Math. Ann. 303, 149-164 (1995) m 9 Springer-Verlag 1995 MacLane homology and topological Hochschild homology Z. Fiedorowicz, T. Pirashvili, R. Schw~inzl, R. Vogt, F. Waldhausen Faehbercich 6/Mathematik, Universit~tOsnabfiick,Postfach 4465, D-49065 Osnabriick, Germany Received: 6 June 1994 / Revised version: 7 October 1994 I Introduction In 1985 B6kstedt [2] introduced the topological Hochschild homology functor THH which is related to Waldhausen's algebraic K-theory of certain A~ rings [26] in a way similar to the relationship of Hochschild homology of rings to Quillen K-theory: there is a generalization of the trace map. If R is a genuine ring, the trace map factors K.(R) ,K~,(R) " , HH, (R) through Bfkstedt's map ft. KS,(R) is Waldhausen's stable K-theory of R [27]. It was a surprise when Pirashvili brought MacLane homology H, ML into the picture, a homology theory for rings developed 30 years earlier by MacLane [15]. He observed [19] that a factors as K~.(R) ~ , HH.(R)