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

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\('\).


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


Zbl 0579.18006
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