×

Gottlieb groups of homogeneous spaces. (English) Zbl 1064.55010

This paper studies the Gottlieb groups of some homogeneous spaces. Namely the sphere \(S^2\), the quaternionic projective \(n\)-space \(\mathbb{H} P^n\) and Stiefel manifolds. In the case of the sphere \(S^2\) the authors show that \(G_n(S^2)=0\) for \(n=1,2\) and it equals \(\pi_n(S^2)\) for \(n>2\). For the quaternionic projective \(n\)-space they estimate the group \(G_k(\mathbb{H} P^n)\) under the hypothesis that \(\pi_k(\mathbb{H} P^n)\) is infinite and \(\pi_k(Sp(n)\times Sp(1))\) is finite. Applications are given for \(k=4n+2\). For Stiefel manifolds several examples are shown where certain Gottlieb groups are infinite. As a consequence some results on homogeneous spaces involving the exceptional Lie groups are obtained. The authors use the main theorem from J. Siegel [Pac. J. Math. 31, 209-214 (1969; Zbl 0183.51701)], which relates the homotopy groups of a Lie group Y and the Gottlieb groups of a homogeneous space as an important tool.

MSC:

55Q10 Stable homotopy groups
55Q35 Operations in homotopy groups
55P99 Homotopy theory

Citations:

Zbl 0183.51701
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Golasinski, J. Mukai, On Gottlieb subgroups of homotopy groups of spheres, preprint; M. Golasinski, J. Mukai, On Gottlieb subgroups of homotopy groups of spheres, preprint · Zbl 1172.55004
[2] Gottlieb, D. H., Evaluation subgroups of homotopy groups, Amer. J. Math., 91, 729-755 (1969) · Zbl 0185.27102
[3] Gottlieb, D. H., Witness, transgressions, and the evaluation map, Indiana Univ. Math. J., 24, 9, 825-836 (1975) · Zbl 0299.55007
[4] Y. Hirato, Some Gottlieb subgroups of homotopy groups of spheres, in preparation; Y. Hirato, Some Gottlieb subgroups of homotopy groups of spheres, in preparation · Zbl 1056.57023
[5] Lang, G. E., Evaluation subgroups of factor spaces, Pacific. J. Math., 42, 701-709 (1972) · Zbl 0244.55018
[6] G. Lupton, S. Smith, Rationalized evaluation subgroups of a map and the rationalized \(G\); G. Lupton, S. Smith, Rationalized evaluation subgroups of a map and the rationalized \(G\) · Zbl 1112.55012
[7] Lundell, A. T., Concise tables of James numbers and some homotopy classical Lie groups and associated homogeneous spaces, (Algebraic Topology (1990). Algebraic Topology (1990), Lecture Notes in Math., vol. 1509 (1992), Springer: Springer Berlin), 250-272 · Zbl 0752.57020
[8] Lee, K. Y.; Woo, M. H., The G-sequence and the \(ω\)-homology of a CW-pair, Topology Appl., 52, 3, 221-236 (1993) · Zbl 0792.55006
[9] Lee, K. Y.; Woo, M. H., Cyclic morphisms in the category of pairs and generalized G-sequences, J. Math. Kyoto Univ., 38, 2, 271-285 (1998) · Zbl 0920.55014
[10] Lee, K. Y.; Woo, M. H., Cocyclic morphisms and dual G-sequences, Topology Appl., 116, 1, 123-136 (2001) · Zbl 0996.55010
[11] Mimura, M., Homotopy theory of Lie groups, (Handbook of Algebraic Topology (1995), Elsevier Science: Elsevier Science Amsterdam), 951-991 · Zbl 0867.57035
[12] Mimura, M.; Toda, H., Homotopy groups of symplectic groups, J. Math. Kyoto Univ., 3, 251-273 (1964) · Zbl 0129.15405
[13] Oprea, J., Gottlieb Groups, Group Actions, Fixed Points and Rational Homotopy, Lecture Notes Series, vol. 29 (1995), Seoul National Univ. Research Inc. Math. Global Analysis Research Center: Seoul National Univ. Research Inc. Math. Global Analysis Research Center Seoul · Zbl 0834.55001
[14] Siegel, J., G-spaces, \(W\)-spaces and \(H\)-spaces, Pacific J. Math., 31, 209-214 (1970) · Zbl 0183.51701
[15] Smith, S. B., Rational evaluation subgroups, Math. Z., 221, 3, 387-400 (1996) · Zbl 0855.55009
[16] Toda, H., On unstable homotopy groups of spheres and classical groups, Proc. Nat. Acad. Sci. USA, 46, 1102-1105 (1960) · Zbl 0099.38904
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.