Covering Space of Triangulable Space

Question

Assuming a triangulable space is one homeomorphic to a simplical complex.

How can one prove that any covering space of a triangulable space is triangulable?

I know that one can lift the homeomorphisms on each simplex to the covering space by using lift theorems and the fact that simplex are simply connected. But I can't seem to find a way to paste this "local" homeomorphisms into a global homeomorphisms or to describe the simplical complex associated to the covering space.

My question is motivated by the result relating euler characteristic of a covering space with the one of its base space.

Answer 1

What you can always do if you are lazy (since you just care for "triangulable" and not for an induced or inherited simplicial structure), you can choose a barycentric subdivision so that neighborhods of the simplices lie entirely in one of (previously fixed) the trivializing open sets.

Answer 2

I don’t know about simplicial complexes but I can tell you (or just future visitors, considering the age of this post) about simplicial sets. Every simplicial complex is apparently the geometric realisation of a simplicial set, and simplicial sets are fundamental (apparently?) to algebraic topology, more so than these ‘complexes’.

In “Calculus of Fractions and Homotopy Theory”, the authors demonstrate to a good level of detail that, for any simplicial set $X$, the category of covering spaces of the space $|X|$ is equivalent to the category of simplicial coverings of $X$. In particular, there is a claim that for any covering map $p:Y\to|X|$, there is a simplicial set $K$ so that $Y\cong|K|$ over $|X|$ (they are isomorphic as covering spaces of $X$).

Specifically, if $\eta_X:X\to\mathsf{S}|X|$ is the adjunction unit, then $K$ is the pullback of $\mathsf{S}p:\mathsf{S}Y\to\mathsf{S}|X|$ along $\eta_X$. The map $K\to X$ is a simplicial covering that realises to a topological covering map $|K|\to|X|$, and the homeomorphism with $Y$ is taken by the composite $|K|\to|\mathsf{S}Y|\to Y$ where the latter map is the counit $\epsilon_Y$ and the former is the realisation of the pullback projection map. You check this composite is a homeomorphism by observing it places the fibres of every vertex of $X$ in $|K|,Y$ in bijection; noting covering maps surject on all connected components that they touch, and that this composite is a covering map since the base space $|X|$ is locally connected, you can deduce from bijecting the fibres for vertices that it is a one-sheeted cover i.e. a homeomorphism.

As an aside, for any locally path connected $Y$ admitting a simply connected covering space, we find the coverings of $Y$ are equivalent to the coverings of $\mathsf{S}Y$.

As another aside, these notes sketch an equivalence between the subgroups of the fundamental group of a connected simplicial set and coverings of that simplicial set. Taking realisations then gives you a nice “algorithm” for building, e.g., universal covers of ‘simplicial complexes’,’delta-sets’ and realisations of simplicial sets. It is enlightening to carry out this computation by hand for simple cases (e.g. the Möbius strip): it felt a little bit like magic, to be honest, the way everything automatically fits together in the correct way.