Mathematics > Geometric Topology
[Submitted on 31 Dec 2017 (v1), last revised 21 Apr 2019 (this version, v6)]
Title:Plato's cave and differential forms
View PDFAbstract: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.
Submission history
From: Fedor Manin [view email][v1] Sun, 31 Dec 2017 18:28:44 UTC (42 KB)
[v2] Sat, 24 Feb 2018 17:33:51 UTC (43 KB)
[v3] Sun, 18 Mar 2018 18:53:25 UTC (44 KB)
[v4] Thu, 3 May 2018 18:16:13 UTC (46 KB)
[v5] Sat, 29 Sep 2018 03:45:25 UTC (51 KB)
[v6] Sun, 21 Apr 2019 15:20:18 UTC (53 KB)
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