Let $M^n,N^n$ be compact, orientable, smooth manifolds. If $f:N\to M$ is a nonzero degree smooth map, prove that $f^*:H^k(M)\to H^k(N)$ is injective for all $k=0,...,n$.
For $k=n$ and $\omega\in\Omega^n(M)$ with $f^*[\omega]=$, we have: $$0=\int_Nf^*[\omega]=\deg(f)\int_M[\omega]$$
And since $\deg(f)\neq 0$, we get $\int_M[\omega]=0\Rightarrow [\omega]=$.
My initial idea for $k<n-1$ was to take embbeded submanifolds $S^k\subset M^n, T^k\subset N^n$ of dimension $k$ so that if $\omega\in\Omega^k(M)$ is such that $f^*[\omega]=$, then $$0=\int_Tf^*[\omega]=\deg(f)\int_S[\omega]$$
then I could use the same argument above.
But this implicitly assumes that $f|_S$ maps $S$ to $T$, which is not necessarily true, so it doesn't really work, now I'm stuck.