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The Steenrod problem of realizing polynomial cohomology rings

2008; Wiley; Volume: 1; Issue: 4 Linguagem: Inglês

10.1112/jtopol/jtn021

1753-8424

Kasper K. S. Andersen, Jesper Grodal,

Advanced Topics in Algebra

Journal of TopologyVolume 1, Issue 4 p. 747-760 Original article The Steenrod problem of realizing polynomial cohomology rings Kasper K. S. Andersen, Corresponding Author Kasper K. S. Andersen [email protected] Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Bygning 1530, DK-8000 Aarhus C, DenmarkSearch for more papers by this authorJesper Grodal, Corresponding Author Jesper Grodal [email protected] Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, DenmarkSearch for more papers by this author Kasper K. S. Andersen, Corresponding Author Kasper K. S. Andersen [email protected] Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Bygning 1530, DK-8000 Aarhus C, DenmarkSearch for more papers by this authorJesper Grodal, Corresponding Author Jesper Grodal [email protected] Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, DenmarkSearch for more papers by this author First published: 23 December 2016 https://doi.org/10.1112/jtopol/jtn021Citations: 5 2000 Mathematics Subject Classification 55N10 (primary), 55R35, 55R40 (secondary) The second named author was partially supported by NSF grant DMS-0354633, the Alfred P. Sloan Foundation, and the Danish Natural Science Research Council. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract In this paper, we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R = ℤ, our result states that the only polynomial cohomology rings over ℤ that can occur are tensor products of copies of H * ( ℂ P ∞ ; ℤ ) ≅ ℤ [ x 2 ] , H * ( B SU ( n ) ; ℤ ) ≅ ℤ [ x 4 , x 6 , … , x 2 n ] , and H * ( B Sp ( n ) ; ℤ ) ≅ ℤ [ x 4 , x 8 , … , x 4 n ] , confirming an old conjecture. Our classification extends Notbohm's solution for R = 𝔽p, p odd. Odd degree generators, excluded above, only occur if R is an 𝔽2-algebra and in that case the recent classification of 2-compact groups by the authors can be used instead of the present paper. Our proofs are short and rely on the general theory of p-compact groups, but not on classification results for these. References 1J. F. Adams, C. W. Wilkerson, Finite H-spaces and algebras over the Steenrod algebra, Ann. of Math., 1980, 111(2), no. 1, 95–143; erratum, Ann. of Math. (2) 113 (1981) no. 3, 621–622 2J. Aguadé, A note on realizing polynomial algebras, Israel J. Math., 1981, 38(1–2), 95– 99 3J. 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