Real connective K-theory and the quaternion group
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- by Dilip Bayen and Robert R. Bruner PDF
- Trans. Amer. Math. Soc. 348 (1996), 2201-2216 Request permission
Abstract:
Let $ko$ be the real connective K-theory spectrum. We compute $ko_*BG$ and $ko^*BG$ for groups $G$ whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients $t(ko)^G_*$ of the $G$ fixed points of the Tate spectrum $t(ko)$ for $G = Sl_2(3)$ and $G = Q_8$. The results provide a counterexample to the optimistic conjecture of Greenlees and May [ J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Memoirs AMS 543 (1995)], Conj. 13.4, by showing, in particular, that $t(ko)^G$ is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.References
- J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0402720
- D. W. Anderson, E. H. Brown Jr., and F. P. Peterson, Pin cobordism and related topics, Comment. Math. Helv. 44 (1969), 462β468. MR 261613, DOI 10.1007/BF02564545
- Dilip Bayen, The connective real K-homology of finite groups, Thesis, Wayne State University, 1994.
- Donald M. Davis and Mark Mahowald, The spectrum $(P\wedge b\textrm {o})_{-\infty }$, Math. Proc. Cambridge Philos. Soc. 96 (1984), no.Β 1, 85β93. MR 743704, DOI 10.1017/S030500410006196X
- Boris Botvinnik and Peter Gilkey, The eta invariant and metrics of positive scalar curvature, Math. Annalen (to appear).
- Boris Botvinnik and Peter Gilkey, Metrics of positive scalar curvature on spherical space forms, Can. J. Math (to appear).
- Boris Botvinnik, Peter Gilkey and Stephan Stolz, The Gromov-Lawson-Rosenberg conjecture for manifolds with a spherical space form fundamental group, Preprint of IHES, November, 1995.
- J. P. C. Greenlees and Hal Sadofsky, The Tate spectrum of $v_n$-periodic complex oriented theories, Math. Zeit. (to appear).
- J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Memoirs AMS 543 (1995).
- John Martino and Stewart Priddy, Classification of $BG$ for groups with dihedral or quarternion Sylow $2$-subgroups, J. Pure Appl. Algebra 73 (1991), no.Β 1, 13β21. MR 1121628, DOI 10.1016/0022-4049(91)90103-9
- J. P. May and R. J. Milgram, The Bockstein and the Adams spectral sequences, Proc. Amer. Math. Soc. 83 (1981), no.Β 1, 128β130. MR 619997, DOI 10.1090/S0002-9939-1981-0619997-8
- M. Mahowald and R. James Milgram, Operations which detect Sq4 in connective $K$-theory and their applications, Quart. J. Math. Oxford Ser. (2) 27 (1976), no.Β 108, 415β432. MR 433453, DOI 10.1093/qmath/27.4.415
- Zbigniew Fiedorowicz and Stewart Priddy, Homology of classical groups over finite fields and their associated infinite loop spaces, Lecture Notes in Mathematics, vol. 674, Springer, Berlin, 1978. MR 513424, DOI 10.1007/BFb0062824
- Stephan Stolz, Splitting certain $M\, \textrm {Spin}$-module spectra, Topology 33 (1994), no.Β 1, 159β180. MR 1259520, DOI 10.1016/0040-9383(94)90040-X
- S. Stolz and J. Rosenberg, A stable version of the GromovβLawson conjecture, The Δech Centennial, Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995.
- Robert E. Stong, Determination of $H^{\ast } (\textrm {BO}(k,\cdots ,\infty ),Z_{2})$ and $H^{\ast } (\textrm {BU}(k,\cdots ,\infty ),Z_{2})$, Trans. Amer. Math. Soc. 107 (1963), 526β544. MR 151963, DOI 10.1090/S0002-9947-1963-0151963-5
Additional Information
- Dilip Bayen
- Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
- Email: dbayen@math.wayne.edu
- Robert R. Bruner
- Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
- Email: rrb@math.wayne.edu
- Received by editor(s): August 10, 1994
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 2201-2216
- MSC (1991): Primary 19L41, 19L47, 19L64, 55N15, 55R35, 55Q91, 55M05
- DOI: https://doi.org/10.1090/S0002-9947-96-01516-4
- MathSciNet review: 1329527