#### Volume 23, issue 6 (2019)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Plato's cave and differential forms

### Fedor Manin

Geometry & Topology 23 (2019) 3141–3202
##### 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 $C\left(X,Y\right)$ and $p\left(X,Y\right)$ such that any two homotopic $L$–Lipschitz maps have a $C{\left(L+1\right)}^{p}$–Lipschitz homotopy (and if one of the maps is 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.

##### Keywords
rational homotopy theory, quantitative topology, Lipschitz functional, geometry of mapping spaces
##### Mathematical Subject Classification 2010
Primary: 53C23, 55P62
##### Publication
Revised: 22 March 2019
Accepted: 21 April 2019
Published: 1 December 2019
Proposed: Yasha Eliashberg
Seconded: Benson Farb, Leonid Polterovich
##### Authors
 Fedor Manin Department of Mathematics Ohio State University Columbus, OH United States Department of Mathematics University of California, Santa Barbara Santa Barbara, CA United States