Example of covering space with homology non-isomorphic to base space's homology

Question

Let $X \to Y$ be a covering with finite number of sheets, where $X$ and $Y$ are connected simplicial complexes. Can you provide an example when homology groups of $X$ and $Y$ with coefficients in $\mathbb{Z}_2$ are non-isomorphic?

Answer 1

Almost any example will do. For instance, if $Y$ is a wedge of two circles then it has a 2-sheeted covering space $X$ that is homotopy equivalent to a wedge of three circles. (If you want a picture of how this works, see example (1) on page 58 of Hatcher's Algebraic Topology.) Then $H_1(X;\mathbb{Z}_2)\cong\mathbb{Z}_2^3$ but $H_1(Y;\mathbb{Z}_2)\cong\mathbb{Z}_2^2$.

Answer 2

Trying to answer my own question after receiving some useful hints.

Let $\Sigma_g$ be a surface of genus $g$, then consider a covering $\Sigma_m \to \Sigma_g$, where $m = n(g-1)+1$ and $n$ is a number of sheets. The first homology groups are $H_1 (\Sigma_m) = \mathbb{Z}_2^{2m}$, $H_1 (\Sigma_g) = \mathbb{Z}_2^{2g}$ and they are non-isomorphic.