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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 and are finite complexes with simply connected, then there are constants and such that any two homotopic –Lipschitz maps have a –Lipschitz homotopy (and if one of the maps is constant, can be taken to be ). We hope that it will lead more generally to a better understanding of the space of maps from to in this setting.
Manin, Fedor 1
@article{GT_2019_23_6_a8, author = {Manin, Fedor}, title = {Plato{\textquoteright}s cave and differential forms}, journal = {Geometry & topology}, pages = {3141--3202}, publisher = {mathdoc}, volume = {23}, number = {6}, year = {2019}, doi = {10.2140/gt.2019.23.3141}, url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2019.23.3141/} }
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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