Let and be finite complexes. When is a nilpotent space, it has a rationalization which is well understood. Early on it was found that the induced map 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 is in some sense orthogonal to rational homotopy theory but is nevertheless an invariant of the rational homotopy type of in at least some cases. The notion of complexity is geometric, and we also prove a conjecture of Gromov regarding the number of mapping classes that have Lipschitz constant at most .
"Integral and rational mapping classes." Duke Math. J. 169 (10) 1943 - 1969, 15 July 2020. https://doi.org/10.1215/00127094-2020-0012