Date: May 14, 2013 7:36 PM
Author: Bacle H
Subject: Top Homology of Manifold with Boundary is Zero. True? Why?

Hi, All:

I'm trying to see if it is true that the top homology of

an n-manifold with boundary is zero. I have tried some examples;

a closed annulus ( homotopic to S^1 , so H^2(S^1)=0 , R^n with

a boundary copy of some R^m m<n retracts to R^m , etc.)

It seems strange that a manifold would "lose its orientability"

if we capped-in a boundary. It seems like this boundary would bound

all the n-cycles that had no boundary before the boundary was capped-in;

otherwise, how is the quotient Cycles/Boundaries suddenly zero?

I tried some MAyer-Vietoris, but got nowhere. Any Ideas?