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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@ARTICLE{kuhnAQ,
author = {Kuhn, Nicholas J.},
title = {Localization of {A}ndré-{Q}uillen-{G}oodwillie towers, and the periodic homology of infinite loopspaces},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {201},
year = {2006},
number = {2},
pages = {318--378},
issn = {0001-8708},
mrclass = {55N22 (18G55 55N20 55P43 55P47 55P60)},
mrnumber = {2211532},
mrreviewer = {J. P. C. Greenlees},
doi = {10.1016/j.aim.2005.02.005},
url = {https://doi.org/10.1016/j.aim.2005.02.005},
zblnumber = {1103.55007},
} -
[kuhntelescopic]
N. J. Kuhn, "A guide to telescopic functors," Homology Homotopy Appl., vol. 10, iss. 3, pp. 291-319, 2008.
@ARTICLE{kuhntelescopic,
author = {Kuhn, Nicholas J.},
title = {A guide to telescopic functors},
journal = {Homology Homotopy Appl.},
fjournal = {Homology, Homotopy and Applications},
volume = {10},
year = {2008},
number = {3},
pages = {291--319},
issn = {1532-0073},
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mrnumber = {2475626},
mrreviewer = {Keith Peter Johnson},
doi = {10.4310/HHA.2008.v10.n3.a13},
url = {https://doi.org/10.4310/HHA.2008.v10.n3.a13},
zblnumber = {1169.55007},
} -
[kuhnpereira]
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@ARTICLE{kuhnpereira,
author = {Kuhn, Nicholas J. and Pereira, Lu\'ıs Alexandre},
title = {Operad bimodules and composition products on {A}ndré-{Q}uillen filtrations of algebras},
journal = {Algebr. Geom. Topol.},
fjournal = {Algebraic \& Geometric Topology},
volume = {17},
year = {2017},
number = {2},
pages = {1105--1130},
issn = {1472-2747},
mrclass = {55P43 (18D50)},
mrnumber = {3623683},
mrreviewer = {Javier J. Gutiérrez},
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url = {https://doi.org/10.2140/agt.2017.17.1105},
zblnumber = {1362.55008},
} -
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} -
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[mahowaldImJ]
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[mathew]
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series = {Springer Proc. Math. Stat.},
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year = {2018},
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url = {https://doi.org/10.1007/978-3-319-94033-5_11},
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[mccarthy]
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@INCOLLECTION{mccarthy,
author = {McCarthy, Randy},
title = {Dual calculus for functors to spectra},
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1999)},
series = {Contemp. Math.},
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} -
[moerdijkweiss]
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} -
[rationalhomotopy]
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author = {Quillen, Daniel},
title = {Rational homotopy theory},
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fjournal = {Annals of Mathematics. Second Series},
volume = {90},
year = {1969},
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[rezk]
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[riehlverity]
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[sullivan]
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} -
[thompson]
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[thompsonunstablesphere]
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} -
@BOOK{wang,
author = {Wang, Guozhen},
title = {Unstable Chromatic Homotopy Theory},
note = {thesis (Ph.D.)--Massachusetts Institute of Technology},
publisher = {ProQuest LLC, Ann Arbor, MI},
year = {2015},
pages = {(no paging)},
mrclass = {Thesis},
mrnumber = {3427198},
url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:0831043},
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} -
[zhu]
Y. Zhu, "Morava $E$-homology of Bousfield-Kuhn functors on odd-dimensional spheres," Proc. Amer. Math. Soc., vol. 146, iss. 1, pp. 449-458, 2018.
@ARTICLE{zhu,
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journal = {Proc. Amer. Math. Soc.},
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year = {2018},
number = {1},
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doi = {10.1090/proc/13727},
url = {https://doi.org/10.1090/proc/13727},
zblnumber = {1422.55032},
}