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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References:

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