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The Steenrod problem of realizing polynomial cohomology rings

Andersen, Kasper LU and Grodal, Jesper (2008) In Journal of Topology 1(4). p.747-760
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 $H^*(CP^\infty;Z)\cong Z[x_2]$,

$H^*(BSU(n);Z)\cong Z[x_4, x_6, \ldots, x_{2n}]$, and

$H^*(BSp(n);Z)\cong Z[x_4, x_8, \ldots, x_{4n}]$, confirming an old conjecture. Our classification extends Notbohm's solution for $R=F_p$, p odd. Odd degree generators, excluded above, only occur if R is an $F_2$-algebra... (More)
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 $H^*(CP^\infty;Z)\cong Z[x_2]$,

$H^*(BSU(n);Z)\cong Z[x_4, x_6, \ldots, x_{2n}]$, and

$H^*(BSp(n);Z)\cong Z[x_4, x_8, \ldots, x_{4n}]$, confirming an old conjecture. Our classification extends Notbohm's solution for $R=F_p$, p odd. Odd degree generators, excluded above, only occur if R is an $F_2$-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. (Less)
Please use this url to cite or link to this publication:
author
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Topology
volume
1
issue
4
pages
747 - 760
publisher
Oxford University Press
ISSN
1753-8424
DOI
10.1112/jtopol/jtn021
language
English
LU publication?
no
id
950c2789-c24a-42c3-97ed-4e6b2fa882fe (old id 3412529)
alternative location
http://jtopol.oxfordjournals.org/content/1/4/747
date added to LUP
2013-04-09 17:29:48
date last changed
2016-06-29 09:11:34
@article{950c2789-c24a-42c3-97ed-4e6b2fa882fe,
  abstract     = {In this paper, we completely classify which graded polynomial<br/><br>
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 $H^*(CP^\infty;Z)\cong Z[x_2]$,<br/><br>
$H^*(BSU(n);Z)\cong Z[x_4, x_6, \ldots, x_{2n}]$, and<br/><br>
$H^*(BSp(n);Z)\cong Z[x_4, x_8, \ldots, x_{4n}]$, confirming an old conjecture. Our classification extends Notbohm's solution for $R=F_p$, p odd. Odd degree generators, excluded above, only occur if R is an $F_2$-algebra and in that case the recent classification of<br/><br>
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<br/><br>
p-compact groups, but not on classification results for these.},
  author       = {Andersen, Kasper and Grodal, Jesper},
  issn         = {1753-8424},
  language     = {eng},
  number       = {4},
  pages        = {747--760},
  publisher    = {Oxford University Press},
  series       = {Journal of Topology},
  title        = {The Steenrod problem of realizing polynomial cohomology rings},
  url          = {http://dx.doi.org/10.1112/jtopol/jtn021},
  volume       = {1},
  year         = {2008},
}