## Poincare duality [closed]

How is following map isomorphism by Poincare duality?

It seems reasonable to suppose that w_1 means the first Stiefel-Whitney class and therefore that the coefficients of these cohomology groups are supposed to be $\mathbb Z/2$. Some hypotheses on M and p will be needed, because the map surely isn't an isomorphism in general. Offhand, I can't think of reasonable hypotheses, but presumably the source of this question contained some hypotheses. – Andreas Blass Nov 9 2011 at 22:35
There's no reason to think that cup product with w1 will be an isomorphism; after all, $w_1$ could be 0, and even if it is non-0, it will generally fail to be a zero-divisor on $H∗(M)$. Please rethink your question! – Charles Rezk Nov 9 2011 at 22:37