
Re: compact 3manifold cohomology problem
Posted:
Apr 26, 2006 3:55 PM


In article <rIC3g.944927$xm3.375291@attbi_s21>, "bobovski" <bobovski@spamlessgmail.com> wrote:
> <magya_bloom@yahoo.com> wrote in message > news:1146025092.929575.84110@u72g2000cwu.googlegroups.com... > > reposting the problem: for a compact nonorientable 3manifold X, how > > do we prove that cohomology in degree 1, H^1(X;Z) is nonzero? 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 nonorientable > 3mainfolds should finish this off rather easily (shouldn't there be a > nonorientable 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 3manifolds with only irreducible factors.

