Quantitative null-cobordism

For a given null-cobordant Riemannian n 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 n . In the appendix the bound is improved to one that is O ( L 1 + ε ) O(L^{1+\varepsilon }) for every ε > 0 \varepsilon >0 .

This construction relies on another of independent interest. Take X X and Y Y to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y 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 L -Lipschitz maps f , g : X → Y f,g:X \to Y are homotopic via a C L CL -Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y Y .