A zoo of growth functions of mapping class sets

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 $L^r$.