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Scalable spaces

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Abstract

Scalable spaces are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality. They are also characterized by particularly nice behavior from the point of view of quantitative homotopy theory. Among other results, we show that spaces which are formal but not scalable provide counterexamples to Gromov’s long-standing conjecture on distortion in higher homotopy groups.

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Notes

  1. This is essentially the inverse function of the notion used in [12], but accords with the notion of distortion used in geometric group theory.

  2. While the minimal model is unique up to isomorphism, such an isomorphism need not preserve this.

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Acknowledgements

We would like to thank Robert Young, who pointed out the reference [22], as well as Robin Elliott and Shmuel Weinberger for other useful comments. We also thank the anonymous referee for a large number of corrections as well as clarifying questions and remarks which greatly improved the exposition and led us to several discoveries. The second author was supported by the NSF via the individual grant DMS-2001042.

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Authors and Affiliations

  1. Institute for Advanced Study, Princeton, MJ, United States

    Aleksandr Berdnikov

  2. Department of Mathematics, University of California, Santa Barbara, CA, United States

    Fedor Manin

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  1. Aleksandr Berdnikov
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  2. Fedor Manin
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Correspondence to Fedor Manin.

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Berdnikov, A., Manin, F. Scalable spaces. Invent. math. 229, 1055–1100 (2022). https://doi.org/10.1007/s00222-022-01118-9

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