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Topic: compact 3-manifold cohomology problem
Replies: 10   Last Post: Apr 28, 2006 10:51 AM

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Chan-Ho Suh

Posts: 425
Registered: 12/10/04
Re: compact 3-manifold cohomology problem
Posted: Apr 26, 2006 3:55 PM
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In article <rIC3g.944927$xm3.375291@attbi_s21>,
"bobovski" <> wrote:

> <> wrote in message

> > reposting the problem: for a compact non-orientable 3-manifold X, how
> > do we prove that cohomology in degree 1, H^1(X;Z) is non-zero? Poincare
> > duality does not yiled the answer, and i wonder if the euler
> > characteristic has something to do with it?
> >

> As I mentioned earlier, I think the prime decomposition for non-orientable
> 3-mainfolds should finish this off rather easily (shouldn't there be a
> non-orientable S^2 bundle over S^1 as a summand?).

No, not if it's irreducible, as it is if say, it's universal cover is
R^3. So the prime decomposition doesn't help you, except to reduce the
problem to 3-manifolds with only irreducible factors.

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