Projectivity of the Whitehead product in spheres
HTML articles powered by AMS MathViewer
- by WΓͺn Hsiung Lin PDF
- Proc. Amer. Math. Soc. 110 (1990), 527-534 Request permission
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 $i \leq 10$.References
- J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20β104. MR 141119, DOI 10.2307/1970147 β, Operations of the $n$th kind in $K$-theory and what we donβt know about $\mathbb {R}{p^\infty }$, London Math. Soc. Lecture Note Ser., no. 11, Cambridge University, 1974, pp. 1-11.
- M. F. Atiyah, Thom complexes, Proc. London Math. Soc. (3) 11 (1961), 291β310. MR 131880, DOI 10.1112/plms/s3-11.1.291
- M. G. Barratt, J. D. S. Jones, and M. E. Mahowald, The Kervaire invariant problem, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp.Β 9β22. MR 711039, DOI 10.1090/conm/019/711039 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.
- Edgar H. Brown and Ralph L. Cohen, The Adams spectral sequence of $\Omega ^2S^3$ and Brown-Gitler spectra, Algebraic topology and algebraic $K$-theory (Princeton, N.J., 1983) Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp.Β 101β125. MR 921474
- Edgar H. Brown Jr. and Samuel Gitler, A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology 12 (1973), 283β295. MR 391071, DOI 10.1016/0040-9383(73)90014-1
- E. H. Brown Jr. and F. P. Peterson, On the stable decomposition of $\Omega ^{2}S^{r+2}$, Trans. Amer. Math. Soc. 243 (1978), 287β298. MR 500933, DOI 10.1090/S0002-9947-1978-0500933-4
- Γ. Kh. Braun Jr. and F. P. Peterson, The Brown-Gitler spectrum, the space $\Omega ^{2}S^{3}$ and the elements $\eta _{j}\in \Pi _{2^{j}}$, Trudy Mat. Inst. Steklov. 154 (1983), 38β43 (Russian). Topology (Moscow, 1979). MR 733824
- F. R. Cohen, M. E. Mahowald, and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp.Β 225β228. MR 520543
- Donald M. Davis, The antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc. 44 (1974), 235β236. MR 328934, DOI 10.1090/S0002-9939-1974-0328934-1
- Donald M. Davis and Mark Mahowald, Classification of the stable homotopy types of stunted real projective spaces, Pacific J. Math. 125 (1986), no.Β 2, 335β345. MR 863530
- Stanley O. Kochman, Stable homotopy groups of spheres, Lecture Notes in Mathematics, vol. 1423, Springer-Verlag, Berlin, 1990. A computer-assisted approach. MR 1052407, DOI 10.1007/BFb0083795
- Mark Mahowald, A new infinite family in ${}_{2}\pi _{*}{}^s$, Topology 16 (1977), no.Β 3, 249β256. MR 445498, DOI 10.1016/0040-9383(77)90005-2
- Mark Mahowald and Martin Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967), 349β369. MR 214072, DOI 10.1016/0040-9383(67)90023-7
- R. James Milgram and Peter Zvengrowski, Even Whitehead squares are not projective, Canadian J. Math. 29 (1977), no.Β 5, 957β962. MR 448348, DOI 10.4153/CJM-1977-096-9
- R. James Milgram, J. Strutt, and P. Zvengrowski, Projective stable stems of spheres, Bol. Soc. Mat. Mexicana (2) 22 (1977), no.Β 2, 48β57. MR 526797
- Duane Randall, Projectivity of the Whitehead square, Proc. Amer. Math. Soc. 40 (1973), 609β611. MR 356047, DOI 10.1090/S0002-9939-1973-0356047-0
- V. P. Snaith, A stable decomposition of $\Omega ^{n}S^{n}X$, J. London Math. Soc. (2) 7 (1974), 577β583. MR 339155, DOI 10.1112/jlms/s2-7.4.577
- Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
- J. H. C. Whitehead, On the groups $\pi _r(V_{n,m})$ and sphere-bundles, Proc. London Math. Soc. (2) 48 (1944), 243β291. MR 12226, DOI 10.1112/plms/s2-48.1.243
Additional Information
- © Copyright 1990 American Mathematical Society
- 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