Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces

We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index. Introduction. The shape index is a time-discrete analog of Conley’s homotopy index for flows. It was constructed by Robbin and Salamon [RS] for isolated invariant sets of a diffeomorphism on a smooth manifold. In [Mr1] the author presented a cohomological Conley index for isolated invariant sets of homeomorphisms (see also [MR]). Its construction was based on a functor, called the Leray functor and introduced in [Mr1, Sect. 4]. It turns out that there are at least three other functors which can be used in the construction instead of the Leray functor. They provide various Conley indices but with the same basic properties. One of such functors is the inverse limit functor, which can be used to construct the shape index. In the present paper, we propose a general scheme for constructing Conley indices. This enables us to unify the results in [RS] and [Mr1], to get rid of smoothness and injectivity in the construction of the shape index and to compare the shape index with the cohomological Conley index. The scheme also provides other Conley indices: the homology index, the homotopy group index, the shape group index. In contrast to [RS] and [Mr1], in this paper we work with locally defined maps. This is important because of forthcoming applications to differential equations, where one has to deal with t-translations or Poincaré maps which are often defined only in some open subset of the given space. 1991 Mathematics Subject Classification: 34C35, 54H20, 58F35.