Hopf invariants for reduced products of spheres
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- by Hans Joachim Baues PDF
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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 \[ H_m^n:{\pi _{mn - 1}}(S_{m - 1}^n) \to {\mathbf {Z}}\] 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
- J. F. Adams and P. J. Hilton, On the chain algebra of a loop space, Comment. Math. Helv. 30 (1956), 305–330. MR 77929, DOI 10.1007/BF02564350
- J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. MR 141119, DOI 10.2307/1970147
- Hans Joachim Baues, Der Pontryagin-Ring von Quotienten eines Torus, Math. Z. 134 (1973), 221–228 (German). MR 334201, DOI 10.1007/BF01214095
- Hans Joachim Baues, Hindernisse in dem Produkt von Suspensionen, Math. Ann. 200 (1973), 11–23 (German). MR 343267, DOI 10.1007/BF01578289
- K. A. Hardie, A proof of the Nakaoka-Toda formula, Pacific J. Math. 14 (1964), 1249–1254. MR 172288
- K. A. Hardie, Higher Whitehead products, Quart. J. Math. Oxford Ser. (2) 12 (1961), 241–249. MR 141110, DOI 10.1093/qmath/12.1.241
- K. A. Hardie, On the Hopf-Toda invariant, Trans. Amer. Math. Soc. 112 (1964), 43–54. MR 166784, DOI 10.1090/S0002-9947-1964-0166784-8 H. Hopf, Über die Abbilungen von Sphären auf Sphären niedrigerer Dimension, Fund. Math. 25 (1935), 427-440.
- I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170–197. MR 73181, DOI 10.2307/2007107
- I. M. James, Note on cup-products, Proc. Amer. Math. Soc. 8 (1957), 374–383. MR 91467, DOI 10.1090/S0002-9939-1957-0091467-4
- I. M. James, On the homotopy groups of certain pairs and triads, Quart. J. Math. Oxford Ser. (2) 5 (1954), 260–270. MR 68837, DOI 10.1093/qmath/5.1.260
- I. M. James, Filtration of the homotopy groups of spheres, Quart. J. Math. Oxford Ser. (2) 9 (1958), 301–309. MR 100838, DOI 10.1093/qmath/9.1.301
- 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, DOI 10.1073/pnas.46.7.978
- R. James Milgram, Iterated loop spaces, Ann. of Math. (2) 84 (1966), 386–403. MR 206951, DOI 10.2307/1970453
- Minoru Nakaoka and Hirosi Toda, On Jacobi identity for Whitehead products, J. Inst. Polytech. Osaka City Univ. Ser. A 5 (1954), 1–13. MR 65926
- Gerald J. Porter, Higher-order Whitehead products, Topology 3 (1965), 123–135. MR 174054, DOI 10.1016/0040-9383(65)90039-X
- Hans Samelson, A connection between the Whitehead and the Pontryagin product, Amer. J. Math. 75 (1953), 744–752. MR 60819, DOI 10.2307/2372549
- Jean-Pierre Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425–505 (French). MR 45386, DOI 10.2307/1969485
- Albert Shar, $\pi _{mn-}{}_{}{2}(S^{n}_{m-}{}_{}{2})$ contains an element of order $m$, Proc. Amer. Math. Soc. 34 (1972), 303–306. MR 292079, DOI 10.1090/S0002-9939-1972-0292079-8
- Albert Shar, The homotopy groups of spaces whose cohomology is a $Z_{p}$ truncated polynomial algebra, Proc. Amer. Math. Soc. 38 (1973), 172–178. MR 310877, DOI 10.1090/S0002-9939-1973-0310877-X
- N. E. Steenrod, Cohomology invariants of mappings, Ann. of Math. (2) 50 (1949), 954–988. MR 31231, DOI 10.2307/1969589
- N. E. Steenrod, Cohomology operations, Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. MR 0145525
- Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
Additional Information
- © Copyright 1976 American Mathematical Society
- 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