Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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 $ A_n$-maps and $ C_n$-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 $ A_2$-map if and only if $ n$ is a solution to a certain quadratic congruence. In this paper, $ A_3$-power maps are studied. For the Lie group Sp(l) it is shown that the $ A_3$-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 $ A_3$-powers. The proofs reflect the combinatorial nature of the obstructions involved.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, The sphere, considered as an $ H$-space $ \operatorname{mod} \,p$, Quart. J. Math. Oxford. Ser. (2) 12 (1961), 52-60. MR 0123323 (23:A651)
  • [2] -, On the groups $ J(X)$. IV, Topology 5 (1966), 21-71. MR 0198470 (33:6628)
  • [3] M. Arkowitz and C. R. Curjel, On the maps of $ H$-spaces, Topology 6 (1967), 137-148. MR 0230306 (37:5868)
  • [4] R. Bott, A note on the Samelson product in the classical groups, Comment. Math. Helv. 34 (1960), 249-256. MR 0123330 (23:A658)
  • [5] S. Feder and S. Gitler, Mappings of quaternionic projective space, Bol. Soc. Mat. Mexicana 18 (1973), 33-37. MR 0336740 (49:1513)
  • [6] E. M. Friedlander, Exceptional isogenies and the classifying spaces of simple Lie groups, Ann. of Math. (2) 101 (1975), 510-520. MR 0391078 (52:11900)
  • [7] M. Fuchs, Verallgemeinerte Homotopie-homomorphismen und klassifizierende raume, Math. Ann. 161 (1965), 197-230. MR 0195090 (33:3295)
  • [8] J. R. Hubbuck, On homotopy commutative $ H$-spaces, Topology 8 (1969), 119-126. MR 0238316 (38:6592)
  • [9] -, Homotopy homomorphisms of Lie groups, London Math. Soc. Lecture Notes Ser. No. 11, Cambridge Univ. Press, 1974, pp. 33-41. MR 0336746 (49:1519)
  • [10] D. M. Kan, A relation between CW complexes and free c.s.s. groups, Amer. J. Math. 81 (1959), 512-520. MR 0111036 (22:1901)
  • [11] A. L. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 978-981. MR 0132543 (24:A2383)
  • [12] C. A. McGibbon, Multiplicative properties of power maps. I, Quart. J. Math. Oxford (2) 31 (1980), 341-350. MR 587096 (82c:55009)
  • [13] -, Generalized Samelson products in the unitary group, Notices Amer. Math. Soc. 26 (1979), Abstract 770-G9, A-588.
  • [14] J. W. Milnor, Construction of universal bundles. I, Ann. of Math. (2) 63 (1956), 272-284. MR 0077122 (17:994b)
  • [15] J. C. Moore, Algèbres d'Eilenberg Maclane et homotopie, Séminaire H. Cartan, Paris, 1955.
  • [16] -, Some applications of homology theory to homotopy problems, Ann. of Math. (2) 58 (1953), 325-350. MR 0059549 (15:549a)
  • [17] J. P. Serre, Groupes d'homotopie et classes de groupes abélians, Ann. of Math. 58 (1953), 258-294. MR 0059548 (15:548c)
  • [18] J. D. Stasheff, Homotopy associativity of $ H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293-312. MR 0158400 (28:1623)
  • [19] -, $ H$-spaces from a homotopy point of view, Lecture Notes in Math., vol. 161, Springer, Berlin, 1970. MR 0270372 (42:5261)
  • [20] M. Sugawara, A condition that a space is group-like, Math. J. Okayama Univ. 6 (1957), 109-129. MR 0086303 (19:160c)
  • [21] -, On the homotopy-commutativity of groups and loop spaces, Mem. Coll. Sci. Univ. Koyoto Ser. A 33 (1960/61), 257-269. MR 0120645 (22:11394)
  • [22] D. Sullivan, Geometric topology. I: Localization, periodicity, and Galois symmetry, M.I.T. Notes (1970).
  • [23] H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies, no. 49, Princeton Univ. Press, Princeton, N. J., 1962. MR 0143217 (26:777)
  • [24] F. D. Williams, Higher homotopy-commutativity, Trans. Amer. Math. Soc. 139 (1969), 191-206. MR 0240818 (39:2163)
  • [25] A. Zabrodsky, Hopf spaces, North-Holland Mathematics Studies, no. 22, North-Holland, Amsterdam, 1976. MR 0440542 (55:13416)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55P45, 22E20

Retrieve articles in all journals with MSC: 55P45, 22E20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0675065-6
Keywords: $ A_n$-map, $ C_n$-form, projective $ n$-space, localization of spaces, homotopy commutativity
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society