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

Total and local topological indices are constructed for various types ofcontinuous 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 orbeing asymptotically compact etc.); condensing maps ($k$-set contraction)with respect to Kuratowski's or Hausdorff's measure of non-compactness onFinsler 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 indexis in fact internal while the construction is produced through transition tothe enveloping space. Examples of applications are given.