Two results on the 2-local 𝐸𝐻𝑃 spectral sequence

Barratt, M. G.; Cohen, F.; Gray, B.; Mahowald, M.; Richter, W.

doi:10.1090/S0002-9939-1995-1246514-4

Two results on the $2$-local $EHP$ spectral sequence
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by M. G. Barratt, F. Cohen, B. Gray, M. Mahowald and W. Richter

Proc. Amer. Math. Soc. 123 (1995), 1257-1261

DOI: https://doi.org/10.1090/S0002-9939-1995-1246514-4

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Abstract:

The ${E_2}$-term of the 2-local EHP spectral sequence is shown to be a $\mathbb {Z}/2$ module. 4 is the order of the identity map on the double loop space of the fiber $W(n)$ of the double suspension ${E^2}:{S^{2n - 1}} \to {\Omega ^2}{S^{2n + 1}}$.

References

F. R. Cohen, A course in some aspects of classical homotopy theory, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 1–92. MR 922923, DOI 10.1007/BFb0078738
F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. (2) 110 (1979), no. 3, 549–565. MR 554384, DOI 10.2307/1971238
Brayton Gray, Unstable families related to the image of $J$, Math. Proc. Cambridge Philos. Soc. 96 (1984), no. 1, 95–113. MR 743705, DOI 10.1017/S0305004100061971
Brayton Gray, On the iterated suspension, Topology 27 (1988), no. 3, 301–310. MR 963632, DOI 10.1016/0040-9383(88)90011-0
John R. Harper, A proof of Gray’s conjecture, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 189–195. MR 1022681, DOI 10.1090/conm/096/1022681
I. M. James, On the suspension sequence, Ann. of Math. (2) 65 (1957), 74–107. MR 83124, DOI 10.2307/1969666
Mark Mahowald, The image of $J$ in the $EHP$ sequence, Ann. of Math. (2) 116 (1982), no. 1, 65–112. MR 662118, DOI 10.2307/2007048
Mark E. Mahowald and Douglas C. Ravenel, The root invariant in homotopy theory, Topology 32 (1993), no. 4, 865–898. MR 1241877, DOI 10.1016/0040-9383(93)90055-Z
Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042

The H-space squaring map on

factors through the double suspension

Paul Selick, $2$-primary exponents for the homotopy groups of spheres, Topology 23 (1984), no. 1, 97–99. MR 721456, DOI 10.1016/0040-9383(84)90029-6