Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the third homology of $ SL_2$ and weak homotopy invariance

Authors: Kevin Hutchinson and Matthias Wendt
Journal: Trans. Amer. Math. Soc. 367 (2015), 7481-7513
MSC (2010): Primary 20G10; Secondary 14F42
Published electronically: November 12, 2014
MathSciNet review: 3378837
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The goal of the paper is to achieve - in the special case of the linear group $ SL_2$ - some understanding of the relation between group homology and its $ \mathbb{A}^1$-invariant replacement. We discuss some of the general properties of the $ \mathbb{A}^1$-invariant group homology, such as stabilization sequences and Grothendieck-Witt module structures. Together with very precise knowledge about refined Bloch groups, these methods allow us to deduce that in general there is a rather large difference between group homology and its $ \mathbb{A}^1$-invariant version. In other words, weak homotopy invariance fails for $ SL_2$ over many families of non-algebraically closed fields.

References [Enhancements On Off] (What's this?)

  • [AF12a] A. Asok and J. Fasel, A cohomological classification of vector bundles on smooth affine threefolds. To appear in Duke Math. J., arXiv:1204.0770v5.
  • [AF12b] A. Asok and J. Fasel, Algebraic vector bundles on spheres. To appear in J. Top., arXiv:1204.4538v3.
  • [Bas74] Hyman Bass, Clifford algebras and spinor norms over a commutative ring, Amer. J. Math. 96 (1974), 156-206. MR 0360645 (50 #13092)
  • [BT73] H. Bass and J. Tate, The Milnor ring of a global field, Algebraic $ K$-theory, II: ``Classical'' algebraic $ K$-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972) Springer, Berlin, 1973, pp. 349-446. Lecture Notes in Math., Vol. 342. MR 0442061 (56 #449)
  • [GJ99] Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. MR 1711612 (2001d:55012)
  • [Hor02] Jens Hornbostel, Constructions and dévissage in Hermitian $ K$-theory, $ K$-Theory 26 (2002), no. 2, 139-170. MR 1931219 (2003i:19002),
  • [Hov99] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134 (99h:55031)
  • [HT09] Kevin Hutchinson and Liqun Tao, The third homology of the special linear group of a field, J. Pure Appl. Algebra 213 (2009), no. 9, 1665-1680. MR 2518168 (2010j:20067),
  • [HT10] Kevin Hutchinson and Liqun Tao, Homology stability for the special linear group of a field and Milnor-Witt $ K$-theory, Doc. Math. Extra volume: Andrei A. Suslin sixtieth birthday (2010), 267-315. MR 2804257 (2012f:19012)
  • [Hut13a] Kevin Hutchinson, A refined Bloch group and the third homology of $ \rm SL_2$ of a field, J. Pure Appl. Algebra 217 (2013), no. 11, 2003-2035. MR 3057074,
  • [Hut13b] Kevin Hutchinson, A Bloch-Wigner complex for $ \mathrm {SL}_2$, J. K-Theory 12 (2013), no. 1, 15-68. MR 3126634,
  • [Hut13c] Kevin Hutchinson, Scissors congruence groups and the third homology of $ SL_2$ of local rings and fields. Preprint, arXiv:1309.5010.
  • [Jar83] J. F. Jardine, On the homotopy groups of algebraic groups, J. Algebra 81 (1983), no. 1, 180-201. MR 696133 (85d:18009),
  • [Kar73] Max Karoubi, Périodicité de la $ K$-théorie hermitienne, Algebraic $ K$-theory, III: Hermitian $ K$-theory and geometric applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 301-411. Lecture Notes in Math., Vol. 343 (French). MR 0382400 (52 #3284)
  • [Kar83] M. Karoubi, Homology of the infinite orthogonal and symplectic groups over algebraically closed fields, Invent. Math. 73 (1983), no. 2, 247-250. An appendix to the paper: ``On the $ K$-theory of algebraically closed fields'' by A. Suslin. MR 714091 (85h:18008b),
  • [Kar84] Max Karoubi, Relations between algebraic $ K$-theory and Hermitian $ K$-theory, Proceedings of the Luminy conference on algebraic $ K$-theory (Luminy, 1983), 1984, pp. 259-263. MR 772061 (86c:18007),
  • [KM97] Sava Krstić and James McCool, Free quotients of $ {\rm SL}_2(R[x])$, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1585-1588. MR 1376995 (97g:20055),
  • [Knu01] Kevin P. Knudson, Homology of linear groups, Progress in Mathematics, vol. 193, Birkhäuser Verlag, Basel, 2001. MR 1807154 (2001j:20070)
  • [KV69] Max Karoubi and Orlando Villamayor, Foncteurs $ K^{n}$ en algèbre et en topologie, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A416-A419 (French). MR 0251717 (40 #4944)
  • [Lam05] T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR 2104929 (2005h:11075)
  • [Mir08] B. Mirzaii, Third homology of general linear groups, J. Algebra 320 (2008), no. 5, 1851-1877. MR 2437633 (2009j:20063),
  • [Mor04] Fabien Morel, Sur les puissances de l'idéal fondamental de l'anneau de Witt, Comment. Math. Helv. 79 (2004), no. 4, 689-703 (French, with English and French summaries). MR 2099118 (2005f:19004),
  • [Mor11] Fabien Morel, On the Friedlander-Milnor conjecture for groups of small rank, Current developments in mathematics, 2010, Int. Press, Somerville, MA, 2011, pp. 45-93. MR 2906371
  • [Mor12] Fabien Morel, $ \mathbb{A}^1$-algebraic topology over a field, Lecture Notes in Mathematics, vol. 2052, Springer, Heidelberg, 2012. MR 2934577
  • [MV99] Fabien Morel and Vladimir Voevodsky, $ {\bf A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45-143 (2001). MR 1813224 (2002f:14029)
  • [Mos] L.-F. Moser, $ \mathbb{A}^1$-locality results for linear algebraic groups. Preprint, 2011.
  • [Neu99] Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G. Harder. MR 1697859 (2000m:11104)
  • [Qui69] Daniel Quillen, Cohomology of groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 47-51. MR 0488054 (58 #7627a)
  • [Qui73] Daniel Quillen, Finite generation of the groups $ K_{i}$ of rings of algebraic integers, Algebraic $ K$-theory, I: Higher $ K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 179-198. Lecture Notes in Math., Vol. 341. MR 0349812 (50 #2305)
  • [Ros02] Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. MR 1876657 (2003d:11171)
  • [Sch09] Marco Schlichting, Hermitian $ K$-theory of exact categories, J. K-Theory 5 (2010), no. 1, 105-165. MR 2600285 (2011b:19007),
  • [Sus90] A. A. Suslin, $ K_3$ of a field, and the Bloch group, Trudy Mat. Inst. Steklov. 183 (1990), 180-199, 229 (Russian). Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217-239; Galois theory, rings, algebraic groups and their applications (Russian). MR 1092031 (91k:19003)
  • [Wen11] Matthias Wendt, Rationally trivial torsors in $ \mathbb{A}^1$-homotopy theory, J. K-Theory 7 (2011), no. 3, 541-572. MR 2811715,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20G10, 14F42

Retrieve articles in all journals with MSC (2010): 20G10, 14F42

Additional Information

Kevin Hutchinson
Affiliation: School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Matthias Wendt
Affiliation: Fakultät Mathematik, Universität Duisburg-Essen, Thea-Leymann-Strasse 9, 45127, Essen, Germany

Keywords: Weak homotopy invariance, group homology
Received by editor(s): October 18, 2013
Received by editor(s) in revised form: April 25, 2014
Published electronically: November 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society