# Some Questions on Covering maps.

I want to know a few things about covering spaces/maps. First of all, it is true that covering maps induce injective homomorphisms between fundamental groups. Is it also true that covering maps produce injective homomorphisms between 1st homology groups? Also, is it true that covering maps preserve orientation whenever the two spaces in question are manifolds?

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The first homology group of $\Bbb R$ is zero, while the first homology group of the circle $\Bbb S^1$ is isomorphic to $\Bbb Z$, so injectivity fails in homology. –  Olivier Bégassat Apr 14 '13 at 1:39
@OlivierBégassat, that example trivially satisfies injectivity... –  Chris Gerig Apr 14 '13 at 2:25
The question essentially comes down to if a homomorphism $h:G \rightarrow H$ is injective, is $h^{ab}:G^{ab} \rightarrow H^{ab}$ injective? –  gary Apr 14 '13 at 2:33
@ChrisGerig Indeed, I got confused ^^ –  Olivier Bégassat Apr 14 '13 at 2:44
Take $C_4$ inside $D_8$. –  user641 Apr 14 '13 at 5:04

For a counterexample, considering the double cover $\mathbb{T}^2\rightarrow\mathbb{K}$, where $\mathbb{T}^2$ is the torus, and $\mathbb{K}$ is the Klein bottle.