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Higher homotopy associativity in the Harris decomposition of Lie groups

Part of: Homotopy theory Homotopy groups

Published online by Cambridge University Press:  11 September 2019

Daisuke Kishimoto  and
Toshiyuki Miyauchi
Daisuke Kishimoto
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan (kishi@math.kyoto-u.ac.jp)
Toshiyuki Miyauchi
Affiliation:
Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan (miyauchi@math.sci.fukuoka-u.ac.jp)
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Abstract

For certain pairs of Lie groups (G, H) and primes p, Harris showed a relation of the p-localized homotopy groups of G and H. This is reinterpreted as a p-local homotopy equivalence G ≃ (p)H × G/H, and so there is a projection G(p) → H(p). We show how much this projection preserves the higher homotopy associativity.

Keywords

Lie groupmod p decompositionhigher homotopy associativityhigher homotopy commutativity

MSC classification

Primary: 55P60: Localization and completion
Secondary: 55P35: Loop spaces 55Q15: Whitehead products and generalizations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2982 - 3000
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Bott, R.. A note on the Samelson product in the classical groups. Comment. Math. Helv. 34 (1960), 249–256.CrossRefGoogle Scholar
2Hamanaka, H. and Kono, A.. A note on Samelson products and mod p cohomology of classifying spaces of the exceptional Lie groups. Topol. Appl. 157 (2010), 393–400.CrossRefGoogle Scholar
3Harris, B.. On the homotopy groups of the classical groups. Ann. Math. 74 (1961), 407–413.CrossRefGoogle Scholar
4Harris, B.. Suspensions and characteristic maps for symmetric spaces. Ann. Math. 76 (1962), 295–305.CrossRefGoogle Scholar
5Hasui, S., Kishimoto, D., Miyauchi, T. and Ohsita, A.. Samelson products in quasi-p-regular exceptional Lie groups. Homology Homotopy Appl. 20 (2018), 185–208.CrossRefGoogle Scholar
6Hasui, S., Kishimoto, D. and Ohsita, A.. Samelson products in p-regular exceptional Lie groups. Topology Appl. 178 (2014), 17–29.CrossRefGoogle Scholar
7Hasui, S., Kishimoto, D., So, T. S. and Theriault, S.. Odd primary homotopy types of the gauge groups of exceptional Lie groups. Proc. AMS 147 (2019), 1751–1762.CrossRefGoogle Scholar
8Hasui, S., Kishimoto, D. and Tsutaya, M.. Higher homotopy commutativity in localized Lie groups and gauge groups. Homology Homotopy Appl. 21 (2019), 107–128.CrossRefGoogle Scholar
9Hemmi, Y.. Higher homotopy commutativity of H-spaces and the mod p torus theorem. Pacific J. Math. 149 (1991), 95–111.CrossRefGoogle Scholar
10Iwase, N. and Mimura, M.. Higher homotopy associativity. In Algebraic topology (Arcata, CA, 1986). Lecture Notes in Math., vol. 1370, pp. 193–20 (Berlin: Springer, 1989).CrossRefGoogle Scholar
11Kachi, H. and Mukai, J.. Somce homotopy groups of the rotation group R n. Hiroshima Math. J. 29 (1999), 327–345.CrossRefGoogle Scholar
12Kaji, S. and Kishimoto, D.. Homotopy nilpotency in p-regular loop spaces. Math. Z 264 (2010), 209–224.CrossRefGoogle Scholar
13Kishimoto, D.. Homotopy nilpotency in localized SU(n). Homology Homotopy Appl. 11 (2009), 61–79.CrossRefGoogle Scholar
14Kishimoto, D. and Tsutaya, M.. Samelson products in p-regular SO(2n) and its homotopy normality. Glasgow Math. J. 60 (2018), 165–174.CrossRefGoogle Scholar
15Mahowald, M.. A Samelson product in SO(2n). Bol. Soc. Math. Mexicana 10 (1965), 80–83.Google Scholar
16Mimura, M.. The homotopy groups of Lie groups of low rank. J. Math. Kyoto Univ. 6 (1967), 131–176.CrossRefGoogle Scholar
17Mimura, M.. Homotopy theory of Lie groups. In Handbook of Algebraic Topology, pp. 951–991 (Amsterdam: North-Holland, 1995).Google Scholar
18Mimura, M., Nishida, G. and Toda, H.. Mod p decomposition of compact Lie groups. Publ. Res. Inst. Math. Sci. 13 (1977), 627–680.CrossRefGoogle Scholar
19Russhard, A.. Power maps on quasi-p-regular SU(n). Homology Homotopy Appl. 17 (2015), 235–254.CrossRefGoogle Scholar
20Saumell, L.. Higher homotopy commutativity in localized groups. Math. Z 219 (1995), 203–213.CrossRefGoogle Scholar
21Shay, P. B.. mod p Wu Formulas for the Steenrod Algebra and the Dyer-Lashof Algebra. Proc. Amer. Math. Soc. 63 (1977), 339–347.Google Scholar
22Stasheff, J. D.. Homotopy associativity of H-spaces. I, II. Trans. Amer. Math. 108 (1963), 293–312; ibid., 293–312.Google Scholar
23Theriault, S.. Power maps on p-regular Lie groups. Homology Homotopy Appl. 15 (2013), 83–102.CrossRefGoogle Scholar
24Toda, H.. Composition Methods in Homotopy Groups of Spheres. Ann. Math. Stud., vol. 49 (Princeton N.J.: Princeton Univ. Press, 1962).Google Scholar
25Tsutaya, M.. Mapping spaces from projective spaces. Homology Homotopy Appl. 18 (2016), 173–203.CrossRefGoogle Scholar