ABSTRACT

Given an isolated invariant set of a flow on a manifold of dimension m oriented over a PID R, an intersection class pairing of degree −m on the tensor product of the singular homology modules of the forward and reverse time Conley indices of the isolated invariant set with values in the Čech homology of the invariant set is defined. Restriction of the pairing to elements of degree m results in an intersection number pairing that is invariant under continuation along a continuous path of flows and isolated invariant sets. More generally, the unrestricted pairing defines continuous lifts to a space of Čech homology classes along such a path. Further, when the homology modules of the Conley indices are torsion free, the intersection number pairing is non-singular. Also, the pairing associated to an isolated invariant set of a product flow is, modulo torsion, the product (up to sign) of the pairings associated to the factor isolated invariant sets. Intersection classes of lowest and highest dimension are computed for any R-orientable, normally hyperbolic invariant submanifold whose expanding and contracting normal subbundles are also R-orientable. These computations yield, due to dimensional considerations, a complete computation of the intersection class and number pairings for hyperbolic critical points and hyperbolic closed orbits. Application is made in an appendix to the existence of solution of a class of singularly perturbed two-point boundary value problems such problems having provided strong motivation for the present study.

1991 Mathematics Subject Classification. Primary: 58F25, 58F35, 34C35, 55N45. Secondary:

34B15, 34E15.

Key words and phrases. isolated invariant set, Conley index, reverse time Conley index, homology of a Conley index, intersection numbers, intersection number pairing, intersection class pairing, homology intersection number pairing, invariance under continuation, connected simple system, normally hyperbolic invariant submanifold, singularly perturbed non-linear two-point boundary value problem.