Plato’s cave and differential forms

Manin, Fedor¹

¹ Department of Mathematics, Ohio State University, Columbus, OH, United States, Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, United States

Geometry & topology, Tome 23 (2019) no. 6, pp. 3141-3202.

Résumé

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 $C\left(X,Y\right)$ and $p\left(X,Y\right)$ such that any two homotopic $L$–Lipschitz maps have a $C{\left(L+1\right)}^p$–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.

Détail

DOI : 10.2140/gt.2019.23.3141

Classification : 53C23, 55P62
Keywords: rational homotopy theory, quantitative topology, Lipschitz functional, geometry of mapping spaces

Bibliographie

[1] M Amann, Degrees of self-maps of simply connected manifolds, Int. Math. Res. Not. 2015 (2015) 8545 | DOI

[2] P Andrews, M Arkowitz, Sullivan’s minimal models and higher order Whitehead products, Canadian J. Math. 30 (1978) 961 | DOI

[3] M Arkowitz, G Lupton, Rational obstruction theory and rational homotopy sets, Math. Z. 235 (2000) 525 | DOI

[4] A Berdnikov, Lipschitz null-homotopy of mappings S3 → S2, preprint (2018)

[5] D Blanc, Homotopy operations and rational homotopy type, from: "Categorical decomposition techniques in algebraic topology" (editors G Arone, J Hubbuck, R Levi, M Weiss), Progr. Math. 215, Birkhäuser (2004) 47 | DOI

[6] R Body, M Mimura, H Shiga, D Sullivan, p–universal spaces and rational homotopy types, Comment. Math. Helv. 73 (1998) 427 | DOI

[7] R Bott, L W Tu, Differential forms in algebraic topology, 82, Springer (1982) | DOI

[8] E H Brown Jr., R H Szczarba, On the rational homotopy type of function spaces, Trans. Amer. Math. Soc. 349 (1997) 4931 | DOI

[9] M Čadek, M Krčál, J Matoušek, L Vokřínek, U Wagner, Extendability of continuous maps is undecidable, Discrete Comput. Geom. 51 (2014) 24 | DOI

[10] A Calder, J Siegel, On the width of homotopies, Topology 19 (1980) 209 | DOI

[11] G R Chambers, D Dotterrer, F Manin, S Weinberger, Quantitative null-cobordism, J. Amer. Math. Soc. 31 (2018) 1165 | DOI

[12] G R Chambers, F Manin, S Weinberger, Quantitative nullhomotopy and rational homotopy type, Geom. Funct. Anal. 28 (2018) 563 | DOI

[13] R Douglas, Positive weight homotopy types, Illinois J. Math. 27 (1983) 597 | DOI

[14] E Dror Farjoun, Homotopy and homology of diagrams of spaces, from: "Algebraic topology" (editors H R Miller, D C Ravenel), Lecture Notes in Math. 1286, Springer (1987) 93 | DOI

[15] H Edelsbrunner, D R Grayson, Edgewise subdivision of a simplex, Discrete Comput. Geom. 24 (2000) 707 | DOI

[16] H Federer, W H Fleming, Normal and integral currents, Ann. of Math. 72 (1960) 458 | DOI

[17] Y Félix, S Halperin, J C Thomas, Rational homotopy theory, 205, Springer (2001) | DOI

[18] S Ferry, S Weinberger, Quantitative algebraic topology and Lipschitz homotopy, Proc. Natl. Acad. Sci. USA 110 (2013) 19246 | DOI

[19] M Filakovsky, P Franek, U Wanger, S Zhechev, Computing simplicial representatives of homotopy group elements, from: "Proceedings of the twenty-ninth annual ACM–SIAM symposium on discrete algorithms" (editor A Czumaj), SIAM (2018) 1135 | DOI

[20] M Filakovský, L Vokřínek, Are two given maps homotopic? An algorithmic viewpoint, Found. Comput. Math. (2019) | DOI

[21] S M Gersten, Isoperimetric and isodiametric functions of finite presentations, from: "Geometric group theory, I" (editors G A Niblo, M A Roller), London Math. Soc. Lecture Note Ser. 181, Cambridge Univ. Press (1993) 79 | DOI

[22] P A Griffiths, J W Morgan, Rational homotopy theory and differential forms, 16, Birkhäuser (1981)

[23] M Gromov, Homotopical effects of dilatation, J. Differential Geometry 13 (1978) 303 | DOI

[24] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, 152, Birkhäuser (1999) | DOI

[25] M Gromov, Quantitative homotopy theory, from: "Prospects in mathematics" (editor H Rossi), Amer. Math. Soc. (1999) 45

[26] L Guth, Recent progress in quantitative topology, from: "Surveys in differential geometry 2017 : celebrating the 50th anniversary of the Journal of Differential Geometry" (editors H D Cao, J Li, R M Schoen, S T Yau), Surv. Differ. Geom. 22, International (2018) 191 | DOI

[27] A Haefliger, Plongements de variétés dans le domaine stable, from: "Séminaire Bourbaki, 1962/63", Secrétariat mathématique (1962)

[28] R M Hain, Iterated integrals and homotopy periods, 291, Amer. Math. Soc. (1984) | DOI

[29] D Kotschick, On products of harmonic forms, Duke Math. J. 107 (2001) 521 | DOI

[30] J T Maher, The geometry of dilatation and distortion, PhD thesis, The University of Chicago (2004)

[31] F Manin, Asymptotic invariants of homotopy groups, PhD thesis, The University of Chicago (2015)

[32] F Manin, A zoo of growth functions of mapping class sets, J. Topol. Anal. (2018) | DOI

[33] F Manin, S Weinberger, Integral and rational mapping classes, preprint (2018)

[34] M Mimura, H Toda, On p–equivalences and p–universal spaces, Comment. Math. Helv. 46 (1971) 87 | DOI

[35] A Nabutovsky, Non-recursive functions, knots “with thick ropes”, and self-clenching “thick” hyperspheres, Comm. Pure Appl. Math. 48 (1995) 381 | DOI

[36] M Nakaoka, H Toda, On Jacobi identity for Whitehead products, J. Inst. Polytech. Osaka City Univ. Ser. A. 5 (1954) 1

[37] L Scull, Rational S1–equivariant homotopy theory, Trans. Amer. Math. Soc. 354 (2002) 1 | DOI

[38] H Shiga, Rational homotopy type and self-maps, J. Math. Soc. Japan 31 (1979) 427 | DOI

[39] D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977) 269 | DOI