# Manifolds covered by a single disc

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.

• This was answered before on MO (by Petrunin?) but I cannot find the link. Fix any Riemannian metric on $M$ and a point $p\in M$. There is a start-shaped open domain $D$ in the tangent space $T_pM$ such that the exponential map $exp_p$ is a diffeomorphism on $D$ and is a surjection onto $M$ on the closure of $D$. The closure is a disk. – Igor Belegradek Feb 23 '17 at 21:55
• @Igor, but the identifications you get in that construction are not of the form the question wants. For example, if you start with a round sphere, you are collapsing the boundary to a point. – Mariano Suárez-Álvarez Feb 24 '17 at 1:35
• @MarianoSuárez-Álvarez: It does answer "can a (closed) manifold be obtained by identifying points on the boundary of the unit disk". The sentence after that I do not quite understand but I thought the OP was happy with some equivalence relation on the boundary of the disk whose quotient space is $M$. – Igor Belegradek Feb 24 '17 at 2:32
• Are you assuming that $f$ is an involution? – Anton Petrunin Feb 24 '17 at 2:50
• I doubt there is a complete list of obstructions. Homology will give some. For instance, one can show (by verifying vanishing of $H^1$) that among surfaces only $S^2$ and $RP^2$ can be obtained as quotients $D^2/x\sim f(x)$, where $f: \partial D^2\to \partial D^2$ is a continuous map. – Misha Mar 10 '17 at 16:45

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:
• The only issue is that the identification of boundary points is not necessarily generated by $f\colon S^{n-1}\to S^{n-1}$. – Sebastian Goette Feb 25 '17 at 11:18