The Conley index theory was introduced by Charles C. Conley (1933-1984) in [C1] and a major part of the foundations of the theory was developed in Ph. D. theses of his students, see for example [Ch, Ku, Mon]. The Conley index associates the homotopy type of some pointed space to an isolated invariant set of a flow, just as the fixed point index associates an integer number to an isolated set of fixed points of a continuous map. Examples of isolated invariant sets arise naturally in the critical point theory - each isolated critical point of a functional is also an isolated invariant set of its gradient flow. If the critical point is nondegenerate then its Conley index is equal to the homotopy type of the pointed k-sphere, where k is the Morse index of that point. There are other relations to Morse theory, for example a generalization of Morse inequalities can be achieved. The aim of this note is to describe briefly some basic facts of the Conley index theory for (continuous-time) flows. We refer to [C2, Ry, S1, Smo] for a more detailed presentation. We do not touch more advanced topics of the theory: the Conley index as a connected simple system (see [C2, Ku, McM, S1]), connection and transition matrices (see [F1, F2, FM, McM, Mi1, Moe, Re]]), infinite dimensional Conley indices (see [Be, Ry], the Conley index for multivalued flows (see [KM, Mr2]), Conley-type indices for discrete-time flows (see [Mr3, RS, Sz]), equivariant Conley indices (see [Ba, Ge]), and relations to the Floer homology (see [S2]). (The list of bibliography items is far from completeness.) Moreover, we do not present any applications of the index. For some more recent results we refer to [Mi2]. We also refer to the other articles in this Proceedings.
@article{bwmeta1.element.bwnjournal-article-bcpv47i1p21bwm,
author = {Srzednicki, Roman},
title = {On foundations of the Conley index theory},
journal = {Banach Center Publications},
volume = {50},
year = {1999},
pages = {21-27},
zbl = {0942.37002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv47i1p21bwm}
}
Srzednicki, Roman. On foundations of the Conley index theory. Banach Center Publications, Tome 50 (1999) pp. 21-27. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv47i1p21bwm/
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