Multiplicative properties of power maps. II

McGibbon, C. A.

doi:10.1090/S0002-9947-1982-0675065-6

Multiplicative properties of power maps. II

Author: C. A. McGibbon
Journal: Trans. Amer. Math. Soc. 274 (1982), 479-508
MSC: Primary 55P45; Secondary 22E20
DOI: https://doi.org/10.1090/S0002-9947-1982-0675065-6
MathSciNet review: 675065
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The notions of -maps and -forms can be regarded as crude approximations to the concepts of homomorphisms and commutativity, respectively. These approximations are used to study power maps on connected Lie groups and their localizations. For such groups the power map $x \mapsto {x^n}$ is known to be an -map if and only if is a solution to a certain quadratic congruence. In this paper, -power maps are studied. For the Lie group Sp(l) it is shown that the -powers coincide with solutions which are common to the quadratic congruence, mentioned earlier, and another cubic congruence. Other Lie groups, when localized so as to become homotopy commutative, are also shown to have infinitely many -powers. The proofs reflect the combinatorial nature of the obstructions involved.

[1] J. F. Adams, The sphere, considered as an 𝐻-space 𝑚𝑜𝑑𝑝, Quart. J. Math. Oxford Ser. (2) 12 (1961), 52–60. MR 123323, https://doi.org/10.1093/qmath/12.1.52
[2] J. F. Adams, On the groups 𝐽(𝑋). IV, Topology 5 (1966), 21–71. MR 198470, https://doi.org/10.1016/0040-9383(66)90004-8
[3] M. Arkowitz and C. R. Curjel, On maps of 𝐻-spaces, Topology 6 (1967), 137–148. MR 230306, https://doi.org/10.1016/0040-9383(67)90030-4
[4] Raoul Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960), 249–256. MR 123330, https://doi.org/10.1007/BF02565939
[5] S. Feder and S. Gitler, Mappings of quaternionic projective spaces, Bol. Soc. Mat. Mexicana (2) 18 (1973), 33–37. MR 336740
[6] Eric M. Friedlander, Exceptional isogenies and the classifying spaces of simple Lie groups, Ann. Math. (2) 101 (1975), 510–520. MR 0391078, https://doi.org/10.2307/1970938
[7] Martin Fuchs, Verallgemeinerte Homotopie-Homomorphismen und klassifizierende Räume, Math. Ann. 161 (1965), 197–230 (German). MR 195090, https://doi.org/10.1007/BF01361971
[8] J. R. Hubbuck, On homotopy commutative 𝐻-spaces, Topology 8 (1969), 119–126. MR 238316, https://doi.org/10.1016/0040-9383(69)90004-4
[9] J. R. Hubbuck, Homotopy homomorphisms of Lie groups, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), Cambridge Univ. Press, London, 1974, pp. 33–41. London Math. Soc. Lecture Note Ser., No. 11. MR 0336746
[10] Daniel M. Kan, A relation between 𝐶𝑊-complexes and free c.s.s. groups, Amer. J. Math. 81 (1959), 512–528. MR 111036, https://doi.org/10.2307/2372755
[11] Arunas Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 978–981. MR 132543, https://doi.org/10.1073/pnas.46.7.978
[12] C. A. McGibbon, Multiplicative properties of power maps. I, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 123, 341–350. MR 587096, https://doi.org/10.1093/qmath/31.3.341
[13] -, Generalized Samelson products in the unitary group, Notices Amer. Math. Soc. 26 (1979), Abstract 770-G9, A-588.
[14] John Milnor, Construction of universal bundles. I, Ann. of Math. (2) 63 (1956), 272–284. MR 77122, https://doi.org/10.2307/1969609
[15] J. C. Moore, Algèbres d'Eilenberg Maclane et homotopie, Séminaire H. Cartan, Paris, 1955.
[16] John C. Moore, Some applications of homology theory to homotopy problems, Ann. of Math. (2) 58 (1953), 325–350. MR 59549, https://doi.org/10.2307/1969791
[17] Jean-Pierre Serre, Groupes d’homotopie et classes de groupes abéliens, Ann. of Math. (2) 58 (1953), 258–294 (French). MR 59548, https://doi.org/10.2307/1969789
[18] James Dillon Stasheff, Homotopy associativity of 𝐻-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, https://doi.org/10.1090/S0002-9947-1963-0158400-5
[19] James Stasheff, 𝐻-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR 0270372
[20] Masahiro Sugawara, On a condition that a space is an 𝐻-space, Math. J. Okayama Univ. 6 (1957), 109–129. MR 86303
[21] Masahiro Sugawara, On the homotopy-commutativity of groups and loop spaces, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 257–269. MR 120645, https://doi.org/10.1215/kjm/1250775911
[22] D. Sullivan, Geometric topology. I: Localization, periodicity, and Galois symmetry, M.I.T. Notes (1970).
[23] Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
[24] Francis D. Williams, Higher homotopy-commutativity, Trans. Amer. Math. Soc. 139 (1969), 191–206. MR 240818, https://doi.org/10.1090/S0002-9947-1969-0240818-9
[25] Alexander Zabrodsky, Hopf spaces, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1976. North-Holland Mathematics Studies, Vol. 22; Notas de Matemática, No. 59. MR 0440542