley indices of the factor isolated invariant sets; a corresponding result holds for ^{#}𝔏. Also presented is a Kunneth theorem for Conley indices which follows straightforwardly from the standard Kunneth theorem.

1In general, no real square root exists, nor does one exist generically: a non-singular real matrix has a real square root if, and only if, each elementary Jordan block associated to a negative eigenvalue of the matrix appears an even number of times in the Jordan normal form of the matrix.

A proof can be carried out using the real Jordan form of the matrix.

The last chapter, Chapter 9, presents an alternative characterization of 𝔏 using an analogue of the classical cap product representation of intersection products and a cofinal family of isolating block neighborhoods of S. An immediate consequence of this cap product representation is that^{#}𝔏 is a non-singular (hence, non-degenerate) pairing of H̃_{m−k}C (S) and H̃_{k}C∗(S) if both modules are torsion-free. The result is strongly suggested by the results of Chapter 7 and the invariance of ^{#}𝔏 under continuation. The development of the cap product representation, and therefore the proof of non-singularity, currently requires that the minimal smoothness conditions be met by M and the flow. It is unclear if these results can be extended to the case where M and the flow are at best C^{0}. When the minimal smoothness conditions are met, there is a Poincaré duality of forward and reverse time Conley indices which in terms of a block B that is a C^{0} manifold with boundary and which has its exit and entrance sets as submanifolds with boundary is expressed by H^{m−k}(B, B^{∓}) ≃ H_{k}(B, B^{±}). This type of duality was first noted by Montgomery [M] using field coefficients and with all modules Alexander-Spanier cohomology modules, but he gave no proof. Recently, McCord [Mc2] has given a short proof of these duality relations. A second, longer proof comes out in the wash in the course of proving the cap product representation of 𝔏 which is

𝔏(α ⊗ γ) = \underleftarrow{lim}(x_{B} ⌢ γ_{B})

where x_{B} ∈ H^{∗}(B, B^{−}) is the cohomology class dual to the homology class in H_{∗}(B, B^{+}) whose canonical image is α ∈ H̃_{∗}C (S) and where γ_{B} ∈ H_{∗}(B, B^{−}) has canonical image γ ∈ H̃_{∗}C∗(S). The inverse limit is with respect to the inverse system of homology modules and inclusion induced homomorphisms of the cofinal family of isolating block neighborhoods of S directed by reverse inclusion; however, it is far from obvious that the inverse limit of the cap products is well-defined.

As mentioned in the first paragraph of this Introduction, largely motivating the present study is the application of L and^{#}𝔏 and their invariance under continuation to the statement and proof of existence results ensuring the presence of endpoint or interior transition layer behavior for singularly perturbed, vector two-point boundary value problems on the finite interval [a, b] of the form

εu′ = F(u, x, ε) (0.1a) u(a) ∈ E(a, ε), u(b) ∈ E(b, ε) (0.1b)

where differentiation is with respect to x ∈ R, u ∈ R^{m}, and E(λ, ε) (λ = a, b, ε > 0) is the locus of points in the phase space R^{m} satisfying the endpoint condition at λ and where existence of solution is to be proved for 0 < ε ≪ 1. Because of the large motivational role of problem (0.1), in Appendix A to the current study the weakest form of such an existence theorem is stated and proved and then applied to the analysis of Example 3.5 (which illustrates the invariance of intersection numbers under continuation) to yield existence of a non-trivial stationary solution satisfying Neumann boundary conditions of a Fisher equation for the variation in allelic density over space and time of the two alleles present at a single gene locus in a population that is genetically isolated, has sufficiently low genetic diffusion, and is confined to a finite, one-dimensional habitat where heterozygote individuals are assumed to have lower genetic fitness than homozygote individuals of either type. In fact, this problem from population genetics as well as certain defects in the results of [Bg] initiated the author’s general study of problem (0.1).

As there has been a long delay in the publication of this work since its original submission due, on the one hand, to a long refereeing process and, on the other, to the author’s glacial speed in rewriting the originally submitted papers, and since at least one paper closely related to this work has appeared in the interim, viz., McCord’s [Mc2], a few comments on the history of this work seem in order. The two papers originally comprising this work were submitted in April and July of 1990. However, the author first proved the existence of the pairing L in the unpublished paper [K7] which is cited in [Mc2] and possibly played some part in motivating that paper. In any event, the author received a preprint of McCord’s article too late for it to have any substantial influence on the original submissions although it did result in the addition to the first paper submitted of a few remarks on how the method used to construct the pairing L could be used to construct an isomorphism giving the Poincaré duality of Conley indices. In the work at hand those remarks are represented by Proposition 9.12 and Theorem 9.15. Also, in the original submission, 𝔏 was computed only for a hyperbolic critical point of a vectorfield on M whose Hessian at the critical point had m distinct eigenvalues. Thus, as far as content is concerned, the main difference between the originally submitted papers and the present work is that in the former nearly all of the material in Chapters 7 and 9 was not present.

The author would like to thank John Milnor for asking several stimulating questions of the author when he presented the results of Chapters 2 and 3 and Appendix A at an April 1992 meeting in dynamical systems held at Union College, Schenectady, NY which led the author to tackle again (this time successfully) the question of whether or not ^{#}𝔏 was non-degenerate. The author would also like to thank Mark Steinberger for several helpful conversations on the classification of fibre bundles and to thank the Department of Mathematics and Statistics at SUNY-Albany for permitting the author to use their computing facilities in the preparation of this work and for sometimes providing employment. Finally, the author would like to thank Kristoffer H. Rose and Ross Moore for writing and for their help in using the XY-pic diagram package used to great effect throughout this work, but especially in Chapters 8 and 9 and Appendix B, in typesetting the many non-rectangular diagrams.