NK_0 and NK_1 of the groups C_4 and D_4, by Chuck Weibel

We explicitly compute the groups NK_0(Z[G]) for the cyclic group G=C_4 and the dihedral group D_4. We also compute NK_1(Z[C_4]) and show that NK_1(Z[D_4]) is nontrivial. The computation, which is fairly elementary, keeps track of the action of the Verschiebung and Frobenius operators. (Recall that NK_i(R) is the quotient K_i(R[x])/K_i(R).)

These calculations are applied to complete the Lafont-Ortiz calculations of K_0(Z[G]) when G is a hyperbolic 3-simplex reflection group, and the Lueck calculation of the Whitehead groups Wh_0 and Wh_1 for the semidirect product of C_4 by the discrete Heisenberg group.


Chuck Weibel <weibel@math.rutgers.edu>