1
$\begingroup$

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 $π$?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
7
$\begingroup$

No, this is already false if $\pi $ is a Galois covering (i.e. $Y\cong X/G$): the index is the order of the abelianized group $G_{ab}$. Indeed from the exact sequence $$\pi _1(X)\rightarrow \pi _1(Y)\rightarrow G\rightarrow 1$$we get an exact sequence $$H_1(X,\Bbb{Z})\rightarrow H_1(Y,\Bbb{Z})\rightarrow G_{ab}\rightarrow 0\ .$$

Improve this answer
$\endgroup$
1
  • 2
    $\begingroup$ For anyone else momentarily confused by the notation: the $\pi_1$ in this answer refers to the fundamental group whereas the $\pi_*$ in the question indicates the map induced on homology by the covering $\pi: X \to Y$. $\endgroup$
    – Vidit Nanda
    Commented Jan 23, 2014 at 11:01

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .