Degree One Map induces Surjections on Homology

Question

Is the following statement true:

If $f:M\to N$ is a degree one map of compact closed manifolds, then $f$ induces surjections $f^*:H_q(M)\to H_q(N)$.

I found this claimed on http://topospaces.subwiki.org/wiki/Degree_one_map . The $H_1$ case is clear to me, but I don't see how to generalize this.

Answer 1

By Poincare duality every class $x \in H_p(M;R)$ is of the form $[M] \frown \psi$, for $\psi \in H^{n-p}(M; R)$.

Now if $y \in H_p(N; R)$, then we know $y = \psi_y \frown [N]$.

Now generally for a homology class $x \in H_*(M;R)$ and $\psi \in H^*(N;R)$ we have that $f_*(x) \frown \psi = f_*(x \frown f^*\psi)$. Applying this to $x = [M]$ and $\psi = \psi_y$ we find that

$$y = [N] \frown \psi_y = f_*([M] \frown f^* \psi_y),$$ and in particular $y$ is in the image of $[M] \frown f^* \psi_y$.

Therefore, the induced map on homology of a degree-1 map is necessarily surjective.