The Euler characteristic and finiteness obstruction of manifolds with periodic ends

Abstract

Let M be a complete orientable manifold of bounded geometry. Suppose that M has finitely many ends, each having a neighborhood quasi-isometric to a neighborhood of an end of an infinite cyclic covering of a compact manifold. We consider a class of exponentially weighted inner products (\cdot ,\cdot)_k on forms, indexed by k>0. Let \delta_k be the formal adjoint of d for (\cdot ,\cdot)_k. It is shown that if M has finitely generated rational homology, d+\delta_k is Fredholm on the weighted spaces for all sufficiently large k. The index of its restriction to even forms is the Euler characteristic of M. This result is generalized as follows. Let \pi =\pi_1(M) . Take d+\delta_k with coefficients in the canonical C^{*}(\pi) -bundle \psi over M. If the chains of M with coefficients in \psi are C^{*}(\pi) -finitely dominated, then d+\delta_k is Fredholm in the sense of Miscenko and Fomenko for all sufficiently large k. The index in \tilde{K}_0(C^{*}(\pi)) is related to Wall's finiteness obstruction. Examples are given where it is nonzero.