Example of a non surjective and non injective induced homomorphism on $H_1$.

Question

I'm trying to solve the following problem from a past qualifying exam on algebraic topology:

Give an example of a space $X$ and a finite sheeted, connected cover $p:Y\to X$ such that the induced map $p_*:H_1(Y,\mathbb{Z})\to H_1(X,\mathbb{Z})$ is not surjective and not injective. ($H_1$ are the first homology groups)

The Galois correspondence might help but I'm not sure where to start. Any hints are appreciated.

I wonder if such a covering space exists for $S^1\vee S^1$.

Answer 1

Using the correspondence between fundamental group and first homology, along with the correspondence between inclusions of subgroups of the fundamental group and covering spaces, the question reduces to the following:

Find a group $G$ and subgroup $H$ such that upon abelianization the inclusion does not induce an injective map or a surjective map.

Then just construct a space with fundamental group $G$ and take the appropriate cover.

Answer 2

Nonsurjective is relatively simple. For example $f:S^1 \to S^1$ given by $z \mapsto z^2$ corresponds to the map $ \mathbb Z \to \mathbb Z, x \mapsto 2x$ on the first integral homology (try degree or the Hurewicz isomorphism, or any homology theory where you know a generator for $H_1$)

Noninjective is a little trickier, since covering maps are injective on $\pi_1$ you are going to need an example with nonabelian fundamental group.

I think one example is taking the Dihedral group $D_3$ and its presentation complex will give a topological space $X$, and the subgroup consisting of rotations will correspond to some cover $p:Y \to X$, and then you can check what the possible homomorphisms $p_*:\mathbb Z/3 \to \mathbb Z/2 $ (since $\mathbb Z/2$ is the abelianization of $D_3$.)