Is the fundamental group of a compact manifold finitely presented?

Solution 1:

Differentiable manifolds can always be given the structure of PL manifolds, which can be triangulated into simplicial complexes. By shrinking a spanning tree of the 1-skeleton of this simplicial complex, we can obtain a CW complex $X$ with a single $0$-cell. This complex is no longer a manifold, but has the same fundamental group as the original manifold, since quotienting out by a contractible subspace is a homotopy equivalence.

If the manifold is compact, it has a simplicial decomposition with a finite number of cells. This carries over to $X$. But the fundamental group of a $CW$ complex with a single $0$-cell has a presentation with a generator for each $1$-cell and a relation for each $2$-cell. Thus $X$, and therefore the original manifold, has a finitely presented fundamental group.

Solution 2:

One can even do a bit better:

Every compact topological manifold (possibly with boundary) is homotopy-equivalent to a finite CW-complex.

That every ANR (absolute neighborhood retract, and every topological manifold with or without boundary is ANR) is homotopy-equivalent to a CW complex, was known as Borsuk conjecture. This conjectire was proven by West in 1974 (here) and, later on, a simpler proof was given by Chapman here.

In the case of topological manifolds it is also a theorem in the (unreadable) book by Kirby and Siebenmann that every compact manifold is homotopy-equivalent to a finite CW-complex.