Total and local topological indices for maps of Hilbert and Banach manifolds

Abstract

Total and local topological indices are constructed for various types of
continuous maps of infinite-dimensional manifolds and ANR's from a broad class.
In particular the construction covers locally compact maps with compact sets of fixed points
(e.g. maps having a certain finite iteration compact or having compact attractor or
being asymptotically compact etc.); condensing maps ($k$-set contraction)
with respect to Kuratowski's or Hausdorff's measure of non-compactness on
Finsler manifolds; maps, continuous with respect to the topology of weak convergence,
etc.

The characteristic point is that all conditions are formulated in internal terms and the index
is in fact internal while the construction is produced through transition to
the enveloping space. Examples of applications are given.