Proof of relation between Betti numbers of base manifold and its covering space

Question

For a covering $p\colon\widetilde{M}\to M$ with compact $\widetilde{M}$, how to show that $b_i(M)\leq b_i(\widetilde{M})$ for $0<i<n=\dim(M)$.

If I understand this inequality correctly it says that every generator of $i$-th cohomology of $M$ is also a generator for $i$-th cohomology of $\widetilde{M}$. I am looking for a simple differential geometric proof using De Rham cohomology and a sketch using algebraic topology methods of this fact?

Here is a related post: Betti numbers of the orientable double cover is same as non-orientable one

Answer 1

You can define a natural pushforward $p_*$ of differential forms from those on $\tilde{M}$ to those on $M$. The definition is local, so to see what it looks like take a small open subset trivializing the covering map.

What is interesting is that $p_*$ commutes with the exterior derivative $d$, so that it induces a cohomology morphism $H^*(\tilde{M}) \rightarrow H^*(M)$. But then, you can see that $p_*p^*$ is a scalar map on differential forms, so that $p_*$ is surjective in cohomology, which concludes.