Total and local topological indices for maps of Hilbert and Banach manifolds (Q1590119)

This article deals with some general construction of the local and global index for various types of continuous mappings on infinite-dimensional manifolds and ANRs; more precisely, the author analyses the global index (the Lefschetz number) \(\Lambda_f\) of a continuous mapping \(f:M\to M\) with a priori compact set Fix \(f\); this index is defined as the index (rotation) of the vector field \(I-F\) on the boundary \(\dot\Omega\) of an open set \(\Omega\) containing Fix \(f\), where \(F=f\circ R\) and \(R:U\to M\) is a retraction of a neighbourhood of \(M\) in a Hilbert or Banach space \(E\) (it is assumed that \(M\) can be embedded into \(E)\) and its local analogue. This construction covers numerous classes of mappings considered earlier by Yu. G. Borisovich, F. E. Browder, C. C. Fenske, G. Fourier, M. A. Krasnosel'skii and P. P. Zabreiko, J.-P. Penot, B. N. Sadovski, the author of the article, and others. The cases of condensing and weakly continuous mappings are considered separately.