##
**Quantitative nullhomotopy and rational homotopy type.**
*(English)*
Zbl 1408.57023

The starting point is the following statement of M. Gromov [J. Differ. Geom. 13, 303–310 (1978; Zbl 0427.58010)]: “Let \(X\) and \(Y\) be compact, simply connected Riemannian manifolds. Then the number of homotopy classes of maps \(f: X\to Y\) with Lipschitz constant \(\leq L\) is \(O(L^{\alpha})\), where \(\alpha\) depends only on the rational homotopy type of \(X\) and \(Y\).” The motivation of the paper under review is a conjecture by Gromov on an analogous result for homotopies between maps.

Before stating it, one has to consider two invariants for a homotopy: the thickness which is the Lipschitz constant in the space variable and the width which is the Lipschitz constant in the time variable. Gromov’s original conjecture for homotopy maps concerns only the thickness but here the authors address the two invariants in the following conjecture: “Let \(X\) be \(n\)-dimensional. If \(f: X\to Y\) is nullhomotopic with Lipschitz constant \(\leq L\), then it has a nullhomotopy of thickness \(O(L)\) and width \(O(L^q)\), where \(q\) is the minimal depth of a filtration \(0=V_{0}\subset\dots\subset V_{q}\) of the indecomposables in dimension \(\leq n\) of the Sullivan minimal model of \(Y\) with the property that \(dV_{i}\subseteq {\mathbb Q}(V_{i-1})\).”

The case \(q=0\) is proved in [S. Ferry and S. Weinberger, Proc. Natl. Acad. Sci. USA 110, No. 48, 19246–19250 (2013; Zbl 1302.57060)] and the case \(q=1\) in [G. R. Chambers et al., J. Am. Math. Soc. 31, No. 4, 1165–1203 (2018; Zbl 1402.53035)].

In this paper, the authors prove the conjecture for \(q=2\), which includes the simply connected homogeneous spaces, for instance. They also establish that rationally equivalent simply connected finite simplicial complexes admit nullhomotopies of the same shapes and provide a series of examples illustrating the sharpness and the limits of their results.

Before stating it, one has to consider two invariants for a homotopy: the thickness which is the Lipschitz constant in the space variable and the width which is the Lipschitz constant in the time variable. Gromov’s original conjecture for homotopy maps concerns only the thickness but here the authors address the two invariants in the following conjecture: “Let \(X\) be \(n\)-dimensional. If \(f: X\to Y\) is nullhomotopic with Lipschitz constant \(\leq L\), then it has a nullhomotopy of thickness \(O(L)\) and width \(O(L^q)\), where \(q\) is the minimal depth of a filtration \(0=V_{0}\subset\dots\subset V_{q}\) of the indecomposables in dimension \(\leq n\) of the Sullivan minimal model of \(Y\) with the property that \(dV_{i}\subseteq {\mathbb Q}(V_{i-1})\).”

The case \(q=0\) is proved in [S. Ferry and S. Weinberger, Proc. Natl. Acad. Sci. USA 110, No. 48, 19246–19250 (2013; Zbl 1302.57060)] and the case \(q=1\) in [G. R. Chambers et al., J. Am. Math. Soc. 31, No. 4, 1165–1203 (2018; Zbl 1402.53035)].

In this paper, the authors prove the conjecture for \(q=2\), which includes the simply connected homogeneous spaces, for instance. They also establish that rationally equivalent simply connected finite simplicial complexes admit nullhomotopies of the same shapes and provide a series of examples illustrating the sharpness and the limits of their results.

Reviewer: Daniel Tanré (Villeneuve d’ Ascq)

### MSC:

57N65 | Algebraic topology of manifolds |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

55P62 | Rational homotopy theory |

55S36 | Extension and compression of mappings in algebraic topology |

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\textit{G. R. Chambers} et al., Geom. Funct. Anal. 28, No. 3, 563--588 (2018; Zbl 1408.57023)

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