Commensurability (mathematics) - in Topology

In Topology

Two topological spaces are commensurable if they have homeomorphic finite-sheeted covering spaces. Depending on the type of topological space under consideration one might want to use homotopy-equivalences or diffeomorphisms instead of homeomorphisms in the definition. Thus, if one uses homotopy-equivalences, commensurability of groups correspond to commensurability of spaces provided one associates the classifying space to a discrete group. For smooth manifolds, the Gieseking manifold is commensurate to the complement of the figure-eight knot.