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Transactions of the American Mathematical Society

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On The Homotopy Type of $BG$ for
Certain Finite 2-Groups $G$


Authors: Carlos Broto and Ran Levi
Journal: Trans. Amer. Math. Soc. 349 (1997), 1487-1502
MSC (1991): Primary 55R35; Secondary 55R40, 55Q52
DOI: https://doi.org/10.1090/S0002-9947-97-01692-9
MathSciNet review: 1370636
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Abstract: We consider the homotopy type of classifying spaces $BG$, where $G$ is a finite $p$-group, and we study the question whether or not the mod $p$ cohomology of $BG$, as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence, determine the homotopy type of $BG$. This article is devoted to producing some families of finite 2-groups where cohomological information determines the homotopy type of $BG$.


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  • 1. J. Aguadé, Cohomology algebras with two generators, Math. Z. 177 (1981), 289-296. MR 82d:55003
  • 2. J. Aguadé, C. Broto and D. Notbohm, Homotopy classification of spaces with interesting cohomology and a conjecture of Cooke, part I, Topology 33 (1994), 455-492. MR 95i:55006
  • 3. J. Aguadé, C. Broto and M. Santos, Fake three connected coverings of Lie groups, Duke Math. J. 80 (1995), 91-103. MR 96j:57050
  • 4. D. Benson and J. Carlson, Cohomology of extraspecial groups, Bull. London Math. Soc. 24 (1992), 209-235. MR 93b:20087
  • 5. A.K. Bousfield, Homological localization towers for groups and $\Pi $-modules, Mem. Amer. Math. Soc. 186 (1977). MR 56:5688
  • 6. A.K. Bousfield and D. Kan, Localizations, completions and homotopy limits, Lecture Notes in Math. 304, Springer Verlag (1972). MR 51:1825
  • 7. W. Browder, Torsion in $H$-spaces Ann. Math. 74 (1961), 24-51. MR 23:A2201
  • 8. W. Dwyer, Strong convergence of the Eilenberg-Moore spectral sequence, Topology 13 (1973), 255-265. MR 52:15464
  • 9. L. Evens and S. B. Priddy, The cohomology of the semidihedral groups, Contemp. Math. 37 (1985), 61-72. MR 82h:20075
  • 10. Z. Fiedorowicz and S. B. Priddy, Homology of classical groups over finite fields and their associated infinite loop spaces, Lecture Notes in Math. 674, Springer Verlag (1978). MR 80g:55018
  • 11. R. Kane, The Homology of Hopf Spaces, North-Holland, (1988). MR 90f:55018
  • 12. I. Leary; 3-groups are not determined by their integral cohomology rings, Preprint.
  • 13. G. Mislin, Cohomologically central elements and fusion in groups, Algebraic Topology, Homotopy and Group Cohomology, Proceedings, Barcelona 1990, Lecture Notes in Math. 1509, Springer Verlag (1992). MR 94d:55032
  • 14. D. Quillen, The mod-2 cohomology rings of extra special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197-212. MR 44:7582
  • 15. D. J. Rusin, The mod-2 cohomology of metacyclic 2-groups, J. Pure Appl. Algebra 44 (1987), 315-327. MR 88k:20076
  • 16. D. J. Rusin, The cohomology of groups of order 32, Math. Comp. 53 (1989), 359-385. MR 89k:20078

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Additional Information

Carlos Broto
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
Email: broto@mat.uab.es

Ran Levi
Affiliation: Mathematisches Institut, Universität Heidelberg, INF 288, Heidelberg 69120, Germany
Address at time of publication: Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, Illinois 60201
Email: ran@math.nwu.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01692-9
Keywords: Classifying spaces, finite 2-groups, cohomology, Steenrod squares, Bockstein spectral sequence
Received by editor(s): May 19, 1995
Additional Notes: C. Broto is partially supported by DGICYT grant PB94-0725.
R. Levi is supported by a DFG grant.
Article copyright: © Copyright 1997 American Mathematical Society

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