# Integration on an non-orientable manifold [closed]

Suppose $$M_n$$ is a $$n$$ dimensional non-orientable manifold.

I am interesting in knowing whether the following statements are true:

1. A characteristic class $$w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$$ can be integrated on $$M_n$$ only when $$p=2$$.
2. Suppose $$n_1+n_2=n$$, and $$n_i, (i=1,2)$$ are positive integers. $$w_{n_i}^{(p)} \in H^{n_i}(M_n, \mathbb{Z}_p), i=1,2$$. Then $$w_{n_1}^{(p)}\cup w_{n_2}^{(p)}$$ can be integrated on $$M_n$$ only when $$p=2$$.

If not true, is there a counter example?

• What do you mean by (1)? It follows from Poincare duality that for a closed non-orientable manifold of dimension $n$, we have $H^n(M;R) \cong R/2$. If $R = \Bbb Z/p$ for $p$ odd, then this group is just zero, so I suppose under any reasonable interpretation the answer to both (1) and (2) is (for $p$ prime) "yes, only when $p=2$". If you didn't just mean $p$ prime, and $p$ is even, then $(\Bbb Z/p)/2 = \Bbb Z/2$, so the answer to (1) and (2) is "the integral is an element of $\Bbb Z/2$". – Mike Miller Mar 24 at 22:29