How about ANR homology manifolds?
See http://www.maths.ed.ac.uk/~aar/homology/tophom.pdf for an important article on the subject.
If I understand correctly, people expect (or know?) that these ANR homology manifold have transitive homeomorphism groups. The possible local models are indexed by the integers, and the value 0 corresponds to $\mathbb R^n$, i.e., to the notion of topological manifold.