covering space which is a homotopy equivalence How can I prove that a covering space which is a homotopy equivalence is a homeomorphism?From the basic property of lifting homotopies I get that there exists $h:Y \rightarrow X $ such that:
$ p \circ h = id_Y $
Where $p:X \rightarrow Y $ is a covering.
 A: If it is an homotopy equivalence, in particular it induces an isomorphism in fundamental groups. Now use the classification of covering spaces in terms of subgroups of the fundamental group.
A: It is not difficult to show that, given a homotopy equivalence $f:X\to Y$ with inverse $g:Y\to X$, whenever there is a path $\kappa:f(x)\leadsto f(x')$, then there is also a path $\lambda:x\leadsto x'$ such that $f\lambda\simeq\kappa$, for any points $x,x'$ in $X$. If you want to prove this, I recommend first showing that two paths $\kappa,\kappa':y\leadsto y'$ with homotopic images $g\kappa,g\kappa'$ are homotopic. Then, given a path $\kappa:f(x)\leadsto f(x')$, show that $g(\kappa)\simeq gf(\lambda)$ for $\lambda=\sigma_x g(\kappa)\sigma_{x'}^{-1}$, where $\sigma_x(t)=H(x,t)$ and $H$ is the homotopy between $1_X$ and $gf.$
Now, given two points $x$ and $x'$ in your covering space $X$ such that $y=p(x)=p(x')$, there exists a path $\lambda$ from $x$ to $x'$ such that $p\lambda$ is nullhomotopic, since $p:X\to Y$ is a homotopy equivalence. Since $p$ is a covering map, the class $[1_x]$ is the unique class of paths starting at $x$ such that their images under $p$ are homotopic to the identity at $y$. It follows that this is the class of $\lambda$, so $\lambda$ is nullhomotopic and $x=x'$.
The problem has an elegant approach using groupoids. To outline this approach:


*

*A homotopy equivalence $f:X\to Y$ induces an equivalence (as categories) between their fundamental groupoids $\pi f:\pi X \to \pi Y$

*An equivalence of categories is a full functor

*A covering map $p:X\to Y$ induces a covering morphism $\pi p: \pi X \to \pi Y$ of groupoids

*A covering morphism which is full is an isomorphism.


Finally, if a covering map induces an isomorphism of fundamental groupoids, it is injective and thus a homeomorphism.
The first two paragraphs basically just follow this scheme, using the language of topology.
If you want to learn about groupoids and their applications in topology, I recommend the book Topology and Groupoids by Ronald Brown.
