$p$-central groups and Poincaré duality
HTML articles powered by AMS MathViewer
- by Thomas S. Weigel PDF
- Trans. Amer. Math. Soc. 352 (2000), 4143-4154 Request permission
Abstract:
In this note we investigate the mod $p$ cohomology ring of finite $p$-central groups with a certain extension property. For $p$ odd it turns out that the structure of the cohomology ring characterizes this class of groups up to extensions by $p’$-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-$p$-group of rank $n$ is isomorphic to $\Lambda [x_{1},..,x_{n}]$.References
- 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
- 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, DOI 10.1090/S0002-9947-1994-1142778-X
- 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
- 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, DOI 10.1093/qmath/44.2.155
- 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, DOI 10.1090/S0002-9947-97-01692-9
- W.Browder, J.Pakianathan, Cohomology of $p$-power exact groups, preprint.
- L. S. Charlap and A. T. Vasquez, The cohomology of group extensions, Trans. Amer. Math. Soc. 124 (1966), 24–40. MR 214665, DOI 10.1090/S0002-9947-1966-0214665-5
- 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
- J. Duflot, Depth and equivariant cohomology, Comment. Math. Helv. 56 (1981), no. 4, 627–637. MR 656216, DOI 10.1007/BF02566231
- 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, DOI 10.1007/BF02564492
- Peter John Hilton and Urs Stammbach, A course in homological algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York-Berlin, 1971. MR 0346025
- Michel Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603 (French). MR 209286
- J.Martino, S.Priddy, On the cohomology and homotopy of Swan groups, Math. Z. 225 (1997), 277–288.
- John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
- 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 298694, DOI 10.2307/1970770
- Daniel Quillen, A cohomological criterion for $p$-nilpotence, J. Pure Appl. Algebra 1 (1971), no. 4, 361–372. MR 318339, DOI 10.1016/0022-4049(71)90003-X
- T.Weigel, Combinatorial properties of $p$-central groups, preprint.
Additional Information
- Thomas S. Weigel
- Affiliation: Math. Institute, University of Oxford, 24-29 St. Giles, Oxford OX1 3LB, UK
- MR Author ID: 319262
- Email: weigel@maths.ox.ac.uk
- 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’.
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 4143-4154
- MSC (1991): Primary 20J06
- DOI: https://doi.org/10.1090/S0002-9947-99-02385-5
- MathSciNet review: 1621710