A beta family in the homotopy of spheres

Shimomura, Katsumi

doi:10.1090/S0002-9939-2014-12009-0

A beta family in the homotopy of spheres
HTML articles powered by AMS MathViewer

by Katsumi Shimomura PDF

Proc. Amer. Math. Soc. 142 (2014), 2921-2928 Request permission

Abstract:

Let $p$ be a prime number greater than three. In the $p$-component of stable homotopy groups of spheres, Oka constructed a beta family from a $v_2$-periodic map on a four cell complex. In this paper, we construct another beta family in the groups at a prime $p$ greater than five from a $v_2$-periodic map on an eight cell complex.

References

Haynes R. Miller, Douglas C. Ravenel, and W. Stephen Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2) 106 (1977), no. 3, 469–516. MR 458423, DOI 10.2307/1971064
Shichirô Oka, A new family in the stable homotopy groups of spheres, Hiroshima Math. J. 5 (1975), 87–114. MR 380791
Shichirô Oka, A new family in the stable homotopy groups of spheres. II, Hiroshima Math. J. 6 (1976), no. 2, 331–342. MR 418096
Shichirô Oka, Realizing some cyclic $\textrm {BP}_\ast$-modules and applications to stable homotopy of spheres, Hiroshima Math. J. 7 (1977), no. 2, 427–447. MR 474290
Shichirô Oka, Ring spectra with few cells, Japan. J. Math. (N.S.) 5 (1979), no. 1, 81–100. MR 614695, DOI 10.4099/math1924.5.81
Shichirô Oka, Small ring spectra and $p$-rank of the stable homotopy of spheres, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, R.I., 1983, pp. 267–308. MR 711058
Shichirô Oka, Multiplicative structure of finite ring spectra and stable homotopy of spheres, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 418–441. MR 764594, DOI 10.1007/BFb0075582
Shichirô Oka, Derivations in ring spectra and higher torsions in $\textrm {Coker}\,J$, Mem. Fac. Sci. Kyushu Univ. Ser. A 38 (1984), no. 1, 23–46. MR 736944, DOI 10.2206/kyushumfs.38.23
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
Katsumi Shimomura, Note on beta elements in homotopy, and an application to the prime three case, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1495–1499. MR 2578544, DOI 10.1090/S0002-9939-09-10190-9
Katsumi Shimomura, The beta elements $\beta _{tp^2/r}$ in the homotopy of spheres, Algebr. Geom. Topol. 10 (2010), no. 4, 2079–2090. MR 2745666, DOI 10.2140/agt.2010.10.2079
Larry Smith, On realizing complex bordism modules. Applications to the stable homotopy of spheres, Amer. J. Math. 92 (1970), 793–856. MR 275429, DOI 10.2307/2373397