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Subgroup posets, Bredon cohomology and equivariant Euler characteristics

Author: Conchita Martínez-Pérez
Journal: Trans. Amer. Math. Soc. 365 (2013), 4351-4370
MSC (2010): Primary 20J05, 18G35, 18G30, 20J06
Published electronically: February 7, 2013
MathSciNet review: 3055698
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Abstract: For discrete groups $ \Gamma $ with a bound on the order of their finite subgroups, we construct Bredon projective resolutions of the trivial module in terms of projective covers of the chain complex associated to the poset of finite subgroups. We use this to give new results on dimensions of $ \underline {\operatorname {E}}\Gamma $ and to reprove that for virtually solvable groups, $ \underline {\mathrm {cd}}\Gamma =\mathrm {vcd}\Gamma $. We also deduce a formula to compute the equivariant Euler class of $ \underline {\operatorname {E}}\Gamma $ for $ \Gamma $ virtually solvable of type $ \mathrm {FP}_\infty $ and use it to compute orbifold Euler characteristics.

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Additional Information

Conchita Martínez-Pérez
Affiliation: Departamento de Matemáticas, IUMA. Universidad de Zaragoza, 50009 Zaragoza, Spain

Keywords: Bredon cohomology, virtually soluble group, proper classifying space, poset of finite subgroups
Received by editor(s): October 17, 2011
Received by editor(s) in revised form: December 19, 2011
Published electronically: February 7, 2013
Additional Notes: The author was partially supported by BFM2010-19938-C03-03, Gobierno de Aragón and the European Union’s ERDF funds
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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