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Topic: Top Homology of Manifold with Boundary is Zero. True? Why?
Replies: 2   Last Post: May 28, 2013 11:54 PM

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Bacle H

Posts: 283
Registered: 4/8/12
Top Homology of Manifold with Boundary is Zero. True? Why?
Posted: May 14, 2013 7:36 PM
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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?


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