being continuous where N ranges over compact subsets of M and A (AT) inherits
its topology from A. In fact, A(N) is open in A for each compact N c M. The
collection of sets &N(U) where U ranges over open subsets of A(N) and N ranges
over compact subsets of M is a basis for this topology. Also, the natural projection
7r^: S(ip) A is a surjective local homeomorphism with a^ a local inverse on A(N).
Details of the above can be found in [RAII], [C], or [SI].
(B) Throughout this work, whenever g: Y S(ip) is a continuous map, the
components of g are denoted by Sg and \g, i.e., g(y) =: (Sg(y), Xg(y)) where Sg(y)
is an isolated invariant set of the flow (f\ ^ for each y e Y.
B. The Conley Index. The Conley index will be described below as a certain
type of category called a connected simple system. The following notation is used
in describing categories. For any category /C, its class of objects is denoted below
by ob(/C) and for I , 7 e ob(/C), the set of morphisms from X to Y is denoted by
IC(X, Y). Recall that a category /C is a small category if ob(/C) is a set.
A small category /C is a connected simple system if 1C(X,Y)
is a singleton for every X,Y G ob(/C). The unique element of K(X,Y) will be
denoted by
Consequently, morphisms in /C satisfy the following identities:
\ a i _ i
x'x'' i xx' _
x'x _
The middle and left identities hold because /C is a category and because /C(X, y ) =
}; the identity on the right holds as a simple consequence of the other two
and uniqueness of inverses. Consequently, any two objects of /C are equivalent.
For /C a connected simple system, the facts that ob(/C) is a set, that for ev-
ery X,X' e ob(/C), K(X,X') =
and that the identities (1.1) hold are
collectively referred to as the connected simple system properties of /C.
As shown in [K3], for any category W, there is a corresponding category, denoted
CSS(W), of connected simple systems that are subcategories of H. The notion of
morphism in CSS(H) is reviewed below in Chapter 5 and is of particular interest to
us in the case H := T*', the homotopy category of topological spaces with base-
point, because a Conley index is by definition an object of
and because
this notion of morphism undergirds the proof of invariance of intersection numbers
under continuation. Other substitutions for TL of interest to us in this context are
QRM. and 8RM\ the former the category of graded left R-modules, the latter the
homotopy category of chain complexes over R.
Let S be an isolated invariant set of a
flow ip in M, set
u t := ^(u, £), and let
denote the positive semi-orbit of u. A description of
the Conley index of S relative to the flow ip follows.
(A) Compact subsets N\ and A^o of M form an index pair (AT1? No) for S relative
to ip if, and only if, the following three conditions are satisfied:
(1) S C intM(ATi \ A^o) and cl (Ni \ NQ) is an isolating neighborhood of S;
(2) i f u e i V o , t 0, and u [0, t] C Nu then u [0, t] C N0;
(3) if u e Nx and £+(u) £ Nu then there exists t 0 so that u [0,t] C Nx
and u-teNo.
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