Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The $ H$-space squaring map on $ \Omega\sp 3S\sp {4n+1}$ factors through the double suspension


Author: William Richter
Journal: Proc. Amer. Math. Soc. 123 (1995), 3889-3900
MSC: Primary 55Q40; Secondary 55Q15, 55Q25
DOI: https://doi.org/10.1090/S0002-9939-1995-1273520-6
MathSciNet review: 1273520
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We compute the first EHP spectral sequence differential followed by the double suspension. We show that $ 2{\pi _ \ast }({S^{4n + 1}}) \subset \operatorname{Im} ({E^2})$, which refines the exponent for $ {\pi _ \ast }({S^{2n + 1}})$ of James and Selick. The proof follows an odd primary program of Gray and Harper, and uses Barratt's theory of unsuspended Hopf invariants and Boardman and Steer's geometric Hopf invariants.


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

  • [1] M. G. Barratt, Track groups (I), Proc. London Math. Soc. 5 (1955), 71-106. MR 0072477 (17:290c)
  • [2] -, On the spectral sequence of an inclusion, Aarhus lecture notes, Aarhus Universitet, 1962.
  • [3] M. G. Barratt, F. Cohen, B. Gray, M. Mahowald, and W. Richter, Two results on the 2-local EHP spectral sequence, Proc. Amer. Math. Soc. 123 (1995), 1257-1261. MR 1246514 (95e:55018)
  • [4] H. J. Baues, Commutator calculus and groups of homotopy classes, London Math. Soc. Lecture Notes, vol. 50, Cambridge Univ. Press, Cambridge, 1981. MR 634675 (83b:55012)
  • [5] J. M. Boardman and B. Steer, On Hopf invariants, Comment. Math. Helv. 42 (1968), 217-224. MR 0221503 (36:4555)
  • [6] F. Cohen, A course in some aspects of classical homotopy theory, Algebraic Topology (Proc. Seattle 1985) (H. Miller and D. Ravenel, eds.), Lecture Notes in Math., vol. 1286, Springer, New York, 1987, pp. 1-92. MR 922923 (89e:55027)
  • [7] T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1964), 295-322. MR 0179791 (31:4033)
  • [8] -, Induced fibrations and cofibrations, Trans. Amer. Math. Soc. 127 (1967), 442-459. MR 0210131 (35:1025)
  • [9] B. Gray, Unstable families related to the image of J, Math. Proc. Cambridge Philos. Soc. 96 (1984), 95-113. MR 743705 (86b:55014)
  • [10] -, On Toda's fibration, Math. Proc. Cambridge Philos. Soc. 97 (1985), 289-298.
  • [11] J. Harper, A proof of Gray's conjecture, Algebraic Topology (M. Mahowald and S. Priddy, eds.), Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 189-195. MR 1022681 (91b:55013)
  • [12] I. M. James, On the suspension sequence, Ann. of Math. (2) 65 (1957), 74-107. MR 0083124 (18:662e)
  • [13] M. Mahowald and W. Richter, $ \Omega SU(n)$ does not split in 2 suspensions, for $ n \geq 3$, Algebraic Topology (Proc. Oaxtepec) (M. Tangora, ed.), Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993. MR 1224917 (94h:55011)
  • [14] J. Moore and J. Neisendorfer, Equivalence of Toda-Hopf invariants, Dedicated to the memory of A. Zabrodsky, Israel J. Math. 66 (1989), 300-318. MR 1017169 (90g:55013)
  • [15] P. Selick, 2-primary exponents for the homotopy groups of spheres, Topology 23 (1984), 97-99. MR 721456 (85k:55011)
  • [16] H. Toda, On the double suspension $ {E^2}$, J. Inst. Polytechnics, Osaka City Univ. 7 (1956), 103-145. MR 0092968 (19:1188g)
  • [17] -, Composition methods in the homotopy groups of spheres, Princeton Univ. Press, Princeton, NJ, 1962. MR 0143217 (26:777)
  • [18] G. W. Whitehead, Elements of homotopy theory, Graduates Texts in Math., vol. 61, Springer, New York, 1980. MR 516508 (80b:55001)
  • [19] A. Zabrodsky, Hopf spaces, Math. Stud., vol. 22, North-Holland, Amsterdam, 1976. MR 0440542 (55:13416)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55Q40, 55Q15, 55Q25

Retrieve articles in all journals with MSC: 55Q40, 55Q15, 55Q25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1273520-6
Keywords: James-Hopf invariants, Cartan formula, exponents, EHP fibrations
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society