We know that covering spaces of many low dimensional CW complexes such as graph (CW structures with only 0-cells and 1-cells) and all compact surfaces has CW structure. [The second fact is due to Uniformation Theorem from theory of Riemann surfaces. Basically, the only simply connected 2-manifold are the sphere, the Euclidean plane and hyperbolic plane.]
So I wonder, as put forth in the title, if this is true in general. That is, if $X$ is connected, locally path-connected and it has a CW complex structure then does its universal covering space also have a CW complex structure?
Best Answer
This fact is statement (N) in Whitehead's paper "Combinatorial homotopy I". The harder part is to prove that the covering complex has the CW-topology. That is also carefully proved in the book by W.S. Massey, "Algebraic topology: an Introduction".
The result on the CW-topology may also be proved as a corollary of general results proved in P.I. Booth and R. Brown, ``Spaces of partial maps, fibred mapping spaces and the compact-open topology'', Gen. Top. Appl. 8 (1978) 181-195. This gives reasonable circumstances under which if $p: E \to B$ is a map, then the pullback functor $p^*$ from spaces over $B$ to spaces over $E$ has a right adjoint, and so preserves colimits. This immediately gives the result, and even implies more general types of result, for certain generalised inductively defined structures.