Geodesic
The beta elements βtp2∕r in the homotopy of spheres
Shimomura, Katsumi1
1Department of Mathematics, Faculty of Science, Kochi University, 2-5-1, Akebono, Kochi 780-8520, Japan
Algebraic and Geometric Topology, Tome 10 (2010) no. 4, pp. 2079-2090
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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 βtp2∕r 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 βtp2∕r for t ≥ 1 and r ≤ p2 − 2 survives to the stable homotopy groups.

DOI : 10.2140/agt.2010.10.2079
Keywords: homotopy of spheres, beta family, Adams–Novikov spectral sequence

Shimomura, Katsumi  1

1 Department of Mathematics, Faculty of Science, Kochi University, 2-5-1, Akebono, Kochi 780-8520, Japan 
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Shimomura, Katsumi. The beta elements βtp2∕r in the homotopy of spheres. Algebraic and Geometric Topology, Tome 10 (2010) no. 4, pp. 2079-2090. doi: 10.2140/agt.2010.10.2079

[1] H R Miller, D C Ravenel, W S Wilson, Periodic phenomena in the Adams–Novikov spectral sequence, Ann. Math. $(2)$ 106 (1977) 469

[2] S Oka, A new family in the stable homotopy groups of spheres, Hiroshima Math. J. 5 (1975) 87

[3] S Oka, A new family in the stable homotopy groups of spheres. II, Hiroshima Math. J. 6 (1976) 331

[4] S Oka, Realizing some cyclic $\mathrm{BP}_{*} $–modules and applications to stable homotopy of spheres, Hiroshima Math. J. 7 (1977) 427

[5] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publ. 347, Amer. Math. Soc. (2004)

[6] K Shimomura, Note on beta elements in homotopy, and an application to the prime three case, Proc. Amer. Math. Soc. 138 (2010) 1495

[7] L Smith, On realizing complex bordism modules. IV. Applications to the stable homotopy groups of spheres, Amer. J. Math. 99 (1977) 418

[8] N Yamamoto, Algebra of stable homotopy of Moore space, J. Math. Osaka City Univ. 14 (1963) 45

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