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Nilpotence in group cohomology

Part of: Fiber spaces and bundles Connections with homological algebra and category theory

Published online by Cambridge University Press:  05 December 2012

Nicholas J. Kuhn
Nicholas J. Kuhn*
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA (njk4x@virginia.edu)
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Abstract

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We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds's recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p-groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p-group P.

Keywords

group cohomologyclassifying spacesinvariant theory

MSC classification

Secondary: 20J06: Cohomology of groups 55R40: Homology of classifying spaces, characteristic classes
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 151 - 175
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Aguadé, J. and Smith, L., On the Mod p Torus Theorem of John Hubbuck, Math. Z. 191 (1986), 325–326.CrossRefGoogle Scholar
2.Benson, D., Modules with injective cohomology, and local duality for a finite group, New York J. Math. 7 (2001), 201–215.Google Scholar
3.Benson, D., Dickson invariants, regularity and computation in group cohomology, Illinois J. Math. 48 (2004), 171–197.CrossRefGoogle Scholar
4.Benson, D. and Greenlees, J. P.C., Commutative algebra for cohomology rings of classifying spaces of compact Lie groups, J. Pure Appl. Alg. 122 (1997), 41–53.CrossRefGoogle Scholar
5.Brodmann, M. P. and Sharp, R. Y., Local cohomology, Cambridge Studies in Advanced Mathematics, Volume 60 (Cambridge University Press, 1998).CrossRefGoogle Scholar
6.Broto, C. and Henn, H.-W., Some remarks on central elementary abelian p-subgroups and cohomology of classifying spaces, Q. J. Math. 44 (1993), 155–163.CrossRefGoogle Scholar
7.Campbell, H. E. A. and Hughes, I. P., Vector invariants of : a proof of a conjecture of Richman, Adv. Math. 126 (1997), 1–20.CrossRefGoogle Scholar
8.Carlson, J. F., Depth and transfer maps in the cohomology of groups, Math. Z. 218 (1995), 461–468.CrossRefGoogle Scholar
9.Carlson, J., Mod 2 cohomology of 2 groups: Magma computer computations (available at www.math.uga.edu/~lvalero/cohointro.html).Google Scholar
10.Carlson, J. F., Townsley, L., Valeri-Elizondo, L. and Zhang, M., Cohomology rings of finite groups Algebras and Applications, Volume 3 (Kluwer, Dordrecht, 2003).CrossRefGoogle Scholar
11.Duflot, J., Depth and equivariant cohomology, Comment. Math. Helv. 56 (1981), 627–637.CrossRefGoogle Scholar
12.Green, D. J., On Carlson's depth conjecture in group cohomology, Math. Z. 244 (2003), 711–723.CrossRefGoogle Scholar
13.Green, D. J. and King, S. A., The cohomology of finite p groups: computer calculations (available at http://users.minet.uni-jena.de/cohomology).Google Scholar
14.Henn, H.-W., Finiteness properties of injective resolutions of certain unstable modules over the Steenrod algebra and applications, Math. Annalen 291 (1991), 191–203.CrossRefGoogle Scholar
15.Henn, H.-W., Lannes, J. and Schwartz, L., Localizations of unstable A-modules and equivariant mod p cohomology, Math. Annalen 301 (1995), 23–68.CrossRefGoogle Scholar
16.Kuhn, N. J., On topologically realizing modules over the Steenrod algebra, Annals Math. 141 (1995), 321–347.CrossRefGoogle Scholar
17.Kuhn, N. J., Primitives and central detection numbers in group cohomology, Adv. Math. 216 (2007), 387–442.CrossRefGoogle Scholar
18.Kuhn, N. J., The nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups, Math. Proc. Camb. Phil. Soc. 144 (2008), 575–602.CrossRefGoogle Scholar
19.Lannes, J., Cohomology of groups and function spaces, Preprint (based on a talk given at the University of Chicago, 1986).Google Scholar
20.Lannes, J., Sur les espaces fonctionnels dont la source est le classifiant d'un p-groupe abélien élémentaire, Publ. Math. IHES 75 (1992), 135–244.CrossRefGoogle Scholar
21.Quillen, D., The spectrum of an equivariant cohomology ring, I, Annals Math. 94 (1971), 549–572.CrossRefGoogle Scholar
22.Schwartz, L., La filtration nilpotente de la catégorie et la cohomologie des espaces de lacets, in Algebraic topology: rational homotopy, Springer Lecture Notes in Mathematics, Volume 1318, pp. 208–218 (Springer, 1988).CrossRefGoogle Scholar
23.Schwartz, L., Modules over the Steenrod algebra and Sullivan's fixed point conjecture, Chicago Lectures in Mathematics (University of Chicago Press, 1994).Google Scholar
24.Symonds, P., On the Casteluovo-Mumford regularity of the cohomology ring of a group, J. Am. Math. Soc. 23 (2010), 1159–1173.CrossRefGoogle Scholar
25.Wehlau, D. L., Invariants for the modular cyclic group of prime order via classical invariant theory, Preprint (arXiv:0912.1107, 2009).Google Scholar