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



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> Is it still true for topological manifold?
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**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 :)



Rugged manifold
The answer is yes -- Morton Brown's mapping theorem says that for every closed (connected) topological $n$-manifold $M$ there is a continuous map $f$ from the $n$-cube $I^n$ onto $M$ which is injective on the interior of the cube (for manifolds with boundary, see Remark 1 below). This was proved in early sixties and can be found in M.Brown,*"A mapping theorem for untriangulated manifolds,"* Topology of 3-manifolds, pp.92-94, M.K.Fort, Jr.(Editor), Prentice-Hall, Englewood Cliffs, N.J.,1962. [MR0158374](http://www.ams.org/mathscinet-getitem?mr=0158374)


Main idea of the proof is simple: "Use local PL structures to expand a small $n$-cell in $M$ gradually, until it becomes the whole manifold." 


This can be realized the "infinite composition" of engulfings of finitely many points at a time. This argument is not too difficult, so I will try to sketch it.


First consider a closed $n$-cell $C$ in $M$ and a finite set $X=\{x\_1,\ldots,x\_k\}$ of points of $M$ disjoint from $C$ which we want to engulf. Assume that each $x\_i$ lies in some open $n$-cell $U\_i$ which intersects $C$. For each $i$, fix a PL structure on $U\_i$, and join $x\_i$ and some point $y\_i\in \partial C$ by a PL arc $\alpha\_i\subset U\_i$ relative to this structure. If we assume $\dim M\geq 3$ (this is not a restriction because the theorem is clear for $\dim M=1$, and easily follows from the surface classification for $\dim M=2$), we may require that $\alpha\_i$'s should be disjoint. The regular neighborhood $Q\_i$ of $\alpha\_i$ within $U\_i$ is a closed $n$-cell, and $Q\_i$'s can be made disjoint. Let $h$ be a homeomorphism of $M$ pushing $y\_i$ towards $x\_i$ within $Q\_i$ for each $i$ which is identity outside $Q\_i$'s. Then $X\subset h(C)$, and we can apply this process again to $h(C)$ and another finite set $X'\subset M\setminus h(C)$. Repeating this process we obtain a sequence of engulfing homeomorphisms $h\_1, h\_2,\ldots$.


We can arrage that the uniform limit $f\colon M\to M$ of the composition of engulfing homeomorphisms $f\_n=h\_n\circ\cdots\circ h\_1$ exists and $f(C)=M$. If we choose sufficiently small $Q\_i$'s in each stage, one can make sure that $f$ is injective on the interior of $C$ (indeed, we can arrange that for each interior point $x$ there is $n$ such that $f\_n(x)=f(x)$).


Remark 1: In Brown's paper, one can find a corollary that "If $M$ is a compact
connected manifold with nonempty boundary $B$, then there is a surjection $f\colon B\times [0,1]\to M$ that restricts to the identity on $B\times 0$ and is injective on $B\times[0,1)$". Notice also that the above theorem can be applied to the **closed** manifold $B$. Then it follows that any compact **topological** $n$-manifold (possibly with boundary) can be obtained by identifying some points in the boundary of the $n$-ball.


Remark 2: [Berlanga's theorem](http://archive.numdam.org/ARCHIVE/CM/CM_1987__63_2/CM_1987__63_2_209_0/CM_1987__63_2_209_0.pdf) extended the Brown's theorem to noncompact manifolds:
This theorem states that for every (connected) $\sigma$-compact $n$-manifold $M$, there is a nice kind of surjection $I^n\to \overline{M}$ similar to Brown's map, where $\overline{M}$ denotes the end compactification of $M$.


