The mod 2 cohomology of the linear groups

over the ring of integers

Authors:
Dominique Arlettaz, Mamoru Mimura, Koji Nakahata and Nobuaki Yagita

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2199-2212

MSC (1991):
Primary 20G10; Secondary 19D55, 20J05, 55R40, 55S10

DOI:
https://doi.org/10.1090/S0002-9939-99-05183-7

Published electronically:
April 8, 1999

MathSciNet review:
1646320

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper completely determines the Hopf algebra structure of the mod 2 cohomology of the linear groups , and as a module over the Steenrod algebra, and provides an explicit description of the generators.

**[Ar1]**D. Arlettaz: Torsion classes in the cohomology of congruence subgroups,*Math. Proc. Cambridge Philos. Soc*.**105**(1989), 241-248. MR**90j:20097****[Ar2]**D. Arlettaz: A note on the mod 2 cohomology of , in: Algebraic Topology Pozna\'{n} 1989, Proceedings,*Lecture Notes in Math.***1474**(1991), 365-370. MR**93g:19005****[Au]**C. Ausoni: Propriétés homotopiques de la K-théorie algébrique des entiers,*Ph.D. thesis, Université de Lausanne*(1998).**[Be]**J. Berrick: An Approach to algebraic K-theory. (Pitman, 1982). MR**84g:18028****[Bok]**M. Bökstedt: The rational homotopy type of , in: Algebraic Topology, Aarhus 1982,*Lecture Notes in Math.***1051**(1984), 25-37. MR**86e:18011****[Bor]**A. Borel: Topics in the homology theory of fibre bundles,*Lecture Notes in Math.***36**(1967). MR**36:4559****[Br]**W. Browder: Algebraic K-theory with coefficients , in: Geometric Applications of Homotopy Theory I, Evanston 1977,*Lecture Notes in Math.***657**(1978), 40-84. MR**80b:18011****[DF]**W. Dwyer and E. Friedlander: Conjectural calculations of general linear group homology, in Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Boulder 1983,*Contemp. Math.***55**Part I (1986), 135-147. MR**88f:18013****[FP]**Z. Fiedorowicz and S. Priddy: Homology of classical groups over finite fields and their associated infinite loop spaces,*Lecture Notes in Math.***674**(1978). MR**80g:55018****[M]**S. Mitchell: On the plus construction for at the prime 2,*Math. Zeitschrift***209**(1992), 205-222. MR**93b:55021****[MT]**M. Mimura and H. Toda: Topology of Lie groups I and II,*Translations of Math. Monographs***91**(AMS 1991). MR**92h:55001****[Q1]**D. Quillen: The mod 2 cohomology rings of extra-special 2-groups and spinor groups,*Math. Ann.***194**(1971), 197-212. MR**44:7582****[Q2]**D. Quillen: On the cohomology and K-theory of the general linear groups over a finite field,*Ann. of Math.***96**(1972), 552-586. MR**47:3565****[RW]**J. Rognes and C. Weibel: Two-primary algebraic K-theory of rings of integers in number fields,*preprint*(1997), http://math.uiuc.edu/K-theory/0220/.**[V]**V. Voevodsky: The Milnor conjecture,*preprint*(1996), http://math.uiuc.edu/K-theory/0170/.**[W]**C. Weibel: The 2-torsion in the K-theory of the integers,*C. R. Acad. Sci. Paris Sér. I***324**(1996), 615-620. MR**98h:19001**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
20G10,
19D55,
20J05,
55R40,
55S10

Retrieve articles in all journals with MSC (1991): 20G10, 19D55, 20J05, 55R40, 55S10

Additional Information

**Dominique Arlettaz**

Affiliation:
Institut de Mathématiques, Université de Lausanne, 1015 Lausanne, Switzerland

Email:
dominique.arlettaz@ima.unil.ch

**Mamoru Mimura**

Affiliation:
Department of Mathematics, Faculty of Science, Okayama University, Okayama, Japan 700

Email:
mimura@math.okayama-u.ac.jp

**Koji Nakahata**

Affiliation:
Institut de Mathématiques, Université de Lausanne, 1015 Lausanne, Switzerland

Email:
koji.nakahata@ima.unil.ch

**Nobuaki Yagita**

Affiliation:
Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

Email:
yagita@mito.ipc.ibaraki.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-99-05183-7

Received by editor(s):
September 15, 1997

Published electronically:
April 8, 1999

Additional Notes:
We would like to thank Christian Ausoni for his helpful comments on Bökstedt’s work \cite{Bok} and the referee for his interesting suggestions. The third author thanks the Swiss National Science Foundation for financial support.

Communicated by:
Ralph Cohen

Article copyright:
© Copyright 1999
American Mathematical Society