Htykuut

Let $p:Erightarrow X$ be a covering map.

Let $U$ be a contractible open subset of $X$.

Is $U$ necessarily an evenly covered open subset, i.e. $p^{-1}(U)$ a disjoint union of open subsets of $E$ homeomorphic to $U$?

I know that if $X$ is path connected and locally path connected, then $U$ is an evenly covered open set. This can be shown by lifting the inclusion $Uhookrightarrow X$ to maps $U hookrightarrow E$.

What if $X$ is just a general topological space, not necessarily locally path connected and path connected?

The answer is "yes". To see this, let us recall the concept of a pullback. Given a pair of maps $(p : E to X,f : Y to X)$, a pullback of $(p,f)$ is given by a pair of maps $(p^* : E^* to Y,f^*: E^* to E)$ such that

(1) $fp^* = pf^*$

(2) For any pair of maps $(u : Z to Y, v : Z to E)$ such that $fu = pv$ there exists a unique map $w : Y to E^*$ such that $p^*w = u$ and $f^*w = v$.

This universal property implies that pullbacks are unique up to canonical homeomorphism. Here is an explicit construction:
$$E^* = { (e,y) in E times Y mid p(e) = f(y) }, p^*(e,y) = y, f^*(e,y) = e .$$
If $f$ is the inclusion map of a subspace $Y subset X$, then we may also take $E^* = p^{-1}(Y) subset E, p^*(e) = p(e), f^*(e) = e$.

It is well-known that pullbacks of covering projections are covering projections. This means that if $p$ is a covering projection, then so is $p^*$. See Pullback of a covering map.

Lemma : Let $p : E to X$ be a covering projection and $f_0, f_1 : Y to X$ be homotopic maps. Then the pullback covering projections $p^*_k : E^*_k to Y$ along $f_k$ are equivalent which means that there exists a homeomorphism $h : E^*_0 to E^*_1$ such that $p^*_1h = p^*_0$.

Proof. Let $H : Y times I to X$ be a homotopy from $f_0$ to $f_1$ and for $k = 0,1$ let $i_k : Y times { k } hookrightarrow Y times I$ denote inclusion. Form the pullback $(p^* : E^* to Y times I, H^* : E^* to E)$ of $(p,H)$ and the pullbacks $(pi_k : E^*_k to Y times { k }, i^*_k : E^*_k hookrightarrow E^*)$ of $(p^*,i_k)$, where $E^*_k = (p^*)^{-1}(Y times { k }) $. Then we may assume that $p^*_k = r_k pi_k : E^*_k to Y$, where $r_k : Y times { k } to Y, r_k(y,k) = y$.

Now the map $i_0 pi_0 : E^*_0 to Y times I$ lifts to $i^*_0 : E^*_0 to E^*$. Hence the homotopy
$$G : E^*_0 times I to Y times I, G(e,t) = (p^*_0(e),t)$$
which satisfies $G_0 = i_0 pi_0$ lifts to a homotopy $G' : E^*_0 times I to E^*$. The map $G'_1 : E^*_0 to E^*$ has the property $G'_1(E^*_0) subset (p^*)^{-1}(Y times { 1 }) = i^*_1(E^*_1)$. Therefore we get a map $h : E^*_0 to E^*_1$ such that $p^*_1h = p^*_0$. By lifting $G' : E^*_1 times I to Y times I, G'(e,t) = (p^*_1(e),1-t)$ we obtain a map $h' : E^*_1 to E^*_0$ such that $p^*_0h' = p^*_1$. It is easy to see that $h' h = id$ (this follows from unique path lifting).

Now let $U subset X$ be open and contractible. This means that the inclusion $i : U to X$ is homotopic to a constant map $c : U to X, c(y) = x_0$. Hence the covering projection $p : p^{-1}(U) to U$ is equivalent to the pullback of $p$ along $c$. But the latter is easily seen to be a trivial covering $U times F to U$ (recall $E^* = { (e,y) in E times U mid p(e) = c(y) = x_0 } = { (e,y) in E times U mid e in p^{-1}(x_0)) } , p^*(e,y) = y$). This means that $U$ is evenly covered.

Note that it suffices to assume that $i : U to X$ is inessential which is weaker than $U$ contractible.