M ar 1 99 9 A Natural Framing for Asymptotically Flat Integral Homology 3 - Sphere Su - Win

For an integral homology 3-sphere embedded asymptotically flatly in an Euclidean space, we find a natural framing extending the standard trivialization on the asymptotically flat part. Suppose M is a 3-dimensional closed smooth manifold which has the same integral homology groups as the 3-sphere S. x0 is a fixed point in M . Embed M in a Euclidean space R such that x0 is the infinite point of the 3-dimensional flat space R × {0} of R and a neighborhood of x0 contains the whole flat space R × {0} except a compact set. Precisely, for any positive number r, let Br denote the closed ball of radius r in R 3 and Nr = (R 3 − Br) × {0}; there exists r0, a positive number, such that Nr0 is contained in M and Nr0 ∪ {x0} is an open neighborhood of x0 in M . Let M = M − {x0}, it is an asymptotically flat 3-dimensional manifold with acyclic homology. The main purpose of this article is to define a natural framing for M . If we identify the tangent spaces of points in the flat part Nr0 with R 3 × {0}, then the tangent bundle of M can be thought as a 3dimensional vector bundle over the closed manifold M0 = M/N s, where s is a number greater than r0 and N s is the closure of Ns; we shall call this vector bundle the tangent bundle T (M0) of M0. And our natural framing is just a trivialization of T (M0), which corresponds to a trivialization of the tangent bundle T (M) whose restriction to the flat part is the standard trivialization on R. Because M0 is a closed 3-manifold, there are countably infinite many