1The authors are supported by Xunta de Galicia grant EM2013/016. The first author is supported by Ministerio de EconomĂa y Competitividad (Spain), grant MTM2016-79661-P (AEI/FEDER, UE, support included). The second author is supported by Ministerio de EconomĂa y Competitividad (Spain), grant MTM2016-80439-P. The third author is supported by Ministerio de EconomĂa y Competitividad (Spain), grants MTM2013-41768-P and MTM2016-78647-P (AEI/FEDER, UE, support included).
Received 25 May 2016
Revised 29 May 2017
Published 5 September 2023
Abstract
Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant between the classical LS cocategory and the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuckâs Torus Theorem, and obtain corresponding results for p-compact groups and p-Noetherian groups.
References
1.Â
Adem, A. and Milgram, R. J., Cohomology of Finite Groups, 2nd ed., Grundlehren der Mathematischen Wissenschaften 309, Springer, Berlin, 2004, viii+324 pp.
Andersen, K. K. S., Bauer, T., Grodal, J. and Pedersen, E. P., A finite loop space not rationally equivalent to a compact Lie group, Invent. Math. 157 (2004), 1â10.
4.Â
Arone, G., Dwyer, W. G. and Lesh, K., Loop structures in Taylor towers, Algebr. Geom. Topol. 8 (2008), 173â210.
5.Â
Arone, G. and Mahowald, M., The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999), 743â788.
6.Â
Badzioch, B., Algebraic theories in homotopy theory, Ann. of Math. 155 (2002), 895â913.
7.Â
Berrick, A. J., An Approach to Algebraic K-Theory, Research Notes in Mathematics 56, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982, iii+108 pp.
8.Â
Berstein, I. and Ganea, T., Homotopical nilpotency, Illinois J. Math. 5 (1961), 99â130.
9.Â
Biedermann, G., Homotopy nilpotent groups and their associated functors, Preprint. arXiv.org/abs/1705.04963.
10.Â
Biedermann, G. and Dwyer, W. G., Homotopy nilpotent groups, Algebr. Geom. Topol. 10 (2010), 33â61.
11.Â
Bousfield, A. K. and Kan, D. M., Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics 304, Springer, Berlin-New York, 1972, v+348Â pp.
12.Â
Castellana, N., Crespo, J. and Scherer, J., Deconstructing Hopf spaces, Invent. Math. 167 (2007), 1â18.
13.Â
Castellana, N., Crespo, J. and Scherer, J., Noetherian loop spaces, J. Eur. Math. Soc. 13 (2011), 1225â1244.
14.Â
Chorny, B. and Scherer, J., Goodwillie calculus and Whitehead products, Forum Math. 27 (2015), 119â130.
15.Â
Dror Farjoun, E., Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics 1622, Springer, Berlin, 1996.
16.Â
Dwyer, W. G. and Farjoun, E. D., Localization and cellularization of principal fibrations, in Alpine Perspectives on Algebraic Topology, Contemp. Math. 504, pp. 117â124, Amer. Math. Soc., Providence, RI, 2009.
17.Â
Dwyer, W. and Wilkerson, C., Homotopy fixed points methods for Lie groups and finite loop spaces, Ann. of Math. 139 (1994), 395â442.
18.Â
Eldred, R., Goodwillie calculus via adjunction and LS cocategory, Homology, Homotopy Appl. 18 (2016), 31â58.
19.Â
Ganea, T., LusternikâSchnirelmann category and cocategory, Proc. Lond. Math. Soc. 10 (1960), 623â639.
20.Â
Goodwillie, T. G., Calculus. II. Analytic functors, K-Theory 5 (1991/92), 295â332.
21.Â
Goodwillie, T. G., Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645â711 (electronic).
22.Â
Grodal, J., The transcendence degree of the mod p cohomology of finite Postnikov systems, in Stable and Unstable Homotopy, Fields Inst. Commun. 19, pp. 111â130, 1998.
23.Â
Grodal, J., The classification of p-compact groups and homotopical group theory, in Proceedings of the International Congress of Mathematicians. Volume II, pp. 973â1001, Hindustan Book Agency, New Delhi, 2010.
Kaji, S. and Kishimoto, D., Homotopy nilpotency in p-regular loop spaces, Math. Z. 264 (2010), 209â224.
31.Â
Lawvere, F. W., Functorial semantics of algebraic theories, Proc. Natl. Acad. Sci. USA 50 (1963), 869â872.
32.Â
Lin, J., A cohomological proof of the torus theorem, Math. Z. 190 (1985), 469â476.
33.Â
Lurie, J., Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press, Princeton, NJ, 2009, xviii+925 pp.
34.Â
McGibbon, C. A., Homotopy commutativity in localized groups, Amer. J. Math. 106 (1984), 665â687.
35.Â
McGibbon, C. A., Infinite loop spaces and Neisendorfer localization, Proc. Amer. Math. Soc. 125 (1997), 309â313.
36.Â
Miller, H., The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984), 39â87.
37.Â
Murillo, A. and Viruel, A., LusternikâSchnirelmann cocategory: A Whitehead dual approach, in Cohomological Methods in Homotopy Theory, Progr. Math. 196, pp. 323â347, BirkhĂ€user Verlag, Basel, 2001.
38.Â
Neisendorfer, J., Localization and connected covers of finite complexes, in The Äech Centennial, Contemp. Math. 181, Boston, MA, 1993, pp. 385â390, Amer. Math. Soc., Providence, RI, 1995.
39.Â
Porter, G. J., Homotopical nilpotence of , Proc. Amer. Math. Soc. 15 (1964), 681â682.
40.Â
Rao, V. K., Homotopy nilpotent Lie groups have no torsion in homology, Manuscripta Math. 92 (1997), 455â462.
41.Â
Rector, D. L., Subgroups of finite dimensional topological groups, J. Pure Appl. Algebra 1 (1971), 253â273.
42.Â
Rotman, J. J., An Introduction to the Theory of Groups, 4th ed., Graduate Texts in Mathematics 148, Springer, New York, 1995, xvi+513 pp.