Dilatation of homotopy classes and norms of cohomology classes

Abstract

For a pointed homotopy class of maps between two length spaces, it is easy to obtain a lower bound on its dilatation (that is, the best Lipschitz constant). However, obtaining upper bounds is harder. When the source space is a path-connected compact Riemannian manifold and the target space is a flat torus, we give an upper bound for the dilatation in terms of the comass norms of the cohomology classes corresponding to the pointed homotopy class. The upper bound obtained is not much larger than the lower bound. Using the lower and upper bounds obtained, we determine the order of the number of homotopy classes into a flat torus with prescribed dilatation, and extend the Burago–Ivanov–Gromov inequality.