Let $\pi : M^n \to N^n$ be a regular and finite sheeted smooth covering map between smooth $n$-manifolds. I know that the induced map $\pi^\ast : H^p_{dR}(N) \to H^p_{dR}(M)$ in de Rham cohomology is injective (also valid with compact support de Rham cohomology). Is it true that $\pi_\ast : H_k(M;\mathbb{Z}) \to H_k(N;\mathbb{Z})$ is surjective in integer homology?
My thoughts: let $\sigma : \Delta^k \to N$ be a $k$-simplex in $N$. Since $\Delta^k$ is simply connected, there exists a lift $\tilde{\sigma} : \Delta^k \to M$: $\pi(\tilde{\sigma})= \sigma$. If $\sigma$ is a cycle, is $\tilde{\sigma}$ also a cycle?
