It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a *good cover,* i.e., a locally finite cover by open balls such that all nonempty intersections of the elements of this cover are still homeomorphic to open balls. The same is true, more generally, for PL manifolds.

**Question:** Do all topological $n$-manifolds admit good covers?

I am especially curious about the case $n=4$.

Here is my guess: a good cover is too close to a handle decomposition for gauge-theoretic obstructions to rule them out in dimension 4. In higher dimensions, I do not even have a guess.

**Edit:** The notion of a good cover has a homotopy analogue, a *homotopy good cover* where the requirement is that all elements of the covering and all their intersections are contractible. Here is a couple of observations about existence of homotopy good coverings:

Every simplicial complex, of course, admits a homotopy good cover; in particular, all topological manifolds which admit triangulations (not necessarily PL) admit homotopy good coverings.

Many (if not all) simply connected closed 4-manifolds admit homotopy good covers; this is a corollary of Freedman's work. Existence of homotopy good covers for all simply connected manifolds essentially hinges on the case of $*CP^2$.