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Journal of Topology 2008 1(4):747-760; doi:10.1112/jtopol/jtn021
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© 2008 London Mathematical Society

The Steenrod problem of realizing polynomial cohomology rings

Kasper K. S. Andersen

Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Bygning 1530, DK-8000 Aarhus C, Denmark kksa@imf.au.dk

Jesper Grodal

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark jg@math.ku.dk

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 Formula , our result states that the only polynomial cohomology rings over Formula that can occur are tensor products of copies of Formula , Formula , and Formula , confirming an old conjecture. Our classification extends Notbohm's solution for Formula , p odd. Odd degree generators, excluded above, only occur if R is an Formula -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.

Received March 8, 2008.

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.



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This Article
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