I asked this question on math.se (<https://math.stackexchange.com/questions/647930/image-of-the-map-on-homology-induced-by-a-covering>), but it have not attracted much of attention.


Let $X$ and $Y$ are two compact connected oriented 2dim smooth manifolds, and $\pi \colon X\to Y$ is an unramified covering of a finite degree. Consider the induced map $\pi\_\*\colon H\_1(X,\mathbb Z)\to H\_1(Y,\mathbb Z)$.


Question: is it true that the image of $\pi\_\*$ is a sublattice of $H\_1(Y,\mathbb Z)$ of index $\#G$, where $G$ is the deck transformation group of $π$?



Image of the map induced on homology by a covering
No, this is already false if $\pi $ is a Galois covering (i.e. $Y\cong X/G$): the index is the order of the *abelianized* group $G\_{ab}$. Indeed from the exact sequence
$$\pi \_1(X)\rightarrow \pi \_1(Y)\rightarrow G\rightarrow 1$$we get an exact sequence
$$H\_1(X,\Bbb{Z})\rightarrow H\_1(Y,\Bbb{Z})\rightarrow G\_{ab}\rightarrow 0\ .$$ 


