The Euler class of a Poincare duality group
The Euler class of a Poincare duality group
This is a revised version of a preprint which previously appeared in 1998. The Euler class of a group G of type FP over a ring R is the element of K_0(RG) given by the alternating sum of the modules in a finite projective resolution for R over RG. (We reserve the term "Wall obstruction" for the image of the Euler class in the reduced K-group.) Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincare duality group over any ring has order at most two. We construct groups that are of type FL over the complex numbers but are not FL over the rationals. We construct group algebras over fields for which K_0 contains torsion, and construct non-free stably-free modules for the group algebras of certain virtually-free groups. The first preprint version of this paper appeared on the Southampton web pages in Spring 1998. There is now (November 2000) a corrected version, which also contains more material than the original.
group cohomology, Poincaré, duality, finiteness conditions
421-448
Leary, Ian J.
57bd5c53-cd99-41f9-b02a-4a512d45150e
2002
Leary, Ian J.
57bd5c53-cd99-41f9-b02a-4a512d45150e
Leary, Ian J.
(2002)
The Euler class of a Poincare duality group.
Proceedings of the Edinburgh Mathematical Society, 45, .
(doi:10.1017/S0013091500001164).
Abstract
This is a revised version of a preprint which previously appeared in 1998. The Euler class of a group G of type FP over a ring R is the element of K_0(RG) given by the alternating sum of the modules in a finite projective resolution for R over RG. (We reserve the term "Wall obstruction" for the image of the Euler class in the reduced K-group.) Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincare duality group over any ring has order at most two. We construct groups that are of type FL over the complex numbers but are not FL over the rationals. We construct group algebras over fields for which K_0 contains torsion, and construct non-free stably-free modules for the group algebras of certain virtually-free groups. The first preprint version of this paper appeared on the Southampton web pages in Spring 1998. There is now (November 2000) a corrected version, which also contains more material than the original.
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Published date: 2002
Keywords:
group cohomology, Poincaré, duality, finiteness conditions
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Local EPrints ID: 29837
URI: http://eprints.soton.ac.uk/id/eprint/29837
PURE UUID: bbd312f7-dc7c-4be0-bad5-e58ca091f431
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Date deposited: 11 May 2006
Last modified: 16 Mar 2024 04:04
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