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Suspension splittings and self-maps of flag manifolds. (English) Zbl 1412.55010

For a compact connected Lie group \(G\) and a maximal torus \(T\subset G\) the authors consider the homogeneous space \(G/T\). They are interested in the set \([G/T,G/T]\) of homotopy classes of self-maps of \(G/T\). It is shown that if \(G\) is simply connected, then there is a bijection between \([G/T,G/T]\) and the Cartesian product \([G/T,G]\times\mathrm{Im}(r)\), where \(r:[G/T,G/T]\rightarrow\mathrm{End}(H^*(G/T))\) sends the homotopy class of a self-map to the induced cohomology ring endomorphism.
In order to study the group \([G/T,G]\) the authors use the identification \([G/T,G]\cong[G/T,\Omega BG]\cong[\Sigma G/T,BG]\) and decompose the suspension \(\Sigma G/T\) into a wedge of some “smaller” spaces. The method of producing these decompositions includes identifying some idempotents in \(\mathrm{End}(H^*(G/T))\), coming from Adams operations on the classifying spaces \(BT\) and \(BG\) and from the action of the Weyl group \(N(T)/T\) on \(G/T\). The decomposition of \(\Sigma G/T\) and the calculation of \([G/T,G]\) are illustrated with the examples in which \(G\) equals \(SU(3)\), \(SU(4)\), \(Sp(2)\) and \(G_2\).

MSC:

55P40 Suspensions
55S37 Classification of mappings in algebraic topology
57T15 Homology and cohomology of homogeneous spaces of Lie groups
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[1] Bergeron, F., Bergeron, N., Howlett, R. B., et al.: A decomposition of the descent algebra of a finite Coxeter group. J. Algebraic Combin., 1, 23-44 (1992) · Zbl 0798.20031 · doi:10.1023/A:1022481230120
[2] Bernstein, I. N., Gelfand, I. M., Gelfand, S. I.: Schubert cells and the cohomology of the spaces G/P, LMS. Cambridge Univ. Press, 69, 115-140 (1982)
[3] Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math., 57, 115-207 (1953) · Zbl 0052.40001 · doi:10.2307/1969728
[4] Borel, A.: Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes. Tôhoku Math. J. (2).13, 216-240 (1961) · Zbl 0109.26101 · doi:10.2748/tmj/1178244298
[5] Bott, R., Samelson, H.: The integral cohomology ring of G/T. Proc. Nat. Acad. Sci. USA, 41, 490-493 (1955) · Zbl 0064.25903 · doi:10.1073/pnas.41.7.490
[6] Chevalley, C.: Sur les décomposition cellulaires des espaces G/B, Algebraic Groups and their Generalizations: Classical Methods (W. Haboush, ed.), Proc. Sympos. Pure Math., 56, Part 1, Amer. Math. Soc., 1-23, 1994 · Zbl 0824.14042
[7] Duan, H., Zhao, X. A.: The classification of cohomology endomorphisms of certain flag manifolds. Pacific J. Math., 192, 93-102 (2000) · Zbl 1017.57015 · doi:10.2140/pjm.2000.192.93
[8] Duan, H., Zhao, X. A.: A unified formula for Steenrod operations in flag manifolds. Compositio Mathematica, 143, 257-270 (2007) · Zbl 1117.55016 · doi:10.1112/S0010437X06002405
[9] Glover, H., Homer, W.: Self-maps of flag manifolds. Trans. Amer. Math. Soc., 267, 423-434 (1981) · Zbl 0479.55014 · doi:10.1090/S0002-9947-1981-0626481-9
[10] Hilton, P., Mislin, G., Roitberg, J.: Localization of Nilpotent Groups and Spaces, Math. Studies No. 15, North-Holland, Amsterdam, 1975 · Zbl 0323.55016
[11] Kaji, S.: Three presentations of torus equivariant cohomology of flag manifolds, in Proceedings of International Mathematics Conference in honour of the 70th Birthday of Professor S. A. Ilori, arXiv.org/1504.01091
[12] Kitchloo, N.: Cohomology splittings of Stiefel manifolds. J. London Math. Soc. (2), 64(2), 457-471 (2001) · Zbl 1030.55008 · doi:10.1112/S0024610701002216
[13] Lascoux, A., Schützenberger, M.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math., 294(13), 447-450 (1982) · Zbl 0495.14031
[14] May, J. P., Ponto, K.: More concise algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2012 · Zbl 1249.55001
[15] Miller, H.: Stable splitting of Stiefel manifolds. Topology, 24(4), 411-419 (1985) · Zbl 0581.55006 · doi:10.1016/0040-9383(85)90012-6
[16] Mimura, M.: The homotopy groups of Lie groups of low rank. J. Math. Kyoto Univ., 6, 131-176 (1967) · Zbl 0171.44101 · doi:10.1215/kjm/1250524375
[17] Mimura, M., Toda, H.: Homotopy groups of SU(3), SU(4) and Sp(2). J. Math. Kyoto Univ., 3, 217-250 (1964) · Zbl 0129.15404 · doi:10.1215/kjm/1250524818
[18] Nishida, G., Yang, Y.: On a p-local stable splitting of U(n). J. Math. Kyoto Univ., 41(2), 387-401 (2001) · Zbl 1006.55006 · doi:10.1215/kjm/1250517639
[19] Papadima, S.: Rigidity properties of compact Lie groups modulo maximal tori. Math. Ann., 275, 637-652 (1986) · Zbl 0585.57023 · doi:10.1007/BF01459142
[20] Priddy, S., Recent progress on stable splittings (1985)
[21] Stembridge, J. R.: Orthogonal sets of Young symmetrizers. Adv. Appl. Math., 46, 576-582 (2011) · Zbl 1227.05268 · doi:10.1016/j.aam.2009.08.004
[22] Toda, H.: Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton University Press, Princeton 1962 · Zbl 0101.40703
[23] Toda, H., Watanabe, T.: The integral cohomology ring of F4/T and E6/T. J. Math. Kyoto Univ., 14, 257-286 (1974) · Zbl 0289.57025 · doi:10.1215/kjm/1250523239
[24] Ullman, H. E.: An equivariant generalization of the Miller splitting theorem. Algebr. Geom. Topol., 12(2), 643-684 (2012) · Zbl 1396.55011 · doi:10.2140/agt.2012.12.643
[25] Yang, Y.: On a p-local stable splitting of Stiefel manifolds. J. Math. Soc. Japan, 54, 911-921 (2002) · Zbl 1032.55009 · doi:10.2969/jmsj/1191591997
[26] Zhao, X. A.: Cohomology endomorphisms of flag manifolds. Acta Math. Sinica, 44, 1099-1106 (2001) · Zbl 1014.57018
[27] Zhao, X. A.: Maps from a simply connected space to a flag manifold G/T. Acta. Math. Sinica, 20, 1131-1134 (2004) · Zbl 1080.55015 · doi:10.1007/s10114-004-0390-7
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