Algebraic & Geometric Topology

Products of Greek letter elements dug up from the third Morava stabilizer algebra

Ryo Kato and Katsumi Shimomura

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In [Hiroshima Math. J. 12 (1982) 611–626], Oka and the second author considered the cohomology of the second Morava stabilizer algebra to study nontriviality of the products of beta elements of the stable homotopy groups of spheres. In this paper, we use the cohomology of the third Morava stabilizer algebra to find nontrivial products of Greek letters of the stable homotopy groups of spheres: α1γt, β2γt, α1,α1,βpppγtβ1 and β1,p,γt for t with pt(t21) for a prime number p>5.

Article information

Algebr. Geom. Topol., Volume 12, Number 2 (2012), 951-961.

Received: 1 August 2011
Revised: 29 November 2011
Accepted: 8 January 2012
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55Q45: Stable homotopy of spheres
Secondary: 55Q51: $v_n$-periodicity

BP–theory stable homotopy of spheres


Kato, Ryo; Shimomura, Katsumi. Products of Greek letter elements dug up from the third Morava stabilizer algebra. Algebr. Geom. Topol. 12 (2012), no. 2, 951--961. doi:10.2140/agt.2012.12.951.

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  • R L Cohen, Odd primary infinite families in stable homotopy theory, Mem. Amer. Math. Soc. 30, no. 242, Amer. Math. Soc. (1981)
  • H R Miller, D C Ravenel, W S Wilson, Periodic phenomena in the Adams–Novikov spectral sequence, Ann. of Math. 106 (1977) 469–516
  • S Oka, K Shimomura, On products of the $\beta $–elements in the stable homotopy of spheres, Hiroshima Math. J. 12 (1982) 611–626
  • D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Math. 121, Academic Press, Orlando, FL (2000)
  • K Shimomura, On differential of a generalized Adams spectral sequence, J. Fac. Educ. Tottori Univ. (Nat. Sci.) 41 (1992) 119–131
  • H Toda, Algebra of stable homotopy of $Z\sb{p}$–spaces and applications, J. Math. Kyoto Univ. 11 (1971) 197–251
  • A Yamaguchi, The structure of the cohomology of Morava stabilizer algebra $S(3)$, Osaka J. Math. 29 (1992) 347–359