-central groups and Poincaré duality

Author:
Thomas S. Weigel

Journal:
Trans. Amer. Math. Soc. **352** (2000), 4143-4154

MSC (1991):
Primary 20J06

Published electronically:
May 3, 1999

MathSciNet review:
1621710

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we investigate the mod cohomology ring of finite -central groups with a certain extension property. For odd it turns out that the structure of the cohomology ring characterizes this class of groups up to extensions by -groups. For certain examples the cohomology ring can be calculated explicitly. As a by-product one gets an alternative proof of a theorem of M.Lazard which states that the Galois cohomology of a uniformly powerful pro--group of rank is isomorphic to .

**[B]**D. J. Benson,*Representations and cohomology. II*, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR**1156302****[BC]**D. J. Benson and Jon F. Carlson,*Projective resolutions and Poincaré duality complexes*, Trans. Amer. Math. Soc.**342**(1994), no. 2, 447–488. MR**1142778**, 10.1090/S0002-9947-1994-1142778-X**[Be]**F. Rudolf Beyl,*The spectral sequence of a group extension*, Bull. Sci. Math. (2)**105**(1981), no. 4, 417–434 (English, with French summary). MR**640151****[BrH]**Carlos Broto and Hans-Werner Henn,*Some remarks on central elementary abelian 𝑝-subgroups and cohomology of classifying spaces*, Quart. J. Math. Oxford Ser. (2)**44**(1993), no. 174, 155–163. MR**1222371**, 10.1093/qmath/44.2.155**[BrL]**Carlos Broto and Ran Levi,*On the homotopy type of 𝐵𝐺 for certain finite 2-groups 𝐺*, Trans. Amer. Math. Soc.**349**(1997), no. 4, 1487–1502. MR**1370636**, 10.1090/S0002-9947-97-01692-9**[BP]**W.Browder, J.Pakianathan,*Cohomology of -power exact groups*, preprint.**[CV]**L. S. Charlap and A. T. Vasquez,*The cohomology of group extensions*, Trans. Amer. Math. Soc.**124**(1966), 24–40. MR**0214665**, 10.1090/S0002-9947-1966-0214665-5**[DD]**J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal,*Analytic pro-𝑝-groups*, London Mathematical Society Lecture Note Series, vol. 157, Cambridge University Press, Cambridge, 1991. MR**1152800****[Du]**J. Duflot,*Depth and equivariant cohomology*, Comment. Math. Helv.**56**(1981), no. 4, 627–637. MR**656216**, 10.1007/BF02566231**[HP]**Hans-Werner Henn and Stewart Priddy,*𝑝-nilpotence, classifying space indecomposability, and other properties of almost all finite groups*, Comment. Math. Helv.**69**(1994), no. 3, 335–350. MR**1289332**, 10.1007/BF02564492**[HS]**Peter John Hilton and Urs Stammbach,*A course in homological algebra*, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 4. MR**0346025****[La]**Michel Lazard,*Groupes analytiques 𝑝-adiques*, Inst. Hautes Études Sci. Publ. Math.**26**(1965), 389–603 (French). MR**0209286****[MP]**J.Martino, S.Priddy,*On the cohomology and homotopy of Swan groups*, Math. Z.**225**(1997), 277-288. CMP**97:16****[MM]**John W. Milnor and John C. Moore,*On the structure of Hopf algebras*, Ann. of Math. (2)**81**(1965), 211–264. MR**0174052****[QU1]**Daniel Quillen,*The spectrum of an equivariant cohomology ring. I, II*, Ann. of Math. (2)**94**(1971), 549–572; ibid. (2) 94 (1971), 573–602. MR**0298694****[QU2]**Daniel Quillen,*A cohomological criterion for 𝑝-nilpotence*, J. Pure Appl. Algebra**1**(1971), no. 4, 361–372. MR**0318339****[W]**T.Weigel,*Combinatorial properties of -central groups*, preprint.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
20J06

Retrieve articles in all journals with MSC (1991): 20J06

Additional Information

**Thomas S. Weigel**

Affiliation:
Math. Institute, University of Oxford, 24-29 St. Giles, Oxford OX1 3LB, UK

Email:
weigel@maths.ox.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-99-02385-5

Received by editor(s):
February 12, 1997

Received by editor(s) in revised form:
March 28, 1998

Published electronically:
May 3, 1999

Additional Notes:
The author gratefully acknowledges financial support of the ‘Deutsche Forschungsgemeinschaft’ through a ‘Heisenberg Stipendium’.

Article copyright:
© Copyright 2000
American Mathematical Society