Quantitative null-cobordism

For a given null-cobordant Riemannian $n$-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on $n$. In the appendix the bound is improved to one that is $O(L^{1+\varepsilon })$ for every $\varepsilon >0$.

This construction relies on another of independent interest. Take $X$ and $Y$ to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose $Y$ is simply connected and rationally homotopy equivalent to a product of Eilenberg-MacLane spaces, for example, any simply connected Lie group. Then two homotopic $L$-Lipschitz maps $f,g:X \to Y$ are homotopic via a $CL$-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces $Y$.