Duke Mathematical Journal

Integral and rational mapping classes

Fedor Manin and Shmuel Weinberger

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Abstract

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 .

Article information

Source
Duke Math. J., Volume 169, Number 10 (2020), 1943-1969.

Dates
Received: 1 March 2018
Revised: 11 February 2020
First available in Project Euclid: 9 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1591689610

Digital Object Identifier
doi:10.1215/00127094-2020-0012

Subjects
Primary: 55P62: Rational homotopy theory
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Keywords
rational homotopy theory quantitative topology sets of homotopy classes Lipschitz homotopy theory

Citation

Manin, Fedor; Weinberger, Shmuel. Integral and rational mapping classes. Duke Math. J. 169 (2020), no. 10, 1943--1969. doi:10.1215/00127094-2020-0012. https://projecteuclid.org/euclid.dmj/1591689610


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