Rational equivariant cohomology theories with toral support

Greenlees, J P C¹

¹School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Algebraic and Geometric Topology, Tome 16 (2016) no. 4, pp. 1953-2019

Résumé

For an arbitrary compact Lie group G, we describe a model for rational G–spectra with toral geometric isotropy and show that there is a convergent Adams spectral sequence based on it. The contribution from geometric isotropy at a subgroup K of the maximal torus of G is captured by a module over H∗(BWGe(K)) with an action of π0(WG(K)), where WGe(K) is the identity component of WG(K) = NG(K)∕K.

Détail

DOI : 10.2140/agt.2016.16.1953

Classification : 55N91, 55P42, 55P91, 55P92, 55T15
Keywords: rational equivariant spectra, algebraic models, Adams spectral sequence, reduction to torus normalizer

Bibliographie

[1] D Barnes, Classifying dihedral O(2)–equivariant spectra, preprint (2008)

[2] D Barnes, Rational equivariant spectra, PhD thesis (2008)

[3] D Barnes, Rational O(2)–equivariant spectra, preprint (2012)

[4] T Bröcker, T Tom Dieck, Representations of compact Lie groups, 98, Springer (1985)

[5] T Tom Dieck, The Burnside ring of a compact Lie group, I, Math. Ann. 215 (1975) 235

[6] J P C Greenlees, Rational Mackey functors for compact Lie groups, I, Proc. London Math. Soc. 76 (1998) 549

[7] J P C Greenlees, Rational O(2)–equivariant cohomology theories, from: "Stable and unstable homotopy" (editors W G Dwyer, S Halperin, R Kane, S O Kochman, M E Mahowald, P S Selick), Fields Inst. Commun. 19, Amer. Math. Soc. (1998) 103

[8] J P C Greenlees, Rational S1–equivariant stable homotopy theory, 661, Amer. Math. Soc. (1999)

[9] J P C Greenlees, Rational SO(3)–equivariant cohomology theories, from: "Homotopy methods in algebraic topology" (editors J P C Greenlees, R R Bruner, N Kuhn), Contemp. Math. 271, Amer. Math. Soc. (2001) 99

[10] J P C Greenlees, Triangulated categories of rational equivariant cohomology theories, Oberwolfach Rep. 3 (2006) 480

[11] J P C Greenlees, Rational torus-equivariant stable homotopy, I : Calculating groups of stable maps, J. Pure Appl. Algebra 212 (2008) 72

[12] J P C Greenlees, Rational torus-equivariant stable homotopy, II : Algebra of the standard model, J. Pure Appl. Algebra 216 (2012) 2141

[13] J P C Greenlees, Algebraic models of induction and coinduction, preprint (2014)

[14] J P C Greenlees, Rational torus-equivariant stable homotopy, III : Comparison of models, J. Pure Appl. Algebra 220 (2016) 3573

[15] J P C Greenlees, B Shipley, An algebraic model for rational torus-equivariant spectra, preprint (2011)

[16] J P C Greenlees, B Shipley, An algebraic model for free rational G–spectra, Bull. Lond. Math. Soc. 46 (2014) 133

[17] M Kedziorek, Algebraic models for rational G–spectra, PhD thesis (2014)

[18] L Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963) 57