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Integral and rational mapping classes. (English) Zbl 07226653

Summary: 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 regarding the number of mapping classes that have Lipschitz constant at most \(L\).

MSC:

55P62 Rational homotopy theory
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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