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Farrell cohomology and centralizets of elementary abelian p-subgroups

Published online by Cambridge University Press:  24 October 2008

Chun-Nip Lee
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A.

Extract

Let Γ be a discrete group. Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torsion-free subgroup Γ′ of G such that Γ′ has finite cohomological dimension over ℤ. Examples of such groups include finite groups, fundamental group of a finite graph of finite groups, arithmetic groups, mapping class groups and outer automorphism groups of free groups. One of the fundamental problems in topology is to understand the cohomology of these finite vcd-groups.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Adem, A.. On the exponent of cohomology of discrete groups. Bull. London Math. Soc. 21 (1989), 585590.CrossRefGoogle Scholar
[2]Bousfield, A. K. and Kan, D. M.. Homotopy limits, completions and localizations. Lecture Notes in Math. 304 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[3]Brown, K.. Cohomology of Groups. Graduate Texts in Math., vol. 87 (Springer, 1982).CrossRefGoogle Scholar
[4]Cartan, H. and Eilenberg, S.. Homological Algebra (Princeton University Press, 1965).Google Scholar
[5]Dwyer, W. and Wilkerson, C.. A cohomology decomposition theorem. Topology 31 (1992), 433443.CrossRefGoogle Scholar
[6]Greenlees, J. P. C.. Representing Tate cohomology of G-spaces. Proc. Edinburgh Math. Soc. 30 (1987), 435443.CrossRefGoogle Scholar
[7]Henn, H-W.. On the mod p cohomology of profinite groups of positive p rank (preprint).Google Scholar
[8]Henn, H-W.. Centralizers of elementary abelian p subgroups, the Borei construction of the singular locus and applications to the cohomology of discrete groups (preprint).Google Scholar
[9]Jackowski, S. and McClure, J.. Homotopy decomposition of classifying spaces via elementary abelian subgroups. Topology 31 (1992), 113132.CrossRefGoogle Scholar
[10]Lee, C-N.. On the unstable homotopy type of the classifying space of virtually torsion-free groups, to appear in Math. Zeit.Google Scholar
[11]Lee, C-N.. Homotopy decomposition of the classifying space of virtually torsion-free groups and applications, preprint.Google Scholar
[12]Lewis, G., May, J. P. and Steinberger, M.. Equivariant stable homotopy theory. Lecture Notes in Math. 1213 (Springer-Verlag, 1986).CrossRefGoogle Scholar
[13]Manjhekar, R.. The mod p cohomology of GL(2p – 2, ℤ). (Ph.D. thesis, Ohio State University, 1994).Google Scholar
[14]Oliver, R.. Higher limits of functors on categories of elementary abelian p-groups, preprint.Google Scholar
[15]Slominska, J.. Homotopy colimits on E-I-categories, Lecture Notes in Math. 1474 (Springer-Verlag, 1991).Google Scholar
[16]Webb, P.. A local method in group cohomology. Comment. Math. Helvetici 62 (1987), 135167.CrossRefGoogle Scholar