The Euler Characteristic and Finiteness Obstruction of Manifolds with Periodic Ends

The analytic index of the operator d+δ on a compact orientable Riemannian manifold M is the Euler characteristic of M, χ (M) . This paper extends this result to a class of complete noncompact manifolds, those with finitely generated rational homology and finitely many quasi-periodic ends. The latter term means that there is a neighborhood of each end which is quasi-isometric to a neighborhood of an end of an infinite cyclic covering of a smooth compact manifold. One reason for interest in such manifolds is a result stated by Siebenmann [33] and proved by Hughes and Ranicki [10]: if M is a manifold of dimension n > 5 with finitely many ends satisfying a certain tameness condition, then each end has a neighborhood homeomorphic to a neighborhood of an end of an infinite cyclic covering of a compact topological manifold. d+ δ acting on L forms is rarely a Fredholm operator. We consider more generally weighted L spaces. These were first used in index theory on manifolds with asymptotically cylindrical ends by Lockhart and McOwen [18] and Melrose and Mendoza. Let ρ (x) be a smooth nonnegative function on M with bounded gradient which tends to ∞ at ∞. Let k > 0. The weighted inner product on compactly supported smooth forms is (u, v)k = (k u, kv), where (·, ·) is the L inner product. The weighted forms are obtained by completion. In other words, they are the L space of the measure kdx, where dx is the Riemannian measure. In the quasi-periodic case ρ (x) is chosen to change approximately linearly under iterated covering translations. We consider the operator Dk = d + δk, where δk is the formal adjoint of d for the weighted inner product. Dk is essentially self-adjoint. We denote by D̄k the closure of Dk. Let D̄ even k be its restriction to even forms. Let χ and χ be the Euler characteristic of the homology and locally finite homology of M. The first main result follows.