Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the cohomology of the mod p Steenrod algebra

Xiugui Liu and He Wang

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Abstract

Let p be an odd prime greater than seven and A the mod p Steenrod algebra. In this paper we prove that in the cohomology of A the product $h_1 h_n \tilde \delta _{s + 4}\in {\rm Ext}_A^{s + 6, t(s,n) + s} ({\bf Z}_p , {\bf Z}_p)$ is nontrivial for $n \geq 5$, and trivial for $n=3, 4$, where $ \tilde \delta _{s + 4}$ is actually $\tilde \alpha _{s+4}^{(4)}$ described by X. Wang and Q. Zheng, $0 \leq s < p - 4$, $t(s,n) = 2(p-1)[(s + 1) + (s + 3)p + (s + 3)p^2 + (s + 4)p^3 + p^n ].$ We show our results by explicit combinatorial analysis of the (modified) May spectral sequence. The method of proof is very elementary.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 85, Number 9 (2009), 143-148.

Dates
First available in Project Euclid: 5 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.pja/1257430683

Digital Object Identifier
doi:10.3792/pjaa.85.143

Mathematical Reviews number (MathSciNet)
MR2573964

Zentralblatt MATH identifier
1186.55008

Subjects
Primary: 55S10: Steenrod algebra
Secondary: 55T15: Adams spectral sequences

Keywords
Steenrod algebra cohomology Adams spectral sequence May spectral sequence

Citation

Liu, Xiugui; Wang, He. On the cohomology of the mod p Steenrod algebra. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 9, 143--148. doi:10.3792/pjaa.85.143. https://projecteuclid.org/euclid.pja/1257430683


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References

  • J.F. Adams, Stable homotopy and generalised homology, Univ. Chicago Press, Chicago, Ill., 1974.
  • A. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem. Amer. Math. Soc. No. 42 (1962), 112 pp.
  • T. Aikawa, 3-dimensional cohomology of the mod <$>p<$> Steenrod algebra, Math. Scand. 47 (1980), no. 1, 91-115.
  • X. Wang and Q. Zheng, The convergence of <$>\tilde\alpha\sb s\sp {(n)}h\sb 0h\sb k<$>, Sci. China Ser. A 41 (1998), no. 6, 622-628.
  • X. Liu and H. Zhao, On a product in the classical Adams spectral sequence, Proc. Amer. Math. Soc. 137 (2009), no. 7, 2489-2496.