The Steenrod problem of realizing polynomial cohomology rings
(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:
http://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
- 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}, }