The beta elements βtp2∕r in the homotopy of spheres

Shimomura, Katsumi

doi:10.2140/agt.2010.10.2079

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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

DOI: 10.2140/agt.2010.10.2079

Abstract

In In [Ann. Math. (2) 106 (1977) 469–516], Miller, Ravenel and Wilson defined generalized beta elements in the $E_{2}$ –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 $β_{t p^{2} ∕ r}$ for positive integers $t$ and $r$ survive to the homotopy of spheres at a prime $p > 3$ , when $r \leq 2 p - 2$ and $r \leq 2 p$ if $t > 1$ . In this paper, for $p > 5$ , we expand the condition so that $β_{t p^{2} ∕ r}$ for $t \geq 1$ and $r \leq p^{2} - 2$ 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

Volume 10, issue 4 (2010)