In article <rIC3g.944927$xm3.375291@attbi_s21>, "bobovski" <firstname.lastname@example.org> wrote:
> <email@example.com> wrote in message > news:firstname.lastname@example.org... > > 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.