Plato’s cave and differential forms
Geometry & topology, Tome 23 (2019) no. 6, pp. 3141-3202.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/gt.2019.23.3141
Classification : 53C23, 55P62
Keywords: rational homotopy theory, quantitative topology, Lipschitz functional, geometry of mapping spaces

Manin, Fedor 1

1 Department of Mathematics, Ohio State University, Columbus, OH, United States, Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, United States
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Manin, Fedor. Plato’s cave and differential forms. Geometry & topology, Tome 23 (2019) no. 6, pp. 3141-3202. doi : 10.2140/gt.2019.23.3141. http://geodesic.mathdoc.fr/articles/10.2140/gt.2019.23.3141/

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