Infinite series in cohomology: attractors and the Conley index

Abstract

We study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps f : U ⊆ Rd → Rd by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3. To obtain these results we develop a new method that may be of independent interest and involves the summation of power series in cohomology.