Local fixed point index theory for non simply connected manifolds

Abstract

Abstract : Given a compactly fixed map f:U M, where U is an open subset of a manifold M, it is a classical result that one can assign an integer-valued index I(f,U) to this situation with the property that I(f,U) is not = to 0 implies f (and any compactly fixed perturbation of f) has fixed points in U, i.e. solutions to the equations f(x) = x. However, it can still happen that I(f,U) = 0 and f has essential fixed points in U. The objective of this paper is to provide a finer invariant o(f,U), called the local obstruction index, which has the property that o(f,U) is not = to 0 if, and only if, every compactly fixed perturbation of f has fixed points in U. The main objective is to assign to this data a new index, o(f,U), which is not necessarily an integer, which removes this fundamental defect so that now the equation has essential solutions in U. A general formula for computing the value of o(f,U) is also given.