2
$\begingroup$

I just found this document http://www.math.toronto.edu/~drorbn/classes/0405/Topology/CoveringSpaces/CoveringSpaces.pdf in which it is said that we can use the classification of covering spaces theorem :

"Theorem: If $B$ is connected and locally connected with base point $b$ and fundamental group $G=\pi_1(B; b)$, then the map which assigns to every covering $p:X\rightarrow B$ its fiber $p^{-1}(b)$ over the basepoint $b$ induces a functor $\mathcal F$ from the category $C(B)$ of coverings of $B$ to the category $S(G)$ of $G$-sets — sets with a right $G$-action and set maps that respect the $G$ action. If in addition if $B$ is semi-locally simply connected then the functor $\mathcal F$ is an equivalence of categories. (In fact, this is iff)"

to show that "Corollary 8: If $B$ is semi-locally simply connected, then for every subgroup $H$ of $\pi_1(B)$ we can found a connected covering $p:X\rightarrow B$ such as $p_*(\pi_1(X))=H$."

But I don't know how to use this theorem (the equivalence of categories) to get the corollary. Could you give me some advices ? Thanks

$\endgroup$
1
$\begingroup$

The covering you are looking for is the one corresponding to the $G$-set $H\backslash G$ by the theorem. This is the quotient of the universal covering of $B$ (which corresponds to the $G$-set $G$) by the action of $H$.

$\endgroup$
1
  • 1
    $\begingroup$ Thanks, I was writting this answer when you posted yours: The functor $\mathcal F$ associates a covering $p:X\rightarrow B$ to $p^{-1}(b)$ for the monodromy action ($\forall[\gamma]\in\pi_1(B,b),\,\forall x\in p^{-1}(b),\,[\gamma]\cdot x=\tilde\gamma(1)$ where $\tilde\gamma$ is the only lift of $\gamma$ such as $\tilde\gamma(0)=x$). Now we can consider $H$ a subgroup of $\pi_1(B,b)$, then $\pi_1(B,b)$ acts on $\pi_1(B,b)/H$ and the stabilizer of $H$ is the subgroup $H$. But $\mathcal F$ is an equivalence of categories, so it exists a covering whose monodromy action is the one we just defined. $\endgroup$
    – user15542
    Sep 4 '11 at 11:44
3
$\begingroup$

These results are illuminated by the notion of covering morphism of groupoids and the result that:

  1. If $G$ is a groupoid, then there is an equivalence of categories between covering morphisms of $G$ and actions of $G$ on sets.

  2. If $X$ is a suitably nice space, then the fundamental groupoid functor gives an equivalence of categories between covering maps of $X$ and covering groupoids of $\pi_1(X)$.

Note that because this models a map of space by a morphism of groupoids, this model is somewhat nearer to intuition, and is easier to apply, than the corresponding action on sets. These ideas are applied in a recent paper

Jeremy Brazas, Semicoverings: a generalization of covering space theory, Homology, Homotopy and Applications, 14(1), (2012) 33-63. (downloadable)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.