The Lefschetz fixed point theorem for noncompact locally connected spaces

Abstract

Leray's notion of convexoid space is localized and used to show that if f: M- M is a relatively compact map on a locally convex manifold M, and f has no fixed points then its Lefschetz trace is zero. A similar theorem holds for certain ad junction spaces Y Uj Z where Y is Q-simplicial and Z is locally convexoid. A number of other properties of locally convexoid spaces are derived; for example, any neighborhood retract of a locally convexoid space is locally convexoid. The Lefschetz fixed point theorem has long been known to hold for compact manifolds, including compact homological manifolds. By restricting attention to compact maps, Leray extended the Lefschetz theorem to open subsets of Banach spaces and more recently, Browder and Eells have obtained it for infinite dimen- sional manifolds modelled on Banach and Frechet spaces. The Leray, Browder and Eells results were each obtained by a reduction to the Lefschetz theorem for compact spaces. This same technique does not seem to apply to other infinite dimensional manifolds, such as those modelled on Montel spaces. In this paper we will consider this problem in the general context of regular Hausdorff locally connected spaces. For spaces which are locally connected in the sense of Cech, the Lefschetz fixed point theorem will be obtained for compact maps which have finite dimensional image. By localizing Leray's notion of convexoid space, and freeing it from compactness, it will be possible to obtain the Lefschetz theorem for compact maps of locally convex manifolds with no dimensional restrictions. Because of its applicability to noncompact spaces, the notion of Q?simplicial space will be used throughout, along with related techniques. As an auxiliary to these techniques, a relative form of the classical method of acyclic carriers, the method of acyclic pairs of interlaced carriers, will be applied- whenever it will be necessary to construct chain maps,