  In the first part of this paper, we let $G$ be a finitely-generated amenable
group such that $G/[G, G]$ is torsion-free. We suppose that $G$ acts by
homeomorphisms homotopic to the identity on a manifold $M$, and give conditions
on $M$ which imply that such an action must lift to an action on the universal
cover $\tilde{M}$. The circle, all 2-manifolds except the open annulus, and
most compact 3-manifolds satisfy these conditions. The proof uses a dynamical
tool called homological rotation vectors, and Thurston's Geometrization Theorem
in the latter case.
  On manifolds not satisfying our conditions, such actions really may fail to
lift. In the second part, we try to understand the dynamical possibilities in
the simplest case: $G = \mathbb{Z}^2$, and $M = \mathbb{A}$ is the open
annulus. We show that if a $\mathbb{Z}^2$ action homotopic to the identity on
$\mathbb{A}$ fails to lift to a $\mathbb{Z}^2$ action on the plane, and if the
action satisfies one additional condition (which may not be necessary), the
action is essentially similar to the one generated by $\bar{f_0}(\theta, y) =
(\theta + y, y)$ and $\bar{g_0}(\theta, y) = (\theta, y + 1)$.
