The Euler characteristic and finiteness obstruction of manifolds with periodic ends

Details The Euler characteristic and finiteness obstruction of manifolds with periodic ends The Euler characteristic and finiteness obstruction of manifolds with periodic ends

2002-06-09

arxiv.org/abs/math/0206088v3

Asian J. Math., Volume 10, Number 4 (2006), 679-714

Mathematics

Differential Geometry

51 pages. Published version. Two small changes

Scientific paper

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.

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