Does anyone know of an example of a closed manifold whose universal cover does not have the homotopy type of a finite CW complex?
(If the universal cover is also compact, then it is of finite homotopy type. The well known examples of closed manifolds having non-compact universal covers are aspherical manifolds, such as hyperbolic manifolds, tori, and products of these manifolds. In these examples the universal cover is contractile and so still has the homotopy type of a finite CW complex.)
Best Answer
Start with a closed manifold $M$, necessarily of dimension $\ge 3$, with infinite fundamental group whose universal cover has nontrivial homology; the simplest example I can think of is $M = S^1 \times S^2$, whose universal cover $\widetilde{M}$ is $\mathbb{R} \times S^2$.
Now consider the connected sum $M\#M$ of two copies of $M$. Its fundamental group is the free product $\mathbb{Z} \ast \mathbb{Z} \cong F_2$. Its universal cover $\widetilde{M\#M}$ is a bit annoying to describe but basically looks like the Cayley graph of $F_2$ but with the edges replaced by copies of $\widetilde{M}$ and the vertices replaced by the $S^2$'s used to connect-sum the copies of $M$ together. Together with the usual contraction of a tree this implies that its $H_2$ is a countable direct sum of copies of $H_2(\widetilde{M}) \cong \mathbb{Z}$, which is not finitely generated. So $\widetilde{M\#M}$ does not have the homotopy type of a finite CW complex.