Research Announcements on K 3 and K a of the Integers Mod

Quillen [7] defines an algebraic A -̂functor from the category of associative rings to that of positively graded abelian groups, with Kf(R) = ÏI^BGLR ") for i > 1. Kx and K2 correspond respectively to the 'classical' Bass and Milnor definitions. The /^-images of finite fields and their algebraic closures were computed by Quillen in [8]. Since then, there has been only a handful of complete calculations of any of the higher A -̂groups (Kt for i > 2). Lee and Szczarba [4] showed that the Karoubi subgroup Z/48 of K3(Z) was the full group. Evens and Friedlander [3] computed KfZ/p) and K((Fp [t] lit )) for i < 5 and prime p greater than 3. Snaith in [1] and, with Lluis, in [5], fully determined K3(¥ m [t]/(t )) for m > 1 and prime p other than 3. This note summarizes computations of the groups K3(Z/n), and K^(Z/p ) for k > 1 and prime p > 3. These complete the recent partial results on K3(Z/4) by Snaith and on K3(Z/9) by Lluis, and extend the work of Evens and Friedlander. The theorem stated below is consistent with the Karoubi conjecture that for odd primes, BGLZ/p is the homotopy fibre of the difference of Adams operations, ty ty _ 1 . However, Priddy [6] has disproved the conjecture in the cases p > 3 and k — 2. I am most grateful to Victor Snaith for his supervision of the thesis in which these results originally appeared. Details of the proofs can also be found in [H-