## Duke Mathematical Journal

### Integral and rational mapping classes

#### Abstract

Let $X$ and $Y$ be finite complexes. When $Y$ is a nilpotent space, it has a rationalization $Y\to Y_{(0)}$ which is well understood. Early on it was found that the induced map $[X,Y]\to[X,Y_{(0)}]$ 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
Revised: 11 February 2020
First available in Project Euclid: 9 June 2020

https://projecteuclid.org/euclid.dmj/1591689610

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

#### 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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