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Cohomologie de GL_2(Z[i,1/2]) à coefficients dans F_2

Abstract : The aim of this Phd thesis was to compute H*(BGL_2(Z[i,1/2]),F_2). This cohomology ring appears in a certain version of the conjecture of Lichtenbaum and Quillen, asserting that the cohomology modulo 2 of the classifying space of a general linear group over Z[1/2] should be detected by the cohomology of its subgroup of diagonal matrices.

The original idea was to show that this conjecture fails in the special case of the general linear group of rank 4 over Z[1/2], and the cohomology of BGL_2(Z[i,1/2]) should have been the main argument. By computing H*(BGL_2(Z[i,1/2]),F_2), we proved that the conjecture is true in the case of GL_2(Z[i,1/2]).

The calculation of H*(BGL_2(Z[i,1/2]),F_2) depends on the analysis of a certain space Z on which PSL_2(Z[i]) acts in a good way, and the as well as on calculation of H*(BPSL_2(Z[i]),F_2) and H*(BGo,F_2) where Go is a suitable subgroup of PSL_2(Z[i]) such that PSL_2(Z[i,1/2]) is isomorphic to the amalgamated sum PSL_2(Z[i])*_Go PSL_2(Z[i]). One obtains H*(BGL_2(Z[i,1/2]),F_2) by studying some morphisms from H*(BPSL_2(Z[i]),F_2) to H*(BGo,F_2) and some spectral sequences.
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https://tel.archives-ouvertes.fr/tel-00174888
Contributor : Nicolas Weiss <>
Submitted on : Saturday, October 6, 2007 - 9:44:27 AM
Last modification on : Friday, June 19, 2020 - 9:10:04 AM
Long-term archiving on: : Friday, April 9, 2010 - 2:51:14 AM

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Nicolas Weiss. Cohomologie de GL_2(Z[i,1/2]) à coefficients dans F_2. Mathématiques [math]. Université Louis Pasteur - Strasbourg I, 2007. Français. ⟨NNT : 2007STR13084⟩. ⟨tel-00174888⟩

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