@DannyRuberman Just a finite presentation of the fundamental groups; I'll need that for the construction I'm doing

Thanks so much for the responses. If I may "go to the well" one more time, how may I specify a Brieskorn homology sphere in SnapPy?

@HJRW Thanks so much for the Brieskorn homology spheres examples! I really appreciate it!

@HJRW That's right, and the hyperbolic ones, at least (I think?), can't have a $\mathbb{Z} \times \mathbb{Z}$ or higher rank free Abelian group as a subgroup; I had looked up both those facts last night but forgot them. It's truly $\mathbb{Z}$ or nothing, thanks.

One trip to the library later: $DX$ is the Jacobian matrix and the matrix A is $\Omega$-skew iff $\Omega(Au,v) = -\Omega(u,Av)$ for all $u,v \in Z$; thanks so much for all your help.

Nawaf Bou-Rabee: Wonderful addition! That's very helpful.

Thanks! I'll have to look at the book (I'm not sure what $DX$ [induced map on double-tangent bundle?] and $\Omega$-skew mean in this context), but I think that's for what I'm looking. (I love Marsden as an author.)

@MohammadF.Tehrani Excellent citation! (I can't "Vote +1" on a comment)

Also, to answer the question in your last paragraph, look up Bieberbach group en.wikipedia.org/wiki/Space_group#Bieberbach.27s_theorems These are all the fundamental groups of closed, ashperical manifolds with 0 sectional curvature everywhere whose universal covers are isometric to $\mathbb{R}^n$. The Klein bottle is the other example in dimension 2.

The theorem of Stallings en.wikipedia.org/wiki/Simply_connected_at_infinity says that for n ≥ 5, a contractible n-manifold is homeomorphic to $\mathbb{R}^n$ precisely when it is simply connected at infinity. So, the universal cover not being simply connected at infinity appears to be the obstruction for which you are looking. See this post math.stackexchange.com/questions/1986451/… for constructing them. (cont)

This may be naive, but I think the Cartan-Hadamard Theorem and a theorem of Stallings should be relevant. The Cartan-Hadamard Theorem en.wikipedia.org/wiki/Cartan%E2%80%93Hadamard_theorem says that if an complete (which all closed manifolds are), Riemannian manifold has non-positive sectional curvature at each point, then its universal cover is diffeomorphic to $\mathbb{R}^n$. (cont)