Homology of non-orientable manifolds.

Question

Is there any example of non-orientable manifold with boundary M of dimension n>2 with the property that $H_i(M, Z_2)$ vanishes for all $0<i<n$? If it does not exist, is there any construction that shows that $H_i(M, Z_2)$ is different from 0 for some $0<i<n$?

Answer 1

A non-orientable manifold has a connected double cover called the orientation double cover (see proposition 3.25 in Hatcher). Because it's a double cover, it induces an index-two subgroup of $\pi_1(M)$ (the subgroup of loops whose lifts are also loops). Quotienting out by this subgroup gives a nonzero homomorphism $\pi_1(M) \to \Bbb Z/2$, and hence (by the universal coefficient theorem and the fact that $H_1(M;\Bbb Z)$ is the abelianization of $\pi_1(M)$) a nonzero element of $H^1(M;\Bbb Z/2) = \text{Hom}(H_1(M;\Bbb Z);\Bbb Z/2) = \text{Hom}(\pi_1(M);\Bbb Z/2)$. By applying the universal coefficient theorem again with different coefficients we also see that $H^1(M;\Bbb Z/2) = \text{Hom}(H_1(M;\Bbb Z/2),\Bbb Z/2)$, and hence that $H_1(M;\Bbb Z/2)$ is nonzero as desired.