Exotic cohomology for 𝐺𝐿_{𝑛}(ℤ[1/2])

Dwyer, W.

doi:10.1090/S0002-9939-98-04279-8

Exotic cohomology for $\operatorname {GL}_{n}(\mathbb {Z}[1/2])$
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by W. G. Dwyer

Proc. Amer. Math. Soc. 126 (1998), 2159-2167

DOI: https://doi.org/10.1090/S0002-9939-98-04279-8

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Abstract:

We show that for $n=32$ the mod $2$ group cohomology of $\operatorname {GL}_{n}(\mathbb {Z}[1/2])$ is not detected on diagonal matrices.

References

A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4
Gunnar Carlsson, Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. (2) 120 (1984), no. 2, 189–224. MR 763905, DOI 10.2307/2006940
Gunnar Carlsson, Equivariant stable homotopy theory and the finite descent problem for unstable algebraic $K$-theories, Amer. J. Math. 113 (1991), no. 6, 963–973. MR 1137531, DOI 10.2307/2374897
William G. Dwyer and Eric M. Friedlander, Algebraic and etale $K$-theory, Trans. Amer. Math. Soc. 292 (1985), no. 1, 247–280. MR 805962, DOI 10.1090/S0002-9947-1985-0805962-2
W. G. Dwyer and E. M. Friedlander, Conjectural calculations of general linear group homology, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 135–147. MR 862634, DOI 10.1090/conm/055.1/862634
William G. Dwyer and Eric M. Friedlander, Topological models for arithmetic, Topology 33 (1994), no. 1, 1–24. MR 1259512, DOI 10.1016/0040-9383(94)90032-9
W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 106–119. MR 928826, DOI 10.1007/BFb0083003
Eric M. Friedlander, Étale homotopy of simplicial schemes, Annals of Mathematics Studies, No. 104, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 676809
Hans-Werner Henn, Jean Lannes, and Lionel Schwartz, Localizations of unstable $A$-modules and equivariant mod $p$ cohomology, Math. Ann. 301 (1995), no. 1, 23–68. MR 1312569, DOI 10.1007/BF01446619
H.-W. Henn, The cohomology of $SL(3,\mathbb {Z} [1/2])$, preprint (Universität Heidelberg) 1996.
H.-W. Henn, Commutative algebra of unstable $K$-modules, Lannes’ $T$-functor and equivariant $\mathrm {Mod}$-$p$ cohomology, J. Reine Angew. Math. 478 (1996), 189–215.
Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723, DOI 10.1007/978-1-4612-0853-2
C.-N. Lee, On the unstable homotopy type of the classifying space of a virtually torsion-free group, Math. Z. 224 (1997), 235–253.
Stephen A. Mitchell, On the plus construction for $B\textrm {GL}\,\textbf {Z}[\frac 12]$ at the prime $2$, Math. Z. 209 (1992), no. 2, 205–222. MR 1147814, DOI 10.1007/BF02570830
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, On the cohomology and $K$-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972), 552–586. MR 315016, DOI 10.2307/1970825
Jean-Pierre Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169 (French). MR 0385006
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2