The Additivity Theorem in K

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182 Quillen’s [5, Theorem B], Waldhausen’s [7, Lemma 1.4.B], or Gillet-Grayson’s [1, Theorem B0 ]. The hypothesis of all these theorems is that the base change maps between naive combinatorial approximations to the homotopy fibers of a map are all homotopy equivalences, and the result is a fibration sequence incorporating the map. In this paper, we add a theorem (Theorem A^ of Section 2) to the list of theorems usable in a Theorem A style proof that makes it easy to convert Waldhausen’s proof [7, Theorem 1.4.2] of the additivity theorem from a Theorem B style proof to a Theorem A style proof. We also introduce a simplicial version of it (Theorem ^ A 0 of section 3) that can be used to convert the proof of additivity in [3, Theorem 5.1.2], which in turn, is based on Waldhausen’s proof. NOTATION. The nerve of a category C will be denoted by NC. Given a functor C ! g D of small categories, we will use g (an abuse of notation) for the induced map of simplical sets NC ! g ND. For C 2 NpC and 0OiOp, the ith vertex is denoted by Ci, so that C will represent a chain C0 ! C1 ! !Cp of objects and morphisms of C. Let D denote the category of finite nonempty ordered sets, let ½nŠ 2D denote the ordered set f0 < 1 < < ng, and let D n be the simplicial set represented by ½nŠ. We use to mean any simplex of D 0 . For simplicial sets S and T, the external product S T is the bisimplicial set defined by ðS TÞ p;q ¼ Sp Tq. By Yoneda’s lemma, we may identify a simplex t 2 Tn with a map t: D n ! T; for an arrow i: ½mŠ !½nŠ the simplex i ðtÞ will be identified with the composite map ti: D m ! T, so we will usually write ti for i ðtÞ. 2. Converting Waldhausen’s Proof KEVIN CHARLES JONES ET AL. In this section, we show how to convert Waldhausen’s proof of the additivity theorem to a Theorem A style proof. Given a functor C ! g E of small categories and an object E of E, we write g=E for the category [5, p. 93] with objects ðC; gC ! e EÞ, where C is an object of C and gC ! e E is an arrow in E; an arrow is a map C ! C0 making the evident triangle commute. Analogously, Eng will be a category with objects ðE ! e gC; CÞ: The projection functor ðC; gC ! e EÞ 7! C will be denoted by g=E ! p C. THEOREM A. ^ Let D f C ! g E be functors of small categories. If the composite functor fp: g=E ! D is a homotopy equivalence for each object E 2 E then the functor ðf; gÞ: C ! D E is a homotopy equivalence. Proof. Our proof is modeled on Quillen’s proof of Theorem A [5, p. 95], which amounts to the special case where D is trivial.
THE ADDITIVITY THEOREM IN K-THEORY 183 It will suffice to show that the map of bisimplicial sets ðf; gÞ 1 : NC D 0 ! ð ND NEÞ D 0 ðC; Þ # ½ðfC; gCÞ; Š is a homotopy equivalence. First, we define bisimplicial sets Xg and Y, analogous to those introduced by Quillen, as follows, where p; q 2 N, with the evident face and degeneracy maps. X g pq :¼fðC; e; EÞjC 2 NpC; E 2 NqE; e 2 HomEðgCp; E0Þg Ypq :¼ fðD; E 0 ; e; EÞjD 2 NpD; E 0 2 NpE; E 2 NqE; e 2 HomEðE 0 p ; E0Þg Consider the following commutative diagram. The maps are given by these formulas. gðC; e; EÞ ¼ðC; Þ bðC; e; EÞ ¼ðfC; gC; e; EÞ aðC; e; EÞ ¼ðfC; EÞ dðD; E 0 ; e; EÞ ¼½ðD; E 0 Þ; Š cðD; E 0 ; e; EÞ ¼ðD; EÞ It will be enough to show that a; c; g; d are homotopy equivalences. Here is the argument for a and c. Fix q and consider the following diagram of simplicial sets, where h and k are defined to make the diagram commute. Xg q ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ! ffi ‘ Nðg=E0Þ E2NqE ? ya ? q yh ðND x ? NEÞ q ƒƒƒƒƒƒƒƒƒƒƒƒƒƒ! ND NqE ffi c q x ? k ‘ ðND Nð1E=E0ÞÞ ffi Y q ƒƒƒƒƒƒƒƒƒƒƒƒƒƒ! E2NqE Here the horizontal arrows are the obvious isomorphisms of simplicial sets. We point out that h is a disjoint union of homotopy equivalences induced by composite functors g=E0 ! p C ! f D, each of which is a homotopy equivalence by hypothesis, and k is a disjoint union of nerves of maps of the form ð1Þ
Page 1: K-Theory 32: 181-191, 2004. Ó 2004
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The Additivity Theorem in K

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