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Hopf invariants for reduced products of spheres


Author: Hans Joachim Baues
Journal: Proc. Amer. Math. Soc. 59 (1976), 169-174
MSC: Primary 55E25
DOI: https://doi.org/10.1090/S0002-9939-1976-0420616-2
MathSciNet review: 0420616
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ S_m^n$ be the $ m$th reduced product complex of the even dimensional sphere $ {S^n}$. Using 'cup'-products, James defined a Hopf invariant homomorphism

$\displaystyle H_m^n:{\pi _{mn - 1}}(S_{m - 1}^n) \to {\mathbf{Z}}$

such that $ H_2^n$ is the classical Hopf invariant. Extending the result of Adams on $ H_2^n$ we determine the image of $ H_m^n$. Partial calculations were made by Hardie and Shar.

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

  • [1] J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305-330. MR 17, 1119. MR 0077929 (17:1119b)
  • [2] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20-104. MR 25 #4530. MR 0141119 (25:4530)
  • [3] H. J. Baues, Der Pontryagin-Ring von Quotienten eines Torus, Math. Z. 134 (1973), 221-228. MR 48 #12520. MR 0334201 (48:12520)
  • [4] -, Hindernisse in dem Produkt von Suspensionen, Math. Ann. 200 (1973), 11-23. MR 49 #8011. MR 0343267 (49:8011)
  • [5] K. A. Hardie, A proof of the Nakaoka-Toda formula, Pacific J. Math. 14 (1964), 1249-1254. MR 30 #2507. MR 0172288 (30:2507)
  • [6] -, Higher Whitehead products, Quart. J. Math. Oxford Ser. (2) 12 (1961), 241-249. MR 25 #4521. MR 0141110 (25:4521)
  • [7] -, On the Hopf-Toda invariant, Trans. Amer. Math. Soc. 112 (1964), 43-54. MR 29 #4057. MR 0166784 (29:4057)
  • [8] H. Hopf, Über die Abbilungen von Sphären auf Sphären niedrigerer Dimension, Fund. Math. 25 (1935), 427-440.
  • [9] I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170-197. MR 17, 396. MR 0073181 (17:396b)
  • [10] -, Note on cup-products, Proc. Amer. Math. Soc. 8 (1957), 374-383. MR 19, 974. MR 0091467 (19:974a)
  • [11] -, On the homotopy groups of certain pairs and triads, Quart. J. Math. Oxford Ser. (2) 5 (1954), 260-270. MR 16, 948. MR 0068837 (16:948e)
  • [12] -, Filtration of the homotopy groups of spheres, Quart. J. Math. Oxford Ser. (2) 9 (1958), 301-309. MR 20 #7266. MR 0100838 (20:7266)
  • [13] A. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 978-981. MR 24 #A2383. MR 0132543 (24:A2383)
  • [14] R. J. Milgram, Iterated loop spaces, Ann. of Math. (2) 84 (1966), 386-403. MR 34 #6767. MR 0206951 (34:6767)
  • [15] M. Nakaoka and H. Toda, On Jacobi identity for Whitehead products, J. Inst. Polytech. Osaka City Univ. Ser. A. 5 (1954), 1-13. MR 16, 505. MR 0065926 (16:505b)
  • [16] G. J. Porter, Higher-order Whitehead products, Topology 3 (1965), 123-135. MR 30 #4261. MR 0174054 (30:4261)
  • [17] H. Samelson, A connection between the Whitehead and the Pontryagin product, Amer. J. Math. 75 (1953), 744-752. MR 15, 731. MR 0060819 (15:731e)
  • [18] J.-P. Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425-505. MR 13, 574. MR 0045386 (13:574g)
  • [19] A. Shar, $ {\pi _{mn - 2}}(S_{m - 2}^n)$ contains an element of order $ m$, Proc. Amer. Math. Soc. 34 (1972), 303-306. MR 45 #1166. MR 0292079 (45:1166)
  • [20] -, The homotopy groups of spaces whose cohomology is a $ {Z_p}$ truncated polynomial algebra, Proc. Amer. Math. Soc. 38 (1973), 172-178. MR 46 #9975. MR 0310877 (46:9975)
  • [21] N. E. Steenrod, Cohomology invariants of mappings, Ann. of Math. (2) 50 (1949), 954-988. MR 11, 122. MR 0031231 (11:122a)
  • [22] -, Cohomology operations (Lectures by N. E. Steenrod and revised by D. B. A. Epstein), Ann. of Math. Studies, no. 50, Princeton Univ. Press, Princeton, N. J., 1962. MR 26 #3056. MR 0145525 (26:3056)
  • [23] H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies, no. 49, Princeton Univ. Press, Princeton, N. J., 1962. MR 26 #777. MR 0143217 (26:777)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0420616-2
Keywords: Hopf invariant, Whitehead product, reduced product space
Article copyright: © Copyright 1976 American Mathematical Society

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