Volume 23, issue 6 (2019)

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Plato's cave and differential forms

Fedor Manin

Geometry & Topology 23 (2019) 3141–3202

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.

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rational homotopy theory, quantitative topology, Lipschitz functional, geometry of mapping spaces
Mathematical Subject Classification 2010
Primary: 53C23, 55P62
Received: 3 May 2018
Revised: 22 March 2019
Accepted: 21 April 2019
Published: 1 December 2019
Proposed: Yasha Eliashberg
Seconded: Benson Farb, Leonid Polterovich
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