Integral and rational mapping classes

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 $L$.