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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Higher stabilization and higher Freudenthal suspension
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by Jacobson R. Blomquist and John E. Harper PDF
Trans. Amer. Math. Soc. 375 (2022), 8193-8240 Request permission


We prove that the stabilization (resp. iterated suspension) functor participates in a derived adjunction comparing pointed spaces with certain (highly homotopy coherent) homotopy coalgebras, in the sense of Blumberg-Riehl, that is a Dwyer-Kan equivalence after restriction to 1-connected spaces, with respect to the associated enrichments. A key ingredient of our proof, of independent interest, is a higher stabilization theorem (resp. higher Freudenthal suspension theorem) for pointed spaces that provides strong estimates for the uniform cartesian-ness of certain cubical diagrams associated to iterating the space level stabilization map (resp. Freudenthal suspension map)—these technical results provide, in particular, new proofs (with strong estimates) of the stabilization and iterated loop-suspension completion results of Carlsson and the subsequent work of Arone-Kankaanrinta, and Bousfield and Hopkins, respectively, for 1-connected spaces; this is the stabilization (resp. Freudenthal suspension) analog of Dundas’ higher Hurewicz theorem.
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Additional Information
  • Jacobson R. Blomquist
  • Affiliation: Department of Mathematical Sciences, Binghamton University, 4400 Vestal Parkway E, Binghamton, New York 13902
  • MR Author ID: 1308159
  • Email:
  • John E. Harper
  • Affiliation: Department of Mathematics, The Ohio State University, Newark, 1179 University Dr, Newark, Ohio 43055
  • MR Author ID: 880393
  • Email:
  • Received by editor(s): May 24, 2017
  • Received by editor(s) in revised form: September 6, 2020, and June 5, 2022
  • Published electronically: September 2, 2022
  • Additional Notes: The first author was supported in part by the National Science Foundation grants DMS-1510640 and DMS-1547357
    The second author was supported in part by the Simons Foundation: Collaboration Grants for Mathematicians #638247
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 8193-8240
  • MSC (2020): Primary 55P60, 55P99; Secondary 55P40, 55P42, 55P43
  • DOI: