Plato’s cave and differential forms

Manin, Fedor

doi:10.2140/gt.2019.23.3141

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Plato's cave and differential forms

Fedor Manin

Geometry & Topology 23 (2019) 3141–3202

DOI: 10.2140/gt.2019.23.3141

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 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

References

Publication

Received: 3 May 2018

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

Volume 23, issue 6 (2019)