The real cohomology of virtually nilpotent groups

Dekimpe, Karel; Pouseele, Hannes

doi:10.1090/S0002-9947-07-04274-2

The real cohomology of virtually nilpotent groups
HTML articles powered by AMS MathViewer

by Karel Dekimpe and Hannes Pouseele PDF

Trans. Amer. Math. Soc. 359 (2007), 2539-2558 Request permission

Abstract:

In this paper we present a method to compute the real cohomology of any finitely generated virtually nilpotent group. The main ingredient in our setup consists of a polynomial crystallographic action of this group. As any finitely generated virtually nilpotent group admits such an action (which can be constructed quite easily), the approach we present applies to all these groups. Our main result is an algorithmic way of computing these cohomology spaces. As a first application, we prove a kind of Poincaré duality (also in the nontorsion free case) and we derive explicit formulas in the virtually abelian case.

References

L. Auslander and L. Markus, Holonomy of flat affinely connected manifolds, Ann. of Math. (2) 62 (1955), 139–151. MR 72518, DOI 10.2307/2007104
Babakhanian, A. Cohomological methods in group theory, volume 11 of Pure and Applied Mathematics. Marcel Dekker Inc., New York, 1972.
Yves Benoist, Une nilvariété non affine, J. Differential Geom. 41 (1995), no. 1, 21–52 (French, with English summary). MR 1316552
Yves Benoist and Karel Dekimpe, The uniqueness of polynomial crystallographic actions, Math. Ann. 322 (2002), no. 3, 563–571. MR 1895707, DOI 10.1007/s002080200005
Robert Bieri, Gruppen mit Poincaré-Dualität, Comment. Math. Helv. 47 (1972), 373–396 (German). MR 352290, DOI 10.1007/BF02566811
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Dietrich Burde, Affine structures on nilmanifolds, Internat. J. Math. 7 (1996), no. 5, 599–616. MR 1411303, DOI 10.1142/S0129167X96000323
D. Burde and F. Grunewald, Modules for certain Lie algebras of maximal class, J. Pure Appl. Algebra 99 (1995), no. 3, 239–254. MR 1332900, DOI 10.1016/0022-4049(94)00002-Z
Karel Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, Berlin, 1996. MR 1482520, DOI 10.1007/BFb0094472
Karel Dekimpe and Paul Igodt, Polycyclic-by-finite groups admit a bounded-degree polynomial structure, Invent. Math. 129 (1997), no. 1, 121–140. MR 1464868, DOI 10.1007/s002220050160
Karel Dekimpe and Paul Igodt, Polynomial structures on polycyclic groups, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3597–3610. MR 1422895, DOI 10.1090/S0002-9947-97-01924-7
Karel Dekimpe, Paul Igodt, and Kyung Bai Lee, Polynomial structures for nilpotent groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 77–97. MR 1327254, DOI 10.1090/S0002-9947-96-01513-9
David Fried, William Goldman, and Morris W. Hirsch, Affine manifolds with nilpotent holonomy, Comment. Math. Helv. 56 (1981), no. 4, 487–523. MR 656210, DOI 10.1007/BF02566225
G. Hochschild and J.-P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110–134. MR 52438, DOI 10.1090/S0002-9947-1953-0052438-8
Ib Madsen and Jørgen Tornehave, From calculus to cohomology, Cambridge University Press, Cambridge, 1997. de Rham cohomology and characteristic classes. MR 1454127
William S. Massey, Singular homology theory, Graduate Texts in Mathematics, vol. 70, Springer-Verlag, New York-Berlin, 1980. MR 569059, DOI 10.1007/978-1-4684-9231-6
Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531–538. MR 64057, DOI 10.2307/1969716
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
Daniel Segal, Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82, Cambridge University Press, Cambridge, 1983. MR 713786, DOI 10.1017/CBO9780511565953
Martin C. Tangora (ed.), Computers in geometry and topology, Lecture Notes in Pure and Applied Mathematics, vol. 114, Marcel Dekker, Inc., New York, 1989. MR 988688