It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary.

Is it still true for topological manifold?

Comments:

• To proof the smooth case, fix a Riemannian metric and consider exponential map up to cutlocus.

• The question was asked by D. Burago. I made a bet that a complete answer will be given in an hour — please help :)