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:
https://lup.lub.lu.se/record/3412529
- author
- Andersen, Kasper
LU
and Grodal, Jesper
- publishing date
- 2008
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Journal of Topology
- volume
- 1
- issue
- 4
- pages
- 747 - 760
- publisher
- Oxford University Press
- external identifiers
-
- scopus:79957772953
- 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
- 2016-04-01 12:29:25
- date last changed
- 2023-09-25 08:25:44
@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}},
doi = {{10.1112/jtopol/jtn021}},
volume = {{1}},
year = {{2008}},
}