The additivity theorem in \(K\)-theory. (English) Zbl 1077.19001

Jones, Kevin Charles; Kim, Youngsoo; Mhoon, Andrea H.; Santhanam, Rekha; Walker, Barry J.; Grayson, Daniel R.

\(K\)-Theory 32, No. 2, 181-191 (2004).

The authors give new versions, in the style of Quillen’s Theorem A, of F. Waldhausen’s proof [Lect. Notes Math. 1126, 318–419 (1985; Zbl 0579.18006)] of the additivity theorem for algebraic \(K\)-theory. This is carried out by showing some results that allow to convert a theorem B style proof into a theorem A style proof. The first of these results is Theorem  showing conditions such that, for any functors of small categories \(f:C\rightarrow D\) and \(g:C\rightarrow E\), the induced functor \((f,g):C\rightarrow D\times E\) is a homotopy equivalence. This theorem is then rewritten for maps of simplicial sets obtaining the simplicial set versions Theorem Â* and Theorem Â\('\).

Reviewer: Antonio R. Garzón (Granada)

MSC:

19D06	\(Q\)- and plus-constructions
55U10	Simplicial sets and complexes in algebraic topology

Keywords:

additivity theorem; theorem A style; theorem B style; simplicial set

Citations:

Zbl 0579.18006