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A finite loop space not rationally equivalent to a compact Lie group

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We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5.

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Authors and Affiliations

  1. Department of Mathematics, University of Aarhus, Ny Munkegade, Building 530, DK-8000, Århus C, Denmark

    Kasper K. S. Andersen

  2. Sonderforschungsbereich 478 Geometrische Strukturen in der Mathematik, Westfälische Wilhelms-Universität Münster, Hittorfstr. 27, D-48149, Münster, Germany

    Tilman Bauer

  3. Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL, 60637, USA

    Jesper Grodal

  4. Department of Mathematical Sciences, Binghamton University, Binghamton, New York, 13902-6000, USA

    Erik Kjaer Pedersen

Authors
  1. Kasper K. S. Andersen
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  2. Tilman Bauer
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  3. Jesper Grodal
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  4. Erik Kjaer Pedersen
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Corresponding authors

Correspondence to Kasper K. S. Andersen, Tilman Bauer, Jesper Grodal or Erik Kjaer Pedersen.

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Mathematics Subject Classification (2000)

Primary: 55P35; Secondary: 55P15, 55R35

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Andersen, K., Bauer, T., Grodal, J. et al. A finite loop space not rationally equivalent to a compact Lie group. Invent. math. 157, 1–10 (2004). https://doi.org/10.1007/s00222-003-0341-4

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  • DOI: https://doi.org/10.1007/s00222-003-0341-4

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