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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
up vote 3 down vote accepted

Covering maps do induce injective maps on the first homotopy groups.

Covering maps don't induce injective maps on the first homology groups.

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.

This example also shows you can have an orientable manifold covering a non-orientable one.

You cannot, however, have a non-orientable manifold covering an orientable one. [Essentially, the "local homeomoprhism" nature of a covering allows one to lift the orientation from the target to the covering space.]

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