Abstract
We consider the problem how far from being homotopy commutative is a loop space having the homotopy type of the p-completion of a product of finite numbers of spheres. We determine the homotopy nilpotency of those loop spaces as an answer to this problem.
Similar content being viewed by others
References
Andersen K.K.S., Grodal J., Møller J.M., Viruel A.: The classification of p-compact groups for p odd. Ann. Math. 167, 95–210 (2008)
Aguadé J., Smith L.: On the mod p torus theorem of John Hubbuck. Math. Z. 19(2), 325–326 (1986)
Bott R.: A note on the Samelson products in the classical groups. Comment. Math. Helv. 34, 249–256 (1960)
Bousfield A.K., Kan D.M.: Homotopy Limits, Completions and Localizations LNM, vol.304. Springer, Berlin (1972)
Clark A., Ewing J.: The realization of polynomial algebras as cohomology rings. Pacific J. Math. 50, 425–434 (1974)
Dwyer W.G., Wilkerson C.W.: A new finite loop space at the prime two. J. Am. Math. Soc. 6, 37–64 (1993)
Dwyer W.G., Wilkerson C.W.: Homotopy fixed-point methods for Lie groups and finite loop spaces. Ann. Math. 139, 395–442 (1994)
Friedlander E.M.: Exceptional isogenies and the classifying spaces of simple Lie groups. Ann. Math. 101, 510–520 (1975)
Hamanaka, H., Kono, A.: A note on Samelson products in exceptional Lie groups. preprint available at http://www.math.kyoto-u.ac.jp/preprint/2008/15kono.pdf
Harris B.: Suspensions and characteristic maps for symmetric spaces. Ann. Math. 76, 295–305 (1962)
Hubbuck J.: On homotopy commutative H-spaces. Topology 8, 119–126 (1969)
Humphreys J.E.: Reflection groups and Coxeter groups Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
James I.M., Thomas E.: Homotopy-abelian topological groups. Topology 1, 237–240 (1962)
Kishimoto, D.: Homotopy nilpotency in localized SU(n). Homology Homotopy Appl. (to appear)
Kono A., Ōshima H.: Commutativity of the group of self-homotopy classes of Lie groups. Bull. London Math. Soc. 36, 37–52 (2004)
Kumpel P.G.: Mod p-equivalences of mod p H-spaces. Quart. J. Math. 23, 173–178 (1972)
Lin J.P.: Loops of H-spaces with finitely generated cohomology rings. Topology Appl. 60(2), 131–152 (1994)
McGibbon C.: Homotopy commutativity in localized groups. Am. J. Math. 106, 665–687 (1984)
McGibbon C.A.: Higher forms of homotopy commutativity and finite loop spaces. Math. Z. 201(3), 363–374 (1989)
Milnor J.W., Stasheff J.D.: Characteristic Classes, Ann. of Math. Studies, vol. 76. Princeton University Press, Princeton, NJ (1974)
Mimura M., Toda H.: Topology of Lie Groups I II, Translations of Math. Monographs, vol. 91. AMS, Providence, RI (1991)
Mislin G.: Nilpotent groups with finite commutator subgroups, Lecture Notes in Math, vol. 418. Springer, Berlin (1974)
Møller J.: N-determined 2-compact groups II. Fund. Math. 196(1), 1–90 (2007)
Møller J.: N-determined 2-compact groups I. Fund. Math. 195(1), 11–84 (2007)
Serre J.-P.: Groupes d’homotopie et classes des groupes abeliens. Ann. Math. 58, 258–294 (1953)
Toda H.: Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Studies 49. Princeton University Press, Princeton, NJ (1962)
Wilkerson C.W.: K-theory operations and mod p loop spaces. Math. Z. 132, 29–44 (1973)
Zabrodsky A.: Hopf Spaces, North-Holland Mathematics Studies, vol. 22. North-Holland, Amsterdam (1976)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kaji, S., Kishimoto, D. Homotopy nilpotency in p-regular loop spaces. Math. Z. 264, 209–224 (2010). https://doi.org/10.1007/s00209-008-0459-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-008-0459-6