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Projectivity of the Whitehead product in spheres


Author: Wên Hsiung Lin
Journal: Proc. Amer. Math. Soc. 110 (1990), 527-534
MSC: Primary 55Q15; Secondary 55P40, 55Q40
DOI: https://doi.org/10.1090/S0002-9939-1990-1023352-4
MathSciNet review: 1023352
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Abstract: The Whitehead square $ \left[ {{l_{{2^i} - 1}},{l_{{2^i} - 1}}} \right] \in {\pi _{{2^{i + 1}} - 3}}({S^{{2^i} - 1}})$ is shown to be projective for $ i \leq 10$.


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  • [1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960), 20-104. MR 0141119 (25:4530)
  • [2] -, Operations of the $ n$th kind in $ K$-theory and what we don't know about $ \mathbb{R}{p^\infty }$, London Math. Soc. Lecture Note Ser., no. 11, Cambridge University, 1974, pp. 1-11.
  • [3] M. F. Atiyah, Thom complexes, Proc. London. Math. Soc. 11 (1961), 291-310. MR 0131880 (24:A1727)
  • [4] M. G. Barratt, J.D.S. Jones, and M. E. Mahowald, The Kervaire invariant problem, Contemp. Math., Amer. Math. Soc., vol. 19, 1983, pp. 9-22. MR 711039 (85e:55027)
  • [5] A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G Quillen, D. L. Rector, and J. W. Shlessinger, The $ \bmod - p$ lower central series and the Adams spectral sequence. II, Topology 9 (1970), 309-316.
  • [6] E. H. Brown and R. L. Cohen, The Adams spectral sequence of $ {\Omega ^2}{S^3}$ and Brown-Gitler spectra, algebraic topology and algebraic $ k$-theory (W. Browder, ed.), Ann. Math. Studies, no. 113, 1987, pp. 101-125. MR 921474 (89d:55020)
  • [7] E. H. Brown and S. Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973), 283-295. MR 0391071 (52:11893)
  • [8] E. H. Brown and F. P. Peterson, On the stable decomposition of $ {\Omega ^2}{S^{r + 2}}$, Trans. Amer. Math. Soc. 243 (1978), 287-298. MR 0500933 (58:18424)
  • [9] E. H. Brown and F. P. Peterson, The Brown-Gitler spectrum, $ {\Omega ^2}{S^3}$ and $ {\eta _j} \in {\pi _{{2^j}}}(S)$, Trudy Mat. Inst. Steklov. 154 (1983), 38-43. MR 733824 (85b:55013)
  • [10] F. R. Cohen, M. E. Mahowald, and R. J. Milgram, The stable decomposition of the double loop space of a spheres, Proc. Sympos. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1978, pp. 225-228. MR 520543 (80j:55009)
  • [11] D. M. Davis, The antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc. 44 (1974), 235-236. MR 0328934 (48:7276)
  • [12] D. M. Davis and M. E. Mahowald, Classification of stunted real projective space, Pacific J. Math. 125 (1986), 335-345. MR 863530 (88a:55008)
  • [13] S. O. Kochman, A computer assisted approach to the calculation of stable stems (to appear). MR 1052407 (91j:55016)
  • [14] M. E. Mahowald, A new infinite family in $ {}_2\pi _*^S$, Topology 16 (1977), 249-256. MR 0445498 (56:3838)
  • [15] M. E. Mahowald and M. C. Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349-369. MR 0214072 (35:4924)
  • [16] R. J. Milgram and P. Zvengrowski, Even Whitehead squares are not projective, Canad. J. Math. 29 (1977), 957-962. MR 0448348 (56:6655)
  • [17] R. J. Milgram, J. Strutt, and P. Zvengrowsksi, Projective stable stems of spheres, Bol. Soc. Mat. Mexicana (2) 22 (1977), 48-57. MR 526797 (80e:55025)
  • [18] D. Randall, Projectivity of the Whitehead square, Proc. Amer. Math. Soc. 40 (1973), 609-611. MR 0356047 (50:8518)
  • [19] V. P. Snaith, A stable decomposition of $ {\Omega ^n}{S^n}X$, J. London Math. Soc. 2 (1974), 577-583. MR 0339155 (49:3918)
  • [20] H. Toda, Composition methods in the homotopy groups of spheres, Ann. of Math. Stud., vol. 49, Princeton Univ. Press., Princeton, NJ, 1962. MR 0143217 (26:777)
  • [21] J. H. C. Whitehead, On the groups $ {\pi _*}({\upsilon _{n,m}})$ and sphere bundles, Proc. London Math. Soc. 48 (1944), 243-291. MR 0012226 (6:279b)

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DOI: https://doi.org/10.1090/S0002-9939-1990-1023352-4
Article copyright: © Copyright 1990 American Mathematical Society

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