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

DOI: http://dx.doi.org/10.12775/TMNA.2000.002

#### 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.

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.

#### Keywords

Topological index; fixed points; infinite-dimensional manifolds; locally compact maps; condensing maps; weakly continuous maps

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