Subject: Re: Homology of the boundary [for toplist] Date: Sun, 4 Mar 2001 15:37:53 -0800 From: Greg Kuperberg On Mon, Feb 26, 2001 at 06:33:58PM +0100, David Cohen-Steiner wrote: > What are the connections between the homology of a manifold and the > one of its boundary ? > Under which assumptions is the boundary of a contractible manifold > a homology sphere? What are the relevant tools? If you mean a compact manifold, then the basic tools are Poincare duality, the universal coefficient theorem, and the exact sequence of a pair. Suppose that M is the boundary of an oriented n-manifold W. Then the exact sequence of a pair and Poincare duality gives you a beautiful "ladder" diagram: ... -> H_k(M) -> H_k(W) -> H_k(W,M) -> H_{k-1}(M) -> ... | | | | v v v v ... -> H^{n-k-1}(M) -> H^{n-k}(W,M) -> H^{n-k}(W) -> H_{n-k}(M) -> ... The vertical links are the Poincare maps; they are isomorphisms. I keep forgetting whether the squares in this diagram commute or anticommute. It doesn't usually matter much. If W is a "homology R^n", in particular if it is contractible, then the terms for W, and hence (W,M), all vanish except for H_0(W) = H^n(W,M) = \Z. This implies that M is indeed a homology sphere, using no assumptions other than compactness and Poincare duality. In my opinion Poincare duality is a decidedly non-trivial result for topological manifolds, but it is true. For PL manifolds it is relatively easy and it carries over to smooth manifolds once you know that smooth manifolds can be triangulated. -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ \/ * All the math that's fit to e-print *