Universal Covers of Homotopy-Equivalent Spaces

Proposition 123. Let $p:\cX\to X,q:\cY\to Y$ be universal covers where $X,Y$ are path-connected and locally path-connected. If $X\simeq Y$ then $\cX\simeq \cY$.

Lemma 1234. Let $f\in C(X,Y)$. Then $f$ is a homotopy equivalence if there exist maps $g,h\in C(Y,X)$ such that $fg\simeq 1_Y$ and $hg\simeq 1_X$.

Proof of Lemma 1234. 
Given that $fg\simeq 1_Y$ then $h(fg)\simeq h1_Y=h$. But $h(fg)=(hf)g\simeq 1_X g=g$. So $h\simeq g$, thus $1_X\simeq hf\simeq gf$, so $f$ is a homotopy equivalence.

Lemma 1234567.  Let $\fX\stackrel{p}{\to}\cX\stackrel{q}{\to} X$ a sequence such that $q$ and $qp$ are covering maps. If $X$ is locally path-connected then $p$ is a covering map.

Proof of Lemma 1234567. Let $e\in E$. Take a neighborhood $U$ of $q(e)$ which is path-connected and evenly covered with respect to both $q$ and $qp$. Then
$$q^{-1}(U)=\bigsqcup_{i\in I} V_i, (qp)^{-1}(U)=\bigsqcup_{j\in J}W_j$$
such that $q|_{V_i},(q p)|_{W_j}$ are homeomorphisms for each $i,j$. Fix $i$ such that $e\in V:=V_i$. Since $W$ is path-connected, we have $p(W_j)\subseteq V_{i_j}$ for each $j$ and some $i_j\in I$. But since $q|_{V_{i_j}}$ and $qp|_{W_j}$ are homeomorphisms then $p|_{W_j}:W_j\to V_{i_j}$ is a homeomorphism. If $p^{-1}(V)=\varnothing$ we're done. If not, then, since $p^{-1}(V)\subseteq (qp)^{-1}(U)$, if, for some $j$, $p^{-1}(V)\cap W_j\neq \varnothing$, since $p|_{W_j}$ is now a homeomorphism onto $V$ we get $p^{-1}(V)\cap W_j=W_j$. But then $p^{-1}(V)$ is just the union of such $W_j$'s, so $p$ is a covering map.


Proof of Proposition 123. If $X\simeq Y$ then there exist $f:X\to Y$ and $g:Y\to X$ such that $fg\simeq 1_Y$ and $gf\simeq 1_X$.
We have the following diagrams:

and applying the functor $\pi_1$, since $\cX$ and $\cY$ are simply connected we get the commutative diagrams

where $e$ is the trivial group. From this, $(f\circ p)_*=(g\circ q)_*=e=\pi_1(\cX)=\pi_1(\cY)$, so there exists lifts $F,G$ of $fp,gq$, i.e. maps $F:\cX\to\cY, G:\cY\to \cX$ such that the following diagram commutes:

Note that $GF$ is a lift of $gfp$. We also have a homotopy $h:I\times \cX\to X$ between $gfp$ and $p$ which lifts to an homotopy $H$ starting at $GF$.

Note also that the final branch $\sigma=H(1,-)$ has the property that $p\circ \sigma=h(1,-)=p$, i.e. the following triangle commutes

so $\sigma$ is a morphism of $\cX$.
So we have maps
$$\cX\stackrel{\sigma}{\to}\cX\stackrel{p}{\to} X$$
such that $p$ and $p\sigma$ are covering maps. Then, since $X$ is locally path-connected, $\sigma$ is a covering map. But since $\cX$ is simply connected, $\sigma$ is really an isomorphism.


So, since $GF\simeq \sigma$, then $(\sigma^{-1}G)F\simeq 1_\cX$. Analogously there is some isomorphism $\tau:\cY\to\cY$ such that $F(G\tau^{-1})\simeq 1_\cY$. Applying the lemma, we get the result.