TITLE: Manifolds covered by a single disc
QUESTION [6 upvotes]: I feel like this is should be well-known, but I cannot seem to find a straight answer anywhere. Under what conditions can a (closed) manifold be obtained by identifying points on the boundary of the unit disc? Said another way, for which M is is the case that $M$ is diffeomorphic (NOT just homotopy equivalent) to $D^n/x\sim f(x)~$ for some $f: S^{n-1}\to S^{n-1}$? (I don't require f to be smooth, but it shouldn't be too pathological either. Let's say it should be piecewise smooth).  
Is there a smooth or topological invariant that would obstruct this? 
Any references would also be appreciated.

REPLY [7 votes]: The answer (to the first question) is yes in the smooth case: If $M^m$ is closed and compact, then there is a Morse-Smale function on $M$ with a single critical point of index $m$.  Work by Lizhen Qin shows how to give a CW structure on $M$ such that the interiors of the cells correspond to the unstable manifolds of the gradient flow. The conclusion follows from that. The papers where this is done are:
L. Qin, On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds, J. Topol. Anal., 2 (2010), no. 4, 469–526. (preprint here)
As well as:
L. Qin, On the associativity of gluing, . Topol. Anal., 0, 1 (2017). (preprint here)
L. Qin, An application of topological equivalence to Morse theory