Quantitative null-cobordism

TL;DR

This paper investigates how the minimal geometric complexity of null-cobordisms relates to that of the original manifold, establishing a polynomial bound and exploring Lipschitz homotopy properties for certain spaces.

Contribution

It proves a polynomial bound on null-cobordism complexity based on the manifold's complexity and analyzes Lipschitz homotopy equivalences for specific classes of spaces.

Findings

Null-cobordism complexity is at most polynomial in the manifold's complexity.

Homotopic Lipschitz maps into certain spaces are homotopic via Lipschitz homotopies with controlled Lipschitz constants.

Counterexample shows limitations of Lipschitz homotopy extension for broader classes of spaces.

Abstract

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? In [Gro99], Gromov conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on $n$. 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 $f, g : X \rightarrow Y$ are homotopic via a $CL$-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces $Y$.