Two New Families in the Stable Homotopy Groups of Sphere and Moore Spectrum*

Abstract

This paper proves the existence of an order p element in the stable homotopy group of sphere spectrum of degree p ⁿ q + p ^m q + q-4 and a nontrivial element in the stable homotopy group of Moore spectum of degree p ⁿ q + p ^m q + q-3 which are represented by h ₀(h _m b _n-1 - h _n b _m-1) and i _*(h ₀ h _n h _m) in the E ₂-terms of the Adams spectral sequence respectively, where p ≥ 7 is a prime, n ≥ m + 2 ≥ 4; q = 2(p - 1).

Keywords

Stable homotopy groups of spheres / Adams spectral sequence / Toda spectrum / 55Q45

References

[1] Aikawa Math. Scand., 1980, 47: 91

[2] Cohen, R., Odd primary families in stable homotopy theory, Memoirs of Amer. Math. Soc., 242, 1981.

[3] Cohen Topology, 1984, 23: 177

[4] Lin Hiroshima Math. J., 1998, 28: 183

[5] Liulevicius, A., The factorizations of cyclic reduced powers by secondary cohomology operations, Memoirs of Amer. Math. Soc., 42, 1962.

[6] Miller Ann. of Math., 1977, 106: 469

[7] Oka, S., Multiplicative Structure of Finite Ring Spectra and Stable Homotopy of Spheres, Algebraic Topology (Aarhus), Lect. Notes in Math., 1051, 1984.

[8] Ravenel, D. C., Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, Inc., 1986.

[9] Thomas Advances in Math., 1976, 20: 287