I'm going to talk about connections between the geometry of a map and its homotopy type. Suppose that we have a map from the unit m-sphere to the unit n-sphere. We say that the k-dilation of the map is < L if each k-dimensional surface with k-dim volume V is mapped to an image with k-dim volume at most LV. Informally, if the k-dilation of a map is less than a small epsilon, it means the map strongly shrinks each k-dimensional surface. Our main question is: can a map with very small k-dilation still be homotopically non-trivial?

Here are the main results. If k > (m+1)/2, then there are homotopically non-trivial maps from S^m to S^{m-1} with arbitrarily small k-dilation. But if k is at most (m+1)/2, then every homotopically non-trivial map from S^m to S^{m-1} has k-dilation at least c(m) > 0.