Homotoping non zero degree arbitrary maps to a covering map

Question

Let $f:M\rightarrow N$ be a non zero degree smooth map between compact connected oriented manifolds of the same dimension. Must $f$ be homotopic to a covering map ?

This seems too good to be true, So I m expecting counterexamples

Answer 1

For every $d \in \mathbb{Z}$, there is a map $f_d : S^2 \to S^2$ with $\deg f_d = d$. On the other hand, as $S^2$ is simply connected, the only connected covering is $\operatorname{id} : S^2 \to S^2$ which has degree $1$. As degree is a homotopy invariant, we see that the claim is false.