Impact of the rank of a smooth map on the homology groups

Question

Given $f:N\to M$ a smooth map between compact smooth manifolds, suppose the induced map $f_*:H_{l}(N,\mathbb{R})\to H_{l}(M,\mathbb{R})$ is nonzero for some $l\in \mathbb{N}$, then I want to show that the rank of $df$ is $\ge l$ at some point of $N$.

Intuitively I believe it's correct, but I'm not able to make the statement precise. Thanks for your help!

Answer 1

Hint: Consider the induced map on de Rham cohomology.

A full proof is hidden below.

Suppose $df$ has rank less than $l$ at every point. Then for any $l$-form $\omega$ on $M$, $f^*\omega=0$ (the induced map on $1$-forms at a point is just the dual of $df$ and so has rank $<l$, and thus its $l$th exterior power is $0$). It follows that the induced map $f^*:H^l_{dR}(M,\mathbb{R})\to H^l_{dR}(N,\mathbb{R})$ on de Rham cohomology is $0$. By the de Rham theorem, this $f^*$ is dual to $f_*:H_l(N,\mathbb{R})\to H_l(M,\mathbb{R})$, so $f_*$ must be $0$ as well.