INTEGRAL AND RATIONAL MAPPING CLASSES
Open Access Publications from the University of California

## INTEGRAL AND RATIONAL MAPPING CLASSES

• Author(s): Manin, Fedor
• Weinberger, Shmuel
• et al.

## Published Web Location

https://doi.org/10.1215/00127094-2020-0012
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 \cite{GrMS} regarding the number of mapping classes that have Lipschitz constant at most $L$.

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