Shape theory and its applications
by
Sibe Mardešić
University of Zagreb, Bijenicka cesta 30, 10 000 Zagreb, Croatia

This is a survey of ordinary and strong shape theory with reference to some of the applications. It is well known that standard notions of homotopy theory are not adequate to study global properties of spaces with bad local behavior. Shape theory is designed to correct these shortcomings of homotopy theory. When restricted to spaces with good local behavior, like polyhedra, shape theory coincides with homotopy theory, therefore, it can be viewed as the appropriate extension of homotopy theory to general spaces.

The philosophy in constructing the shape categories of topological spaces consists in approximating spaces by inverse systems of polyhedra and in developing a homotopy theory of such systems. It gradually became clear that besides the ordinary shape category Sh(Top), there exists also a finer category SSh(Top), which occupies an intermediate position between the homotopy category Ho(Top) and the category Sh(Top). To construct this strong shape category one needs delicate constructions associating polyhedral inverse systems to spaces, like resolutions and strong expansions. Moreover, the homotopy of systems needed is the rather involved coherent homotopy.

One can expect applications of shape theory in situations concerning global properties of spaces having irregular local behavior. Such spaces naturally appear in many areas of mathematics, e.g. as fibers of mappings, remainders of compactifications, sets of fixed points, attractors of dynamical systems, spectra of operators. Among the nicest applications obtained up to know is the shape-theoretic characterization of compacta, which embed in manifolds as attractors of flows.

Key words: shape, strong shape, inverse system, coherent homotopy, attractor.

AMS(MOS) Subj. Class. 55P55, 54C56, 54B35, 58F12.

Date received: June 15, 1998