10 HENRY L. KURLAND

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.

1.2

DEFINITION.

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

h^Y.

Consequently, morphisms in /C satisfy the following identities:

forX,X',X"eob(/C),

/

n

\ a i _ i

h

x'x'' i xx' _

h

xx"

h

x'x _

(h

xx'\-i

The middle and left identities hold because /C is a category and because /C(X, y ) =

{/i^

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') =

{/i/cx'},

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

CSS(T*/)

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.

1.3

DEFINITION.

Let S be an isolated invariant set of a

Ck

flow ip in M, set

u • t := ^(u, £), and let

(9+(u)

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.