The Cohomology of P(2) in Dimensions Less Than 4 at an Odd Prime

Abstract

Let

P ( 2 ) = P / ( {\xi }_1^{p^2} , {\xi }_2^{p^2} , {\xi }_3^{p^2} , \dots )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P(2)=P/(\xi_{1}^{p^{2}},\xi_{2}^{p^{2}},\xi_{3}^{p^{2}},\ldots)$$\end{document}

, where

P = {\mathbb{F}}_p [ {\xi }_1 , {\xi }_2 , {\xi }_3 , \dots ]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P=\mathbb{F}_{p}[\xi_{1},\xi_{2},\xi_{3},\ldots]$$\end{document}

is the polynomial part of the dual Steenrod algebra and p is an odd prime. In this paper we calculate the cohomology of P(2) in dimensions less than 4 by a May spectral sequence.

Keywords

Chromatic spectral sequence / May spectral sequence / Steenrod algebra / cohomology

References

[1] AndrewsMJ The v ₁-periodic part of the Adams spectral sequence at an odd prime, 2015 Cambridge, MA Massachusetts Institute of Technology

[2] LiuXG, WangXJ. A four-filtrated May spectral sequence and its applications. Acta Math. Sin. (Engl. Ser.), 2008, 24(9): 1507-1524

[3] MillerH. A localization theorem in homological algebra. Math. Proc. Cambridge Philos. Soc., 1978, 84(1): 73-84

[4] MillerH, RavenelDC, WilsonS. Periodic phenomena in the Adams–Novikov spectral sequence. Ann. of Math. (2), 1977, 106(3): 469-516

[5] MilnorJ. The Steenrod algebra and its dual. Ann. of Math. (2), 1958, 67: 150-171

[6] RavenelDC. The cohomology of the Morava stabilizer algebras. Math. Z., 1977, 152(3): 287-297

[7] RavenelDC. Complex Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics, 121, 1986 Orlando, FL Academic Press, Inc.

[8] ShimomuraK, WangXJ. The Adams–Novikov E₂-term for π_*(L₂ S⁰)-term at the prime 2. Math. Z., 2002, 241(2): 271-311

[9] ShimomuraK, WangXJ. The homotopy groups π_*(L ₂ S⁰) at the prime 3. Topology, 2002, 41(6): 1183-1198

[10] ShimomuraK, YabeA. The homotopy groups π_*(L ₂ S⁰). Topology, 1995, 34(2): 261-289

RIGHTS & PERMISSIONS

Peking University