Integral and rational mapping classes

TL;DR

This paper investigates the relationship between integral and rational mapping classes between finite complexes, showing polynomial bounds on the complexity of preimages and confirming a conjecture about Lipschitz constants.

Contribution

It introduces a notion of complexity for rational mapping classes and proves polynomial bounds on the size of preimages, also confirming Gromov's conjecture on Lipschitz bounds.

Findings

Preimages of rational mapping classes have polynomially bounded size.

The complexity notion is geometric and relates to the rational homotopy type.

Confirmed Gromov's conjecture on Lipschitz constants.

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 $L$.