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I asked this question on math.se (http://math.stackexchange.com/questions/647930/image-of-the-map-on-homology-induced-by-a-coveringhttps://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 $π$?

I asked this question on math.se (http://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 $π$?

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 $π$?

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Image of the map induced on homology by a covering

I asked this question on math.se (http://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 $π$?