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The rational cohomology of a $ p$-local compact group


Authors: C. Broto, R. Levi and B. Oliver
Journal: Proc. Amer. Math. Soc. 142 (2014), 1035-1043
MSC (2010): Primary 55R35; Secondary 55R40, 20D20
DOI: https://doi.org/10.1090/S0002-9939-2013-11795-8
Published electronically: September 24, 2013
MathSciNet review: 3148537
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Abstract:

For any prime $ p$, the theory of $ p$-local compact groups is modelled on the $ p$-local homotopy theory of classifying spaces of compact Lie groups and $ p$-compact groups and generalises the earlier concept of $ p$-local finite groups. These objects have maximal tori and Weyl groups, although the Weyl groups need not be generated by pseudoreflections. In this paper, we study the rational $ p$-adic cohomology of the classifying space of a $ p$-local compact group and prove that just as for compact Lie groups, it is isomorphic to the ring of invariants of the Weyl group action on the cohomology of the classifying space of the maximal torus. This is applied to show that unstable Adams operations on $ p$-local compact groups are determined in the appropriate sense by the map they induce on rational cohomology.


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Additional Information

C. Broto
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
Email: broto@mat.uab.cat

R. Levi
Affiliation: Institute of Pure and Applied Mathematics, University of Aberdeen, Fraser Noble Building 138, Aberdeen AB24 3UE, United Kingdom
Email: r.levi@abdn.ac.uk

B. Oliver
Affiliation: Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539 du CNRS, 99, avenue J.-B. Clément, 93430 Villetaneuse, France
Email: bob@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0002-9939-2013-11795-8
Keywords: Classifying space, $p$-completion, $p$-local compact groups, fusion.
Received by editor(s): February 6, 2012
Received by editor(s) in revised form: April 3, 2012
Published electronically: September 24, 2013
Additional Notes: The first author was partially supported by FEDER-MICINN grant MTM 2010-20692.
The second author was partially supported by EPSRC grant EP/I019073/1
The third author was partially supported by UMR 7539 of the CNRS and by project ANR BLAN08-2_338236, HGRT
Communicated by: Brooke Shipley
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.