INTEGRAL AND RATIONAL MAPPING CLASSES

摘要

Let X and Y be finite complexes. When Y is a nilpotent space, it has a rationalization Y →Y_((0)) which is well understood. Early on it was found that the induced map [X,Y] →[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 regarding the number of mapping classes that have Lipschitz constant at most L.