A zoo of growth functions of mapping class sets

TL;DR

This paper explores the diverse possible growth rates of mapping class sets between finite complexes, demonstrating that for any rational number greater than or equal to 4, there exists a pair with growth rate essentially matching that number.

Contribution

It shows that the set of possible growth functions for mapping class sets is very broad, including all rational numbers greater than or equal to 4.

Findings

Existence of pairs with growth rate exactly any rational number ≥ 4

Disproof of Gromov's conjecture on growth rate predictions

Demonstration of the wide variety of growth functions possible

Abstract

Suppose $X$ and $Y$ are finite complexes, with $Y$ simply connected. Gromov conjectured that the number of mapping classes in $[X,Y]$ which can be realized by $L$-Lipschitz maps grows asymptotically as $L^\alpha$, where $\alpha$ is an integer determined by the rational homotopy type of $Y$ and the rational cohomology of $X$. This conjecture was disproved in a recent paper of the author and Weinberger; we gave an example where the `predicted' growth is $L^8$ but the true growth is $L^8\log L$. Here we show, via a different mechanism, that the universe of possible such growth functions is quite large. In particular, for every rational number $r \geq 4$, there is a pair $X,Y$ for which the growth of $[X,Y]$ is essentially $L^r$.