A Counter Example to a Conjecture of Gromov on distortion of higher homotopy groups”

 In a 1978 paper, Gromov explained how the Lipschitz constant of a map S^n -> Y, where Y is a simply connected compact Riemannian manifold, restricts its homotopy class.  He later conjectured that the bounds obtained there are asymptotically sharp.  For example, this would imply that if Y is a closed oriented n-manifold with a disk cut out, then one could always find a C(Y)L-Lipschitz map which is homotopic to L^n times the puncture, even though you can only get
L^{n-1} by winding around the puncture.  I will show that this is not always the case.  If there is time I will also discuss how the same techniques apply to the problem of finding efficient homotopies between Lipschitz maps.