Volume 10, issue 4 (2010)

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The beta elements $\beta_{tp^2/r}$ in the homotopy of spheres

Katsumi Shimomura

Algebraic & Geometric Topology 10 (2010) 2079–2090
Abstract

In In [Ann. Math. (2) 106 (1977) 469–516], Miller, Ravenel and Wilson defined generalized beta elements in the E2–term of the Adams–Novikov spectral sequence converging to the stable homotopy groups of spheres, and in [Hiroshima Math. J. 7 (1977) 427–447], Oka showed that the beta elements of the form βtp2r for positive integers t and r survive to the homotopy of spheres at a prime p > 3, when r 2p 2 and r 2p if t > 1. In this paper, for p > 5, we expand the condition so that βtp2r for t 1 and r p2 2 survives to the stable homotopy groups.

Keywords
homotopy of spheres, beta family, Adams–Novikov spectral sequence
Mathematical Subject Classification 2000
Primary: 55Q45
Secondary: 55Q10
References
Publication
Received: 21 August 2009
Revised: 26 March 2010
Accepted: 2 September 2010
Published: 16 October 2010
Authors
Katsumi Shimomura
Department of Mathematics
Faculty of Science
Kochi University
2-5-1
Akebono
Kochi 780-8520
Japan