Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T17:48:22.814Z Has data issue: false hasContentIssue false
  • English
  • Français

A Bloch-Wigner complex for SL2

Published online by Cambridge University Press:  22 April 2013

Kevin Hutchinson
Kevin Hutchinson*
Affiliation:
School of Mathematical Sciences, University College DublinIrelandkevin.hutchinson@ucd.ie

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We introduce a refinement of the Bloch-Wigner complex of a field F. This refinement is complex of modules over the multiplicative group of the field. Instead of computing K2(F) and Kind3(F) - as the classical Bloch-Wigner complex does - it calculates the second and third integral homology of SL2(F). On passing to F× -coinvariants we recover the classical Bloch-Wigner complex. We include the case of finite fields throughout the article.

Keywords

K-theoryGroup Homology19G9920G10

References

1.Bak, A. & Tang, G., Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups, Adv. Math. 189(1) (2004) 1–37.Google Scholar
2.Bak, A. & Vavilov, N., Presenting powers of augmentation ideals and Pfister forms, K-Theory 20(4) (2000) 299–309, special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part IV.Google Scholar
3.Bloch, S.J., Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, CRM Monograph Series 11, American Mathematical Society, Providence, RI, 2000.Google Scholar
4.Brown, K.S., Cohomology of groups, Graduate Texts in Mathematics 87, Springer-Verlag, New York, 1982.Google Scholar
5.Cartan, H. & Eilenberg, S., Homological algebra, Princeton University Press, Princeton, N. J., 1956.Google Scholar
6.Dupont, J.L. & Sah, C.H., Scissors congruences. II, J. Pure Appl. Algebra 25(2) (1982) 159–195.CrossRefGoogle Scholar
7.Goette, S. & Zickert, C.K., The extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 11 (2007) 1623–1635.CrossRefGoogle Scholar
8.Hutchinson, K., A refined Bloch group and the third homology of SL2 of a field, Journal of Pure and Applied Algebra (2013), doi: 10.1016/j.jpaaa.2013.01.001.Google Scholar
9.Hutchinson, K., A new approach to Matsumoto's theorem, K-Theory 4(2) (1990) 181–200.Google Scholar
10.Hutchinson, K. & Tao, L., The third homology of the special linear group of a field, J. Pure Appl. Algebra 213 (2009) 1665–1680.Google Scholar
11.Hutchinson, K. & Tao, L., Homology stability for the special linear group of a field and Milnor-Witt K-theory, Doc. Math. (Extra Vol.) (2010) 267–315.Google Scholar
12.Levine, M., The indecomposable K3 of fields, Ann. Sci. École Norm. Sup. (4) 22(2) (1989) 255–344.CrossRefGoogle Scholar
13.Matsumoto, H., Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969) 1–62.Google Scholar
14.Mazzoleni, A., A new proof of a theorem of Suslin, K-Theory 35(3-4) (2005) 199–211 (2006).Google Scholar
15.Merkur'ev, A.S. & Suslin, A.A., The group K3 for a field, Izv. Akad. Nauk SSSR Ser. Mat. 54(3) (1990) 522–545.Google Scholar
16.Milnor, J. & Husemoller, D., Symmetric bilinear forms, Springer-Verlag, New York, 1973, ergebnisse der Mathematik und ihrer Grenzgebiete 73.Google Scholar
17.Mirzaii, B., Third homology of general linear groups, J. Algebra 320(5) (2008) 1851–1877.Google Scholar
18.Morel, F., An introduction to $\mathematical double-struck capital A_1\$-homotopy theory, ICTP Lecture Notes 15.Google Scholar
19.Morel, F., Sur les puissances de l’idéal fondamental de l’anneau de Witt, Comment. Math. Helv. 79(4) (2004) 689–703.CrossRefGoogle Scholar
20.Neumann, W.D., Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004) 413–474 (electronic).CrossRefGoogle Scholar
21.Quillen, D., On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. (2) 96 (1972) 552–586.Google Scholar
22.Sah, C.H., Homology of classical Lie groups made discrete. III, J. Pure Appl. Algebra 56(3) (1989) 269–312.Google Scholar
23.Suslin, A.A., Homology of GLn, characteristic classes and Milnor K-theory, In: Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), Lecture Notes in Math. 1046, pages 357–375, Springer, Berlin, 1984.Google Scholar
24.Suslin, A.A., Torsion in K 2 of fields, K-Theory 1(1) (1987) 5–29.Google Scholar
25.Suslin, A.A., K 3 of a field, and the Bloch group, Trudy Mat. Inst. Steklov. 183 (1990) 180–199, 229, translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239, Galois theory, rings, algebraic groups and their applications (Russian).Google Scholar
26.Swan, R.G., The p-period of a finite group, Illinois J. Math. 4 (1960) 341–346.Google Scholar
27.Zagier, D., The dilogarithm function, In: Frontiers in number theory, physics, and geometry. II, pages 3–65, Springer, Berlin, 2007.Google Scholar