January 2021 Lie algebras and $v_n$-periodic spaces
Gijs Heuts
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Ann. of Math. (2) 193(1): 223-301 (January 2021). DOI: 10.4007/annals.2021.193.1.3

Abstract

We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$-periodic homotopy groups. The case $n=0$ corresponds to rational homotopy theory. In analogy with Quillen's results in the rational case, we prove that this $v_n$-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in $T(n)$-local spectra. We also compare it to the homotopy theory of commutative coalgebras in $T(n)$-local spectra, where it turns out there is only an equivalence up to a certain convergence issue of the Goodwillie tower of the identity.

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Gijs Heuts. "Lie algebras and $v_n$-periodic spaces." Ann. of Math. (2) 193 (1) 223 - 301, January 2021. https://doi.org/10.4007/annals.2021.193.1.3

Information

Published: January 2021
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2021.193.1.3

Subjects:
Primary: 55P43 , 55P62 , 55Q51

Keywords: $v_n$-periodic homotopy groups , Bousfield-Kuhn functor , chromatic homotopy theory , spectral Lie algebras

Rights: Copyright © 2021 Department of Mathematics, Princeton University

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Vol.193 • No. 1 • January 2021
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