Plato's cave and differential forms

TL;DR

This paper proves a technical result linking differential algebra maps to space maps, confirming Gromov's conjecture on Lipschitz homotopies between maps of finite complexes, enhancing understanding of their mapping spaces.

Contribution

It establishes that maps of differential algebras closely approximate maps between spaces, proving Gromov's conjecture on Lipschitz homotopies for finite complexes with simply connected targets.

Findings

Maps of differential algebras are closely shadowed by maps between spaces.

Any two homotopic Lipschitz maps have a Lipschitz homotopy with controlled Lipschitz constant.

Confirmed Gromov's conjecture relating Lipschitz maps and homotopies in finite complexes.

Abstract

In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences the geometry of maps between them. The main technical result of this paper supports this intuition: we show that maps of differential algebras are closely shadowed, in a technical sense, by maps between the corresponding spaces. As a concrete application, we prove the following conjecture of Gromov: if $X$ and $Y$ are finite complexes with $Y$ simply connected, then there are constants $C(X,Y)$ and $p(X,Y)$ such that any two homotopic $L$-Lipschitz maps have a $C(L+1)^p$-Lipschitz homotopy (and if one of the maps is a constant, $p$ can be taken to be $2$.) We hope that it will lead more generally to a better understanding of the space of maps from $X$ to $Y$ in this setting.