The 2-torsion in the K-theory of the Integers, by Charles A. Weibel

Using recent results of Voevodsky, Suslin-Voevodsky and Bloch-Lichtenbaum, we completely determine the 2-torsion subgroups of the groups K_j(Z). The 2-torsion is periodic of order 8, starting with K_0(Z). That is, the 2-torsion consists of:

  • the Z/2 summands in degrees 8n+1, 8n+2;
  • the Z/16 summand in degrees 8n+3;
  • the (cyclic) ``Image of J'' in degrees 8n+7.
  • In particular, for each positive number n, the 2-part of the rational number zeta(1-2n) is exactly twice the ratio of order of the 2-torsion in the finite groups K_{4n-2}(Z) and K_{4n-1}(Z).

    This note is in English, preceded by an abridged French version. A description of the 2-torsion in the K-theory of the integers in other number fields will appear in a longer paper with Rognes.

    This has appeared in C. R. Acad. Sci. (Paris), 324 (1997), 615-620.


    Charles A. Weibel <weibel@math.rutgers.edu>