Inclusion of the boundary of the mobius strip into the strip is a homology isomorphism

Question

As in the title, is the inclusion $i : \partial M \rightarrow M$ an isomorphism of the homology groups $H_n (\partial M), H_n(M)$, where $M$ is the Mobius strip?

Answer 1

Well, $M$ is homotopy equivalent to the circle (contract to central line) and $\partial M$ is a circle. But after pushing the inclusion of $\partial M$ to the central line it wraps round twice. So $i$ takes a generator of $H_1(\partial M)$ to twice a generator of $H_1(M)$. The answer is no.