Now showing 1 - 8 of 8
  • Publication
    A Note on Milnor-Witt K-theory and a Theorem of Suslin
    (Taylor and Francis, 2008) ;
    We give a simple presentation of the additive Milnor-Witt K-theory groups KMWn(F) of the field F, for n≥2, in terms of the natural small set of generators. When n = 2, this specialises to a theorem of Suslin which essentially says that KMW2(F)∼=H2(Sp(F),Z).
      444Scopus© Citations 5
  • Publication
    A Bloch-Wigner sequence for SL2
    (Cambridge University Press, 2013-08)
    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 K 2 (F) and K ind 3 (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
      224Scopus© Citations 15
  • Publication
    On the low-dimensional homology of SL_2(k[t,1/t])
    (Elsevier, 2015-03-01)
    We prove analogues of the fundamental theorem of K -theory for the second and the third homology of SL2 over an infinite field. The statements of the theorems involve Milnor–Witt K -theory and refined scissors congruence groups. We use these results to calculate the low-dimensional homology of SL2 of Laurent polynomials over certain fields.
      317Scopus© Citations 9
  • Publication
    A refined Bloch group and the third homology of SL_2 of a field
    (Elsevier, 2013-11)
    We use the properties of the refined Bloch group to study the structure of H3(SL2(F), Z) for a field F. We compute this group up to 2-torsion when F is a local field with finite residue field of odd order, and we show that for any global field F it is not finitely-generated
      298Scopus© Citations 11
  • Publication
    The third homology of the special linear group of a field
    (Elsevier, 2009-09) ;
    We prove that for any infinite field F, the map H-3(SLn(F), Z) -> H-3(SLn+1 (F), Z) is an isomorphism for all n >= 3. When n = 2 the cokernel of this map is naturally isomorphic to 2. K-3(M) (F), where K-n(M)(F) is the nth Milnor K-group of F. We deduce that the natural homomorphism from H-3(SL2(F), Z) to the indecomposable K-3 of F, K-3(F)(ind), is surjective for any infinite field F.
      280Scopus© Citations 14
  • Publication
    The second homology of SL_2 of S-integers
    (Elsevier, 2016-02)
    We calculate the structure of the finitely generated groups H2(SL2(Z[1/m]),Z) when m is a multiple of 6. Furthermore, we show how to construct homology classes, represented by cycles in the bar resolution, which generate these groups and have prescribed orders. When n≥2 and m is the product of the first n primes, we combine our results with those of Jun Morita to show that the projection St(2,Z[1/m])→SL2(Z[1/m]) is the universal central extension. Our methods have wider applicability: The main result on the structure of the second homology of certain rings is valid for rings of S-integers with sufficiently many units. For a wide class of rings A , we construct explicit homology classes in H2(SL2(A),Z), functorially dependent on a pair of units, which correspond to symbols in K2(2,A).
      389Scopus© Citations 4
  • Publication
    On the third homology of SL_2 and weak homotopy invariance
    (American Mathematical Society, 2015-11-12) ;
    The goal of the paper is to achieve - in the special case of the linear group SL2 - some understanding of the relation between group homology and its A1-invariant replacement. We discuss some of the general properties of the A1-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 A1 -invariant version. In other words, weak homotopy invariance fails for SL2 over many families of non-algebraically closed fields.
  • Publication
    Homology stability for the special linear group of a field and Milnor-Witt K-theory
    (Universität Bielefeld, 2010-06) ;
    Let F be a field of characteristic zero and let ft,n be the stabilization homomorphism Hn(SLt(F), Z) → Hn(SLt+1(F), Z). We prove the following results: For all n, ft,n is an isomorphism if t ≥ n + 1 and is surjective for t = n, confirming a conjecture of C-H. Sah. fn,n is an isomorphism when n is odd and when n is even the kernel is isomorphic to I n+1(F), the (n + 1)st power of the fundamental ideal of the Witt Ring of F. When n is even the cokernel of fn−1,n is isomorphic to KMW n (F), the nth Milnor-Witt K-theory group of F. When n is odd, the cokernel of fn−1,n is isomorphic to 2KM n (F), where KM n (F) is the nth Milnor K-group of F.