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 vec-
torfields. 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 manifold. However, the presence of a
minimal amount of smoothness, viz., that the manifold is at least C
and that the
flow is generated by a continuous vectorfield, hereinafter referred to as the min-
imal 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 non-
singularity 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 i?-module, associated to S are the (reduced) singular homology modules
with coefficients in G of its Conley indices in forward and reverse time, respec-
tively denoted by H*C(S;G) and #*C*(S; G). The graded module H*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
5 is a normally hyperbolic invariant submanifold of a smooth flow; the analogous
remarks apply to H*C*(S', G) and any local stable set of S. The precise defini-
tions 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 H*(S) denotes the Cech homology of 5, both with
coefficients in R.
Received by the editor August 30, 1990, and in revised form June 15, 1994.
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