# representing covering space by permutations

When I read the section of representing covering space by permutations of Hatcher's Algebraic Topology,he points that:

n-sheeted covering spaces of $$X$$(path-connected,locallypath-connected,semilocally simply connected) are classified by equivalence classes(cojugate class) of homomorphism $$\pi_1(X,x_0) \to \Sigma_n$$,where $$\Sigma_n$$ is n-symmetric group.

Then does it imply that given a homomorphism $$\pi_1(X,x_0) \to \Sigma_n$$,there is corresponding covering space?If so,how to deal with it?I can't find it in the book.

Consider a representation $$h:\pi_1(M)\rightarrow \Sigma_n$$ and $$U_n=\{1,...,n\}$$ the following $$n$$-cover can be associated: the quotient of $$\hat X\times U_n$$ by the diagonal action of $$\pi_1(M)$$ where $$\hat X$$ is the universal cover of $$X$$.

Conversely given a $$n$$-cover $$p:N\rightarrow M$$ you can associate to it its holonomy obtained by the action of $$\pi_1(M)$$ on the fibre of any element of $$M$$, these actions are conjugate.

• It's worth pointing out that this is an instance of the "Borel construction": given a group $G$ and two $G$-spaces $X$, $Y$, their Borel construction is $X\times Y/\sim$ where $(x,y)\sim(xg, gy)$ (often people require $X$ to be a right $G$-space and $Y$ a left $G$-space for notation's sake). A typical situation is when $X$ is a principal $G$-bundle and $Y$ is an arbitrary $G$-space, and the effect of the construction is "changing the fibres" of $X$ from $G$ to $Y$. In our case the principal $\pi_1M$ bundle is $\hat{X}$ and the new fibres are $Y = U_n$. Commented Jan 27, 2019 at 17:23

One (maybe roundabout) way to see this is with classifying spaces and homotopy theory, at least when $$X$$ is a CW-space. (You will encounter all of these ideas in Hatcher at some point if you have not already.) Tsemo Aristide's answer is certainly cleaner and works for a broader class of spaces, but the concepts here might help in other situations.

"Intuitively" the idea is that since $$B\Sigma_n$$ only has non-vanishing homotopy in degree 1, homotopy classes of maps into it only depend on homotopical information up to degree 2, and in particular homotopy classes of maps to $$B\Sigma_n$$ are the same for spaces that have the same $$2$$-skeleton. Then you have to know that $$X$$ and $$B\pi_1X$$ have the same $$2$$-skeleton, and about how group homomorphisms correspond to maps between classifying spaces. I will elaborate.

An $$n$$-sheeted covering is the same thing as a fibre bundle whose fibre is a set of cardinality $$n$$ and whose structure group is $$\Sigma_n$$; therefore they are classified by homotopy classes of maps $$[X, B\Sigma_n]$$ where $$B\colon Grp \to Top$$ is a classifying space functor. Since $$\Sigma_n$$ is a discrete group it follows that $$B\Sigma_n \sim K(\Sigma_n, 1)$$, the "Eilenberge-Maclane space" defined up to homotopy equivalence by the properties $$\pi_1K(\Sigma_n, 1)\cong \Sigma_n$$ and $$\pi_iK(\Sigma_n, 1)=0$$ for other values of $$i$$. Since the higher homotopy groups of $$B\Sigma_n$$ all vanish, a result from obstruction theory is that

$$[X,B\Sigma_n] \cong [X^{(2)}, B\Sigma_n]$$

where $$X^{(2)}$$ is the $$2$$-skeleton of $$X$$. That is, the (isomorphism class of the) covering space space over $$X$$ is determined by its restriction to the $$2$$-skeleton.

Here's where things get a bit funny: the $$2$$-skeleton of $$X$$ is also the $$2$$-skeleton of a model of $$B\pi_1 X$$. This is because since $$\pi_1X$$ is discrete its classifying space is again an Eilenberg-Maclane space in degree 1, so we can construct a CW model from a group presentation of $$\pi_1X$$ by taking a 1-cell for every generator and attaching 2-cell along every relation, and then adding higher-dimensional cells to kill off any higher homotopy we may have introduced. But the $$2$$-skeleton of $$X$$ determines a presentation of $$\pi_1X$$ so it is also the $$2$$-skeleton of the Eilenberg-Maclane construction. Therefore we get

$$[X,B\Sigma_n] \cong [X^{(2)}, B\Sigma_n] = [(B\pi_1X)^{(2)}, B\Sigma_n]\cong [B\pi_1 X, B\Sigma_n]$$

Now the last step is to establish the correspondence between $$[BG, BH]$$ and conjugacy classes of homomorphisms $$G\to H$$. I will see if I can remember a clean way of showing this and make an edit later... Again, I believe it is also in Hatcher.