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Bulletin of the London Mathematical Society Advance Access originally published online on July 20, 2009
Bulletin of the London Mathematical Society 2009 41(5):782-794; doi:10.1112/blms/bdp028
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© 2009 London Mathematical Society

Finitary group cohomology and Eilenberg–MacLane spaces

Martin Hamilton

Hausdorff Center for Mathematics
Universität Bonn
Landwirtschaftskammer (Neubau)
Endenicher Allee 60
53115 Bonn
Germany

Received 22 January 2008. Revision received 6 November 2008.

We say that a group G has cohomology almost everywhere finitary if and only if the nth cohomology functors of G commute with filtered colimits for all sufficiently large n. In this paper, we show that if G is a group in Kropholler's class LHF with cohomology almost everywhere finitary, then G has an Eilenberg–MacLane space K(G, 1) that is dominated by a CW-complex with finitely many n-cells for all sufficiently large n. It is an open question as to whether this holds for arbitrary G. We also remark that the converse holds for any group G.

2000 Mathematics Subject Classification 20J06, 55P20, 18A22.


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