15 July 2020 Integral and rational mapping classes
Fedor Manin, Shmuel Weinberger
Duke Math. J. 169(10): 1943-1969 (15 July 2020). DOI: 10.1215/00127094-2020-0012


Let X and Y be finite complexes. When Y is a nilpotent space, it has a rationalization Y Y ( 0 ) which is well understood. Early on it was found that the induced map [ X , Y ] [ X , Y ( 0 ) ] on sets of mapping classes is finite-to-one. The sizes of the preimages need not be bounded; we show, however, that, as the complexity (in a suitable sense) of a rational mapping class increases, these sizes are at most polynomial. This “torsion” information about [ X , Y ] is in some sense orthogonal to rational homotopy theory but is nevertheless an invariant of the rational homotopy type of Y in at least some cases. The notion of complexity is geometric, and we also prove a conjecture of Gromov regarding the number of mapping classes that have Lipschitz constant at most L .


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Fedor Manin. Shmuel Weinberger. "Integral and rational mapping classes." Duke Math. J. 169 (10) 1943 - 1969, 15 July 2020. https://doi.org/10.1215/00127094-2020-0012


Received: 1 March 2018; Revised: 11 February 2020; Published: 15 July 2020
First available in Project Euclid: 9 June 2020

zbMATH: 07226653
MathSciNet: MR4118644
Digital Object Identifier: 10.1215/00127094-2020-0012

Primary: 55P62
Secondary: 53C23

Keywords: Lipschitz homotopy theory , quantitative topology , Rational homotopy theory , sets of homotopy classes

Rights: Copyright © 2020 Duke University Press


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Vol.169 • No. 10 • 15 July 2020
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