A zoo of growth functions of mapping class sets

Abstract

Suppose [Formula: see text] and [Formula: see text] are finite complexes, with [Formula: see text] simply connected. Gromov conjectured that the number of mapping classes in [Formula: see text] which can be realized by [Formula: see text]-Lipschitz maps grows asymptotically as [Formula: see text], where [Formula: see text] is an integer determined by the rational homotopy type of [Formula: see text] and the rational cohomology of [Formula: see text]. This conjecture was disproved in a recent paper of the author and Weinberger; we gave an example where the “predicted” growth is [Formula: see text] but the true growth is [Formula: see text]. Here we show, via a different mechanism, that the universe of possible such growth functions is quite large. In particular, for every rational number [Formula: see text], there is a pair [Formula: see text] for which the growth of [Formula: see text] is essentially [Formula: see text].