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The mod \(2\) cohomology of the linear groups over the ring of integers. (English) Zbl 0942.20029

The paper under review represents another of the by-products of Voevodsky’s proof of the Milnor conjecture, via the work of Rognes and Weibel on the 2 torsion in the K-theory of the integers [C. Weibel, C. R. Acad. Sci., Paris, Sér I 324, No. 6, 615-620 (1997; Zbl 0889.11039)].
The authors note that, combining work of M. Bökstedt [“Algebraic topology”, Proc. Conf. Aarhus 1982, Lect. Notes Math. 1051, 25-37 (1984; Zbl 0589.57032)] and S. A. Mitchell [Math. Z. 209, No. 2, 205-222 (1992; Zbl 0773.55006)] with the above K-theoretic result, one can derive a description of the mod 2 cohomology of \(\text{BGL}(\mathbb{Z})^+\) as a Hopf algebra and as a module over the Steenrod algebra.
However, this method does not yield explicit generators and so the authors provide an alternative approach which does. They go on to deduce the cohomology of the infinite special linear group \(\text{SL}(\mathbb{Z})\) and that of the infinite Steinberg group \(\text{St}(\mathbb{Z})\).

MSC:

20G10 Cohomology theory for linear algebraic groups
19D55 \(K\)-theory and homology; cyclic homology and cohomology
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
55S10 Steenrod algebra
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