  In this paper we study the cohomological Conley index of arbitrary isolated
invariant continua for continuous maps $f \colon U \subseteq \mathbb{R}^d \to
\mathbb{R}^d$ by analyzing the topological structure of their unstable
manifold. We provide a simple dynamical interpretation for the first
cohomological Conley index, describing it completely, and relate it to the
cohomological Conley index in higher degrees. A number of consequences are
derived, including new computations of the fixed point indices of isolated
invariant continua in dimensions 2 and 3.
  Our approach exploits certain attractor-repeller decomposition of the
unstable manifold, reducing the study of the cohomological Conley index to the
relation between the cohomology of an attractor and its basin of attraction.
This is a classical problem that, in the present case, is particularly
difficult because the dynamics is discrete and the topology of the unstable
manifold can be very complicated. To address it we develop a new method that
may be of independent interest and involves the summation of power series in
cohomology: if $Z$ is a metric space and $K \subseteq Z$ is a compact, global
attractor for a continuous map $g \colon Z \to Z$, we show how to interpret
series of the form $\sum_{j \ge 0} a_j (g^*)^j$ as endomorphisms of the
cohomology group of the pair $(Z,K)$.
