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

When actions of amenable groups can be lifted to the universal cover