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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?

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  • $\begingroup$ see also mathoverflow.net/q/16632/27004 $\endgroup$ – wonderich Mar 24 at 22:01
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    $\begingroup$ 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 $\Bbb Z/2$". $\endgroup$ – Mike Miller Mar 24 at 22:29