Algebraic & Geometric Topology 11 (2011) 2391–2436 2391

Algebraic K –theory over the infinite dihedral group: an algebraic approach

JAMES FDAVIS QAYUM KHAN ANDREW RANICKI

Two types of Nil-groups arise in the codimension 1 splitting obstruction theory for homotopy equivalences of finite CW–complexes: the Farrell–Bass Nil-groups in the nonseparating case when the fundamental group is an HNN extension and the Waldhausen Nil-groups in the separating case when the fundamental group is an amalgamated free product. We obtain a general Nil-Nil theorem in algebraic K–theory relating the two types of Nil-groups. The infinite dihedral group is a free product and has an index 2 subgroup which is an HNN extension, so both cases arise if the fundamental group surjects onto the infinite dihedral group. The Nil-Nil theorem implies that the two types of the reduced Nilf –groups arising from such a fundamental group are isomorphic. There is also a topological application: in the finite-index case of an amalgamated free product, a homotopy equivalence of finite CW–complexes is semisplit along a separating subcomplex.

19D35; 57R19

Introduction

The infinite dihedral group is both a free product and an extension of the infinite cyclic group Z by the cyclic group Z2 of order 2

D Z2 Z2 Z Ì Z2 1 D  D with Z2 acting on Z by 1. A group G is said to be over D if it is equipped with 1 an epimorphism p G D . We study the algebraic K–theory of RŒG, for any W ! 1 ring R and any group G over D . Such a group G inherits from D an injective 1 1 amalgamated free product structure G G1 H G2 with H an index 2 subgroup D  of G1 and G2 . Furthermore, there is a canonical index 2 subgroup G G with an x 

Published: 5 September 2011 DOI: 10.2140/agt.2011.11.2391 2392 James F Davis, Qayum Khan and Andrew Ranicki

injective HNN structure G H Ì˛ Z for an automorphism ˛ H H . The various x D W ! groups fit into a commutative braid of short exact sequences:

Z 8 8 '

' G H Ì˛ Z D Z2 Z2 x 9 1 D % p 8 8D    9 % " " H G1 G2 G G1 H G2 Z2 D \& D8  9 9  p ı The algebraic K–theory decomposition theorems of Waldhausen for injective amalga- mated free products and HNN extensions give

(1) K .RŒG/ K .RŒH  RŒG1 RŒG2/  D  !  Nilf 1.RŒH RŒG1 H ; RŒG2 H / ; ˚  I (2) K .RŒG/ K .1 ˛ RŒH RŒH /  x D  W ! 1 Nilf 1.RŒH ; ˛/ Nilf 1.RŒH ; ˛ /: ˚  ˚  We establish isomorphisms

1 Nil .RŒH  RŒG1 H ; RŒG2 H / Nil .RŒH ; ˛/ Nil .RŒH ; ˛ /:  I Š  Š  A homotopy equivalence f M X of finite CW–complexes is split along a sub- W ! complex Y X if it is a cellular map and the restriction f N f 1.Y / Y is  jW D ! also a homotopy equivalence. The Nil –groups arise as the obstruction groups to splitting homotopy equivalences of finite CW–complexes for codimension 1 Y X e e e  with injective 1.Y / 1.X /, so that 1.X / is either an HNN extension or an ! amalgamated free product (Farrell–Hsiang, Waldhausen) – see Section 4 for a brief review of the codimension 1 splitting obstruction theory in the separating case of an amalgamated free product. In this papere we introduce the considerably weaker notion of a homotopy equivalence in the separating case being semisplit (Definition 4.4). By way of geometric application we prove in Theorem 4.5 that there is no obstruction to topological semisplitting in the finite-index case.

0.1 Algebraic semisplitting

The following is a special case of our main algebraic result (Theorems 1.1, 2.7) which shows that there is no obstruction to algebraic semisplitting.

Algebraic & Geometric Topology, Volume 11 (2011) Algebraic K–theory over the infinite dihedral group: an algebraic approach 2393

Theorem 0.1 Let G be a group over D , with 1 H G1 G2 G H Ì˛ Z G G1 H G2 : D \  x D  D  (1) For any ring R and n Z the corresponding reduced Nil–groups are isomorphic: 2 1 Niln.RŒH  RŒG1 F; RŒG2 H / Niln.RŒH ; ˛/ Niln.RŒH; ˛ /: I Š Š (2) The inclusion  RŒG RŒG determines induction and transfer maps W x ! !  Kn.RŒG/ Kn.RŒG/ ;  Kn.RŒG/ Kn.RŒG/ : !W x ! W ! x For all integers n 6 1, the Niln.RŒH ; ˛/– Niln.RŒH  RŒG1 H; RŒG2 H /– e ! e I e components of the maps ! and  in the decompositions (2) and (1) are isomor- phisms.

Proof Part (i) is a special case of Theorem 0.4. Part (ii) follows from Theorem 0.4e, Lemma 3.20e and Proposition 3.23.

The n 0 case will be discussed in more detail in Sections 0.2 and 3.1. D Remark 0.2 We do not seriously doubt that a more assiduous application of higher K–theory would extend Theorem 0.1(2) to all n Z (see also Davis, Quinn and 2 Reich[5]).

As an application of Theorem 0.1, we shall prove the following theorem. It shows that the Farrell–Jones Isomorphism Conjecture in algebraic K–theory can be reduced (up to dimension one) to the family of finite-by-cyclic groups, so that virtually cyclic groups of infinite dihedral type need not be considered.

Theorem 0.3 Let € be any group, and let R be any ring. Then, for all integers n < 1, the following induced map of equivariant homology groups, with coefficients in the algebraic K–theory functor KR , is an isomorphism: € € H .Efbc€ KR/ H .Evc€ KR/: n I ! n I Furthermore, this map is an epimorphism for n 1. D This is a special case of a more general fibered version, Theorem 3.29. Theorem 0.3 has been proved for all degrees n by Davis, Quinn and Reich[5]; however our proof here uses only algebraic topology, avoiding the use of controlled topology.

Algebraic & Geometric Topology, Volume 11 (2011) 2394 James F Davis, Qayum Khan and Andrew Ranicki

The original reduced Nil–groups Nil .R/ Nil .R; id/ feature in the decompositions  D  of Bass[2] and Quillen[9]:

K .RŒt/ K .R/ Nilf 1.R/  D  ˚  K .RŒZ/ K .R/ K 1.R/ Nilf 1.R/ Nilf 1.R/:  D  ˚  ˚  ˚  In Section 3 we shall compute severale examplese which require Theorem 0.1:

K .RŒZ2/ K .RŒZ2/ K .RŒZ2 Z2/  ˚  Nilf 1.R/   D K .R/ ˚   K .RŒZ2/ K .RŒZ3/ K .RŒZ2 Z3/  ˚  Nilf 1.R/1   D K .R/ ˚   Wh.G0 Z2/ Wh.G0 Z2/ Wh.G0 Z2 G0 G0 Z2/  ˚  Nilf 0.ZŒG0/    D Wh.G0/ ˚ where G0 Z2 Z2 Z. The point here is that Nil0.ZŒG0/ is an infinite torsion D   abelian group. This provides the first example 3.28 of a nonzero Nil –group in the amalgamated product case and hence the first example of a nonzero obstruction to splitting a homotopy equivalence in the separating case (A). e 0.2 The Nil-Nil Theorem e We establish isomorphisms between two types of codimension 1 splitting obstruction nilpotent class groups, for any ring R. The first type, for separated splitting, arises in the decompositions of the algebraic K–theory of the R–coefficient group ring RŒG of a group G over D , with an epimorphism p G D onto the infinite dihedral 1 W ! 1 group D . The second type, for nonseparated splitting, arises from the ˛–twisted 1 polynomial ring RŒH ˛Œt, with H ker.p/ and ˛ F F an automorphism such D W ! that

G ker. p G Z2/ H Ì˛ Z ; x D ı W ! D where  D Z2 is the unique epimorphism with infinite cyclic kernel. Note: W 1 !

(A) D Z2 Z2 is the free product of two cyclic groups of order 2, whose 1 D  generators will be denoted t1; t2 . 2 2 (B) D t1; t2 t1 1 t2 contains the infinite cyclic group Z t as a 1 D h j D D i D h i subgroup of index 2 with t t1t2 . In fact there is a short exact sequence with a D split epimorphism

 1 / Z / D / Z2 / 1 : f g 1 f g

Algebraic & Geometric Topology, Volume 11 (2011) Algebraic K–theory over the infinite dihedral group: an algebraic approach 2395

More generally, if G is a group over D , with an epimorphism p G D , then: 1 W ! 1 (A) G G1 H G2 is a free product with amalgamation of two groups D  G1 ker.p1 G Z2/; G2 ker.p2 G Z2/ G D W ! D W !  amalgamated over their common subgroup H ker.p/ G1 G2 of index 2 D D \ in both G1 and G2 .

(B) G has a subgroup G ker. p G Z2/ of index 2 which is an HNN x D ı W ! extension G H Ì˛ Z where ˛ H H is conjugation by an element t G x D W ! 2 x with p.t/ t1t2 D . D 2 1

The K–theory of type (A) For any ring S and S –bimodules Ꮾ1; Ꮾ2 , we write the S –bimodule Ꮾ1 S Ꮾ2 as Ꮾ1Ꮾ2 , and we suppress left-tensor products of maps ˝ with the identities id or id . The exact category NIL.S Ꮾ1; Ꮾ2/ has objects Ꮾ1 Ꮾ2 I being quadruples .P1; P2; 1; 2/ consisting of finitely generated ( finitely generated) D projective S –modules P1; P2 and S –module morphisms

1 P1 Ꮾ1P2 ; 2 P2 Ꮾ2P1 W ! W ! such that 21 P1 Ꮾ1Ꮾ2P1 is nilpotent in the sense that W ! k .2 1/ 0 P1 .Ꮾ1Ꮾ2/.Ꮾ1Ꮾ2/ .Ꮾ1Ꮾ2/P1 ı D W !


for some k > 0. The morphisms are pairs .f1 P1 P ; f2 P2 P / such W ! 10 W ! 20 that f2 1  f1 and f1 2  f2 . Recall the Waldhausen Nil–groups ı D 10 ı ı D 20 ı Nil .S Ꮾ1; Ꮾ2/ K .NIL.S Ꮾ1; Ꮾ2//, and the reduced Nil–groups Nil satisfy  I WD  I  Nil .S Ꮾ1; Ꮾ2/ K .S/ K .S/ Nil .S Ꮾ1; Ꮾ2/:  I D  ˚  ˚  I An object .P1; P2; 1; 2/ in NIL.S Ꮾ1; Ꮾ2/ is semisplit if the S –module isomor- I phism 2 P2 Ꮾ2P1 is an isomorphism. W ! Let R be a ring which is an amalgamated free product e e R R1 S R2 D  with R S Ꮾ for S –bimodules Ꮾ which are free S –modules, k 1; 2. The k D ˚ k k D algebraic K–groups were shown by Waldhausen[24; 25; 26] to fit into a long exact sequence

Kn.S/ Nilf n.S Ꮾ1; Ꮾ2/ Kn.R1/ Kn.R2/ Kn.R/


! ˚ I ! ˚ ! Kn 1.S/ Nilf n 1.S Ꮾ1; Ꮾ2/ ! ˚ I !


with Kn.R/ Niln 1.S Ꮾ1; Ꮾ2/ a split surjection. ! I

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For any ring R a based finitely generated free R–module chain complex C has a torsion .C / K1.R/. The torsion of a chain equivalence f C D of based finitely 2 W ! generated free R–module chain complexes is the torsion of the algebraic mapping cone

.f / .Ꮿ.f // K1.R/: D 2 By definition, the chain equivalence is simple if .f / 0 K1.R/. D 2 For R R1 S R2 the algebraic analogue of codimension 1 manifold transversality D  shows that every based finitely generated free R–module chain complex C admits a Mayer–Vietoris presentation

Ꮿ 0 R S D .R R C1/ .R R C2/ C 0 W ! ˝ ! ˝ 1 ˚ ˝ 2 ! ! with Ck a based finitely generated free Rk –module chain complex, D a based finitely generated free S –module chain complex with R –module chain maps R S D C , k k ˝ ! k and .Ꮿ/ 0 K1.R/. This was first proved in[24; 25]; see also Ranicki [18, D 2 Remark 8.7; 19]. A contractible C splits if it is simple chain equivalent to a chain complex (also denoted by C ) with a Mayer–Vietoris presentation Ꮿ with D contractible, in which case C1; C2 are also contractible and the torsion .C / K1.R/ is such that 2 .C / .R R C1/ .R R C2/ .R S D/ im.K1.R1/ K1.R2/ K1.R// : D ˝ 1 C ˝ 2 ˝ 2 ˚ ! By the algebraic obstruction theory of[24] C splits if and only if

.C / im.K1.R1/ K1.R2/ K1.R// ker.K1.R/ K0.S/ Nil0.S Ꮾ1; Ꮾ2// : 2 ˚ ! D ! ˚ I For any ring R the group ring RŒG of an amalgamated free product of groups G D G1 H G2 is an amalgamated free product of rings  RŒG RŒG1 RŒG2 : D RŒH  e If H G1 , H G2 are injective then the RŒH –bimodules RŒG1 H , RŒG2 H  ! ! are free, and Waldhausen[26] decomposed the algebraic K–theory of RŒG as

K .RŒG/ K .RŒH  RŒG1 RŒG2/ Nil 1.RŒF RŒG1 H ; RŒG2 H / :  D  !  ˚  I In particular, there is defined a split monomorphism

A Nil 1.RŒH  RŒG1 H ; RŒG2 H / K .RŒG/ ; W  I !  which for 1 is given by  D e A Nilf 0.RŒH  RŒG1 H ; RŒG2 H / K1.RŒG/ ; W I !    1 2 ŒP1; P2; 1; 2 RŒG RŒH  .P1 P2/; : e 7! ˝ ˚ 1 1

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The K–theory of type (B) Given a ring R and an R–bimodule Ꮾ, consider the tensor algebra TR.Ꮾ/ of Ꮾ over R:

TR.Ꮾ/ R Ꮾ ᏮᏮ : WD ˚ ˚ ˚


The Nil–groups Nil .R Ꮾ/ are defined to be the algebraic K–groups K .NIL.R Ꮾ//  I  I of the exact category NIL.R Ꮾ/ with objects pairs .P; / with P a finitely generated I projective R–module and  P ᏮP an R–module morphism, nilpotent in the sense W ! that for some k > 0, we have

k 0 P ᏮP Ꮾk P : D W ! !


! The reduced Nil–groups Nil are such that  Nil .R Ꮾ/ K .R/ Nil .R Ꮾ/:  I D  ˚  I Waldhausen[26] proved that if Ꮾ is finitely generated projective as a left R–module and free as a right R–module,e then K .TR.Ꮾ// K .R/ Nil 1.R Ꮾ/:  D  ˚ e I There is a split monomorphism

B Nil 1.R Ꮾ/ K .TR.Ꮾ// W  I !  which for 1 is given by  D e B Nil0.R Ꮾ/ K1.TR.Ꮾ// ; ŒP;  ŒTR.Ꮾ/P; 1  : W I ! 7! In particular, for Ꮾ R, we have D e Nil .R R/ Nil .R/; Nilf .R R/ Nilf .R/;  I D   I D  TR.Ꮾ/ RŒt ; K .RŒt/ K .R/ Nilf 1.R/: e D  D  ˚  Relating the K–theory of types (A) and (B) Recall that a small category I is fil- tered if:

For any pair of objects ˛; ˛ in I , there exist an object ˇ and morphisms ˛ ˇ  0 ! and ˛ ˇ in I . 0 ! For any pair of morphisms u; v ˛ ˛ in I , there exists an object ˇ and  W ! 0 morphism w ˛ ˇ such that w u w v. W 0 ! ı D ı Note that any directed poset I is a filtered category. A filtered colimit is a colimit over a filtered category.

Algebraic & Geometric Topology, Volume 11 (2011) 2398 James F Davis, Qayum Khan and Andrew Ranicki

Theorem 0.4 (The Nil-Nil Theorem) Let R be a ring. Let Ꮾ1; Ꮾ2 be R–bimodules. ˛ Suppose that Ꮾ2 colim˛ I Ꮾ2 is a filtered colimit limit of R–bimodules such that D 2 each Ꮾ˛ is a finitely generated projective left R–module. Then, for all n Z, the Nil– 2 2 groups of the triple .R Ꮾ1; Ꮾ2/ are related to the Nil–groups of the pair .R Ꮾ1Ꮾ2/ I I by isomorphisms

Niln.R Ꮾ1; Ꮾ2/ Niln.R Ꮾ1Ꮾ2/ Kn.R/ I Š I ˚ Nilf n.R Ꮾ1; Ꮾ2/ Nilf n.R Ꮾ1Ꮾ2/: I Š I In particular, for n 0 and Ꮾ2 a finitely generated projective left R–module, there are D defined inverse isomorphisms

i Nil0.R Ꮾ1Ꮾ2/ K0.R/ Š Nil0.R Ꮾ1; Ꮾ2/; W I ˚ !  I    12 .ŒP1; 12 P1 Ꮾ1Ꮾ2P1; ŒP2/ P1; Ꮾ2P1 P2; ;.1 0/ W ! 7! ˚ 0

j Nil0.R Ꮾ1; Ꮾ2/ Š Nil0.R Ꮾ1Ꮾ2/ K0.R/; W I ! I ˚ ŒP1; P2; 1 P1 Ꮾ1P2; 2 P2 Ꮾ2P1 .ŒP1; 2 1; ŒP2 ŒᏮ2P1/ : W ! W ! 7! ı The reduced versions are the inverse isomorphisms

i Nilf 0.R Ꮾ1Ꮾ2/ Š Nilf 0.R Ꮾ1; Ꮾ2/;ŒP1; 12 ŒP1; Ꮾ2P1; 12; 1 W I ! I 7! j Nilf 0.R Ꮾ1; Ꮾ2/ Š Nilf 0.R Ꮾ1Ꮾ2/;ŒP1; P2; 1; 2 ŒP1; 2 1 W I ! I 7! ı with i .P1; 12/ .P1; Ꮾ2P1; 12; 1/ semisplit.  D

Proof This follows immediately from Theorem 1.1 and Theorem 2.7.

Remark 0.5 Theorem 0.4 was already known to Pierre Vogel in 1990[22].

1 Higher Nil–groups

In this section, we shall prove Theorem 0.4 for nonnegative degrees.

Quillen[17] defined the K–theory space KᏱ BQ.Ᏹ/ of an exact category Ᏹ. WD The space BQ.Ᏹ/ is the geometric realization of the simplicial set N Q.Ᏹ/, which is  the nerve of a certain category Q.Ᏹ/ associated to Ᏹ. The algebraic K–groups of Ᏹ are defined for Z as  2 K .Ᏹ/  .KᏱ/  WD 

Algebraic & Geometric Topology, Volume 11 (2011) Algebraic K–theory over the infinite dihedral group: an algebraic approach 2399 using a nonconnective delooping for 6 1. In particular, the algebraic K–groups of  a ring R are the algebraic K–groups K .R/ K .PROJ.R//  WD  of the exact category PROJ.R/ of finitely generated projective R–modules. The NIL–categories defined in the Introduction all have the structure of exact categories.

Theorem 1.1 Let Ꮾ1 and Ꮾ2 be bimodules over a ring R. Let j be the exact functor

j NIL.R Ꮾ1; Ꮾ2/ NIL.R Ꮾ1Ꮾ2/;.P1; P2; 1; 2/ .P1; 2 1/: W I ! I 7! ı (1) If Ꮾ2 is finitely generated projective as a left R–module, then there is an exact functor

i NIL.R Ꮾ1Ꮾ2/ NIL.R Ꮾ1; Ꮾ2/;.P; / .P; Ꮾ2P; ; 1/ W I ! I 7! such that i.P; / .P; Ꮾ2P; ; 1/ is semisplit, j i 1, and i and j induce D ı D   inverse isomorphisms on the reduced Nil-groups

Nil .R Ꮾ1Ꮾ2/ Nil .R Ꮾ1; Ꮾ2/:  I Š  I ˛ (2) If Ꮾ2 colim˛ I Ꮾ2 is a filtered colimit of bimodules each of which is finitely D 2 generated projective as a left R–module, then there is a unique exact functor i so that the following diagram commutes for all ˛ I : 2 i - NILe.R Ꮾ1Ꮾ2/ eNIL.R Ꮾ1; Ꮾ2/ I I 6 6

˛ i˛- ˛ NIL.R Ꮾ1Ꮾ / NIL.R Ꮾ1; Ꮾ /: I 2 I 2 Then j i 1 and i and j induce inverse isomorphisms on the reduced ı D   Nil-groups Nil .R Ꮾ1Ꮾ2/ Nil .R Ꮾ1; Ꮾ2/:  I Š  I Proof (1) Note that there are split injections of exact categories

PROJ.R/ PROJ.R/ NIL.R Ꮾ1; Ꮾ2/;.P1; P2/ .P1; P2; 0; 0/  ! I 7! PROJ.R/ NIL.R Ꮾ1Ꮾ2/;.P/ .P; 0/ ! eI e 7! which underlie the definition of the reduced Nil groups. Since both i and j take the image of the split injection to the image of the other split injection, they induce maps i and j on the reduced Nil groups. Since j i 1, it follows that j i 1.   ı D  ı  D

Algebraic & Geometric Topology, Volume 11 (2011) 2400 James F Davis, Qayum Khan and Andrew Ranicki

In preparation for the proof that i j 1, consider the following objects of  ı  D NIL.R Ꮾ1; Ꮾ2/: I x .P1; P2; 1; 2/ WD     0
x0 P1; Ꮾ2P1 P2; ; 1 2 WD ˚ 1

x .P1; Ꮾ2P1; 2 1; 1/ 00 WD ı a .0; P2; 0; 0/ WD a .0; Ꮾ2P1; 0; 0/ 0 WD with x semisplit. Note that .i j /.x/ x . Define morphisms 00 ı D 00  0 f 1; x x WD 1 W ! 0  
f 1; 1 2 x x 0 WD W 0 ! 00    2 g 0; a x WD 1 W ! 0   g 0; 1 0
x a 0 WD W 0 ! 0

h .0; 2/ a a : WD W ! 0 There are exact sequences

 f g  0 1 . g h / 0 x a x a 0 a 0 ! ˚ ! 0 ˚ ! 0 ! g f 0 a x 0 x 0 : ! ! 0 ! 00 ! Define exact functors F ; F ; G; G NIL.R Ꮾ1; Ꮾ2/ NIL.R Ꮾ1; Ꮾ2/ by 0 00 0W I ! I F .x/ x ; F .x/ x ; G.x/ a ; G .x/ a : 0 D 0 00 D 00 D 0 D 0 Thus we have two exact sequences of exact functors

0 1 G F G G 0 ! ˚ ! 0 ˚ ! 0 ! 0 G F F 0 : ! ! 0 ! 00 ! Recall j i 1, and note i j F . By Quillen’s Additivity Theorem[17, page 98, ı D ı D 00 Corollary 1], we obtain homotopies KF 1 KG and KF KG KF . Then 0 ' C 0 0 ' C 00 Ki Kj KF 1 .KG KG/; ı D 00 ' C 0

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where the subtraction uses the loop space structure. Observe that both G and G0 send NIL.R Ꮾ1; Ꮾ2/ to the image of PROJ.R/ PROJ.R/. Thus i j 1 as desired. I   ı  D (2) It is straightforward to show that tensor product commutes with colimits over a category. Moreover, for any object x .P1; P2; 1 P1 Ꮾ1P2; 2 P2 Ꮾ2P1/, D W ! W ! since P2 is finitely generated, there exists ˛ I such that 2 factors through a map ˛ 2 P2 Ꮾ P1 , and similarly for short exact sequences of nil-objects. We thus obtain ! 2 induced isomorphisms of exact categories: ˛ colim NIL.R Ꮾ1Ꮾ2 / NIL.R Ꮾ1Ꮾ2/ ˛ I I ! I 2 ˛ colim NIL.R Ꮾ1; Ꮾ2 / NIL.R Ꮾ1; Ꮾ2/: ˛ I I ! I 2 This justifies the existence and uniqueness of the functor i . By Quillen’s colimit observation[17, Section 2, Equation (9), page 20], we obtain induced weak homotopy equivalences of K–theory spaces: ˛ colim K NIL.R Ꮾ1Ꮾ2 / K NIL.R Ꮾ1Ꮾ2/ ˛ I I ! I 2 ˛ colim K NIL.R Ꮾ1; Ꮾ2 / K NIL.R Ꮾ1; Ꮾ2/: ˛ I I ! I 2 The remaining assertions of part (2) then follow from part (1).

Remark 1.2 The proof of Theorem 1.1 is best understood in terms of finite chain complexes x .P1; P2; 1; 2/ in the category NIL.R Ꮾ1; Ꮾ2/, assuming that Ꮾ2 D I is a finitely generated projective left R–module. Any such x represents a class

X1 r Œx . 1/ Œ.P1/r ;.P2/r ; 1; 2 Nil0.R Ꮾ1; Ꮾ2/: D 2 I r 0 D The key observation is that x determines a finite chain complex x .P ; P ;  ;  / 0 D 10 20 10 20 in NIL.R Ꮾ1; Ꮾ2/ which is semisplit in the sense that  P Ꮾ2P is a chain I 20 W 20 ! 10 equivalence, and such that

(3) Œx Œx  Nil0.R Ꮾ1; Ꮾ2/: D 0 2 I Specifically, let P P1 , P ᏹ.2/, the algebraic mapping cylinder of the chain 10 D 20 D map 2 P2 Ꮾ2P1 , and let W ! 0 0 1

0 @ 0 A P 0 P1 Ꮾ1P 0 ᏹ.1Ꮾ1 2/ 1 D W 1 D e ! 2 D ˝ 1
 1 0 2 P ᏹ.2/ Ꮾ2P1 ; 20 D W 20 D !

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so that P =P2 Ꮿ.2/ is the algebraic mapping cone of 2 . Moreover, the proof 20 D of (3) is sufficiently functorial to establish not only that the following maps of the reduced nilpotent class groups are inverse isomorphisms:

i Nilf 0.R Ꮾ1Ꮾ2/ Nilf 0.R Ꮾ1; Ꮾ2/;.P; / .P; Ꮾ2P; ; 1/ W I ! I 7! j Nilf 0.R Ꮾ1; Ꮾ2/ Nilf 0.R Ꮾ1Ꮾ2/;Œx Œx0 ; W I ! I 7! but also that there exist isomorphisms of Niln for all higher dimensions n > 0, as shown above. In order to prove Equation (3), note that x fits into the sequence .1;u/ .0;v/ (4) 0 / x / x0 / y / 0 with y .0; Ꮿ.2/; 0; 0/ D 001 e u @0A P2 P 0 ᏹ.2/ D W ! 2 D 1 1 0 0 v P ᏹ.2/ Ꮿ.2/ D 0 1 0 W 20 D !

X1 r and Œy . / Œ0;.Ꮾ2P1/r 1 .P2/r ; 0; 0 0 Nilf 0.R Ꮾ1; Ꮾ2/: D ˚ D 2 I r 0 D The projection ᏹ.2/ Ꮾ2P1 defines a chain equivalence ! x .P1; Ꮾ2P1; 2 1; 1/ ij .x/ 0 ' ı D so that Œx Œx0 Œy ŒP1; Ꮾ2P1; 2 1; 1 ij Œx Nilf 0.R Ꮾ1; Ꮾ2/: D D ı D 2 I Now suppose that x is a 0–dimensional chain complex in NIL.R Ꮾ1; Ꮾ2/, that is, an I object as in the proof of Theorem 1.1. Let x0; x00; a; a0; f; f 0; g; g0; h be as defined there. The exact sequence of (4) can be written as the short exact sequence of chain complexes a a

g h f  g0  0 / x / x0 / a0 / 0 : The first exact sequence of the proof of Theorem 1.1 is now immediate:  f g  0 1 . g0 h / 0 / x a / x a / a / 0 : ˚ 0 ˚ 0 The second exact sequence is self-evident:

g f 0 0 / a / x0 / x00 / 0 :

Algebraic & Geometric Topology, Volume 11 (2011) Algebraic K–theory over the infinite dihedral group: an algebraic approach 2403

2 Lower Nil–groups

2.1 Cone and suspension rings

Let us recall some additional structures on the tensor product of modules.

Originating from ideas of Karoubi and Villamayor[11], the following concept was studied independently by S M Gersten[8] and J B Wagoner[23] in the construction of the nonconnective K–theory spectrum of a ring.

Definition 2.1 (Gersten, Wagoner) The cone ring ƒZ is the subring of .! !/–  matrices over Z such that each row and column have only a finite number of nonzero entries. The suspension ring †Z is the quotient ring of ƒZ by the two-sided ideal of matrices with only a finite number of nonzero entries. For each n N , define the rings 2 n 0 † Z †Z Z Z †Z with † Z Z : WD „ ˝ ƒ‚


˝ … D n copies

n n For a ring R and for n N , define the ring † R † Z Z R. 2 WD ˝

Roughly speaking, the suspension should be regarded as the ring of “bounded modulo n compact operators.” Gersten and Wagoner showed that Ki.† R/ is naturally isomor- phic to Ki n.R/ for all i; n Z, in the sense of Quillen when the subscript is positive, 2 in the sense of Grothendieck when the subscript is zero, and in the sense of Bass when the subscript is negative.

n n n For an R–bimodule Ꮾ, define the † R–bimodule † Ꮾ † Z Z Ꮾ. WD ˝

Lemma 2.2 Let R be a ring. Let Ꮾ1; Ꮾ2 be R–bimodules. Then, for each n N , 2 there is a natural isomorphism of †nR–bimodules:

n n n tn † .Ꮾ1Ꮾ2/ † Ꮾ1 †nR † Ꮾ2 ; s .b1 b2/ .s b1/ .1†nZ b2/: W ! ˝ ˝ ˝ 7! ˝ ˝ ˝

Proof By transposition of the middle two factors, note that

n n n n n n † Ꮾ1 † R † Ꮾ2 .† Z Z Ꮾ1/ .† Z ZR/ .† Z Z Ꮾ2/ ˝ D ˝ ˝ ˝ ˝ is isomorphic to

n n n n .† Z †nZ † Z/ Z .Ꮾ1Ꮾ2/ † Z Z .Ꮾ1Ꮾ2/ † .Ꮾ1Ꮾ2/: ˝ ˝ D ˝ D

Algebraic & Geometric Topology, Volume 11 (2011) 2404 James F Davis, Qayum Khan and Andrew Ranicki

2.2 Definition of lower Nil–groups

Definition 2.3 Let R be a ring. Let Ꮾ be an R–bimodule. For all n N , define 2 n n Nil n.R Ꮾ/ Nil0.† R † Ꮾ/ I WD I n n Nilf n.R Ꮾ/ Nilf 0.† R † Ꮾ/: I WD I

Definition 2.4 Let R be a ring. Let Ꮾ1; Ꮾ2 be R–bimodules. For all n N , define 2 n n n Nil n.R Ꮾ1; Ꮾ2/ Nil0.† R † Ꮾ1;† Ꮾ2/ I WD I n n n Nilf n.R Ꮾ1; Ꮾ2/ Nilf 0.† R † Ꮾ1;† Ꮾ2/: I WD I

The next two theorems follow from the definitions and[26, Theorems 1,3].

Theorem 2.5 (Waldhausen) Let R be a ring and Ꮾ be an R–bimodule. Consider the tensor ring 2 3 TR.Ꮾ/ R Ꮾ Ꮾ Ꮾ : WD ˚ ˚ ˚ ˚


Suppose Ꮾ is finitely generated projective as a left R–module and free as a right R–module. Then, for all n N , there is a split monomorphism 2

B Nil n.R Ꮾ/ K1 n.TR.Ꮾ// W I ! given for n 0 by the map D

B Nil0.R Ꮾ/ K1.TR.Ꮾ// ; ŒP;  Œ TR.Ꮾ/P; 1   ; W I ! 7! y where  is defined using  ande multiplication in TR.Ꮾ/. y Furthermore, there is a natural decomposition

K1 n.TR.Ꮾ// K1 n.R/ Nil n.R Ꮾ/: D ˚ I

For example, the last assertion of the theorem follows from the equations

n K1 n.TR.Ꮾ// K1.† TR.Ꮾ// D n K1.T†nR.† Ꮾe// D n n n K1.† R/ Nilf 0.† R † Ꮾ/ D ˚ I K1 n.R/ Nilf n.R Ꮾ/: D ˚ I

Algebraic & Geometric Topology, Volume 11 (2011) Algebraic K–theory over the infinite dihedral group: an algebraic approach 2405

Theorem 2.6 (Waldhausen) Let R; A1; A2 be rings. Let R Ai be ring monomor- ! phisms such that Ai R Ꮾi for R–bimodules Ꮾi . Consider the pushout of rings D ˚

A A1 R A2 D  R .Ꮾ1 Ꮾ2/ .Ꮾ1Ꮾ2 Ꮾ2Ꮾ1/ .Ꮾ1Ꮾ2Ꮾ1 Ꮾ2Ꮾ1Ꮾ2/ : D ˚ ˚ ˚ ˚ ˚ ˚ ˚

Suppose each Ꮾi is free as a right R–module. Then, for all n N , there is a split 2 monomorphism

A Nil n.R Ꮾ1; Ꮾ2/ K1 n.A/; W I ! given for n 0 by the map D

Nil0.R Ꮾ1; Ꮾ2/ K1.A/; I !    1 2 ŒP1; P2; 1e; 2 .AP1/ .AP2/; y ; 7! ˚ 1 1 y where i is defined using i and multiplication in Ai for i 1; 2. y D Furthermore, there is a natural Mayer–Vietoris type exact sequence:

@ K1 n.R/ K1 n.A1/ K1 n.A2/


! ! ˚ ! K1 n.A/ @ K n.R/ Nilf n.R Ꮾ1; Ꮾ2/ ! !


I

2.3 The isomorphism for lower Nil–groups

Theorem 2.7 Let R be a ring. Let Ꮾ1; Ꮾ2 be R–bimodules. Suppose that Ꮾ2 ˛ ˛ D colim˛ I Ꮾ2 is a filtered colimit of R–bimodules Ꮾ2 , each of which is a finitely gen- erated projective2 left R–module. Then, for all n N , there is an induced isomorphism 2

Nil n.R Ꮾ1Ꮾ2/ K n.R/ Nil n.R Ꮾ1; Ꮾ2/: I ˚ ! I

Proof Let n N . By Lemma 2.2 and Theorem 1.1, there are induced isomorphisms 2 n n n Nil n.R Ꮾ1Ꮾ2/ K n.R/ Nil0.† R † .Ꮾ1Ꮾ2// K0† .R/ I ˚ D I ˚ n n n n Nil0.† R † Ꮾ1 †nR † Ꮾ2/ K0† .R/ ! I ˝ ˚ n n n Nil0.† R † Ꮾ1;† Ꮾ2/ Nil n.R Ꮾ1; Ꮾ2/: ! I D I

Algebraic & Geometric Topology, Volume 11 (2011) 2406 James F Davis, Qayum Khan and Andrew Ranicki

3 Applications

We indicate some applications of our main theorem 0.4. In Section 3.1 we prove Theorem 0.1(ii), which describes the restrictions of the maps ! ! K .RŒG/ K .RŒG/ ;  K .RŒG/ K .RŒG/ W  x !  W  !  x to the Nil –terms, with  G G the inclusion of the canonical index 2 subgroup G W x ! x for any group G over D . In Section 3.2 we give the first known example of a nonzero Nil–group occurring in the1 K–theory of an integral group ring of an amalgamated free product. In Section 3.3 we sharpen the Farrell–Jones Conjecture in K–theory, replacing the family of virtually cyclic groups by the smaller family of finite-by-cyclic groups. In Section 3.4 we compute the K .RŒ€/ for the modular group € PSL2.Z/. e  D 3.1 Algebraic K –theory over D 1 The overall goal here is to show that the abstract isomorphisms i and j coincide !  with the restrictions of the induction and transfer maps ! and  in the group ring setting.

3.1.1 Twisting We start by recalling the algebraic K–theory of twisted polynomial rings.

Statement 3.1 Consider any (unital, associative) ring R and any ring automorphism ˛ R R. Let t be an indeterminate over R such that W ! rt t˛.r/.r R/: D 2 For any R–module P , let tP tx x P be the set with left R–module structure WD f j 2 g tx ty t.x y/; r.tx/ t.˛.r/x/ tP : C D C D 2 Further endow the left R–module tR with the R–bimodule structure R tR R tR ;.q; tr; s/ t˛.q/rs :   ! 7! The Nil–category of R with respect to ˛ is the exact category defined by NIL.R; ˛/ NIL.R tR/: WD I The objects .P; / consist of any finitely generated projective R–module P and any nilpotent morphism  P tP tRP . The Nil–groups are written W ! D Nil .R; ˛/ Nil .R tR/; Nilf .R; ˛/ Nilf .R tR/;  WD  I  WD  I so that Nil .R; ˛/ K .R/ Nilf .R; ˛/ :  D  ˚ 

Algebraic & Geometric Topology, Volume 11 (2011) Algebraic K–theory over the infinite dihedral group: an algebraic approach 2407

Statement 3.2 The tensor algebra on tR is the ˛–twisted polynomial extension of R

X1 k TR.tR/ R˛Œt t R : D D k 0 D Given an R–module P there is induced an R˛Œt–module

R˛Œt R P P˛Œt ˝ D P j whose elements are finite linear combinations j1 0 t xj (xj P ). Given R–modules D 2 P; Q and an R–module morphism  P tQ, define its extension as the R˛Œt– W ! module morphism

X1 j X1 j  t P˛Œt Q˛Œt ; t xj t .xj /: y D W ! 7! j 0 j 0 D D Statement 3.3 Bass[2], Farrell and Hsiang[6] and Quillen[9] give decompositions

Kn.R˛Œt/ Kn.R/ Nilf n 1.R; ˛/ D ˚ 1 1 Kn.R˛ 1 Œt / Kn.R/ Nilf n 1.R; ˛ / D ˚ 1 1 Kn.R˛Œt; t / Kn.1 ˛ R R/ Nilf n 1.R; ˛/ Nilf n 1.R; ˛ /: D W ! ˚ ˚ In particular for n 1, by Theorem 2.5, there are defined split monomorphisms D  C Nilf 0.R; ˛/ K1.R˛Œt/ ; ŒP;  ŒP˛Œt; 1 t B W ! 7! 1 1  1 1   Nilf 0.R; ˛ / K1.R˛ 1 Œt / ; ŒP;  P˛ 1 Œt ; 1 t  B W ! 7!
1 1 B  C  Nilf 0.R; ˛/ Nilf 0.R; ˛ / K1.R˛Œt; t / ; D C B B W ˚ !    1 1 t1 0 .ŒP1; 1; ŒP2; 2/ .P1 P2/˛Œt; t ; 1 : 7! ˚ 0 1 t 2 These extend to all integers n 6 1 by the suspension isomorphisms of Section 2.

3.1.2 Scaling Next, consider the effect an inner automorphism on ˛.

Statement 3.4 Suppose ˛; ˛ R R are automorphisms satisfying 0W ! ˛ .r/ u˛.r/u 1 R .r R/ 0 D 2 2 for some unit u R, and that t is an indeterminate over R satisfying 2 0 rt t ˛ .r/.r R/: 0 D 0 0 2

Algebraic & Geometric Topology, Volume 11 (2011) 2408 James F Davis, Qayum Khan and Andrew Ranicki

Denote the canonical inclusions 1 1 1 C R˛Œt R˛Œt; t  ; R˛ 1 Œt  R˛Œt; t  ; W ! W ! 1 1 1 0C R˛ Œt 0 R˛ Œt 0; t 0  ; 0 R˛ 1 Œt 0  R˛ Œt 0; t 0  : W 0 ! 0 W 0 ! 0 Statement 3.5 The various polynomial rings are related by scaling isomorphisms

ˇC R˛Œt R˛ Œt 0 ; t t 0u u W ! 0 7! 1 1 1 1 1 ˇu R˛ 1 Œt  R˛ 1 Œt 0  ; t u t 0 W ! 0 7! 1 1 ˇu R˛Œt; t  R˛ Œt 0; t 0  ; t t 0u W ! 0 7! satisfying the equations

1 ˇu C 0C ˇC R˛Œt R˛ Œt 0; t 0  ı D ı u W ! 0 1 1 ˇu 0 ˇu R˛ 1 Œt  R˛ Œt 0; t 0  : ı D ı W ! 0 Statement 3.6 There are corresponding scaling isomorphisms of exact categories

ˇ NIL.R; ˛/ NIL.R; ˛ /;.P; / .P; t ut 1 P t P/ uCW ! 0 7! 0 W ! 0 ˇ NIL.R; ˛ 1/ NIL.R; ˛ 1/;.P; / .P; t 1ut P t 1P /; uW ! 0 7! 0 W 0 ! 0 0 where we mean

.t ut 1/.x/ t .uy/ with .x/ ty 0 WD 0 D .t 1ut/.x/ t 1.uy/ with .x/ t 1y : 0 WD 0 D Statement 3.7 For all n 6 1, the various scaling isomorphisms are related by equations

.ˇuC/ BC B0C ˇuC Nilf n 1.R; ˛/ Kn.R˛ Œt 0/  ı D ı W ! 0 1 1 .ˇ/   0 ˇ Niln 1.R; ˛ / Kn.R 1 Œt 0 / u B B u f ˛0  ı D ı W  ! ˇuC 0 1 .ˇu/ B B0 Nilf n 1.R; ˛/ Nilf n 1.R; ˛ /  ı D ı 0 ˇu W ˚ 1 Kn.R˛ Œt 0; t 0 / : ! 0 3.1.3 Group rings We now adapt these isomorphisms to the case of group rings RŒG of groups G over the infinite dihedral group D . In order to prove Lemma 3.20 1 and Proposition 3.23, the overall idea is to transform information about the product t2t1 1 1 arising from the transposition Ꮾ2 Ꮾ1 into information about the product t t ˝ 2 1 arising in the second Nil –summand of the twisted Bass decomposition. We continue to discuss the ingredients in a sequence of statements.

Algebraic & Geometric Topology, Volume 11 (2011) e Algebraic K–theory over the infinite dihedral group: an algebraic approach 2409

Statement 3.8 Let F be a group, and let ˛ F F be an automorphism. Recall that W ! the injective HNN extension F Ì˛ Z is the set F Z with group multiplication  m .x; n/.y; m/ .˛ .x/y; m n/ F Ì˛ Z : WD C 2 n Then, for any ring R, writing t .1F ; 1/ and .x; n/ t x F Ì˛ Z, we have D D 2 1 RŒF Ì˛ Z RŒF˛Œt; t  : D

Statement 3.9 Consider any group G G1 F G2 over D , where D  1 F G1 G2 G F Ì˛ Z F Ì˛ Z G G1 F G2 : D \  x D D 0  D  Fix elements t1 G1 F , t2 G2 F , and define elements 2 2 1 1 t t1t2 G ; t t2t1 G ; u .t / t F : WD 2 x 0 WD 2 x WD 0 2 Define the automorphisms

1 ˛1 F F ; x .t1/ xt1 W ! 7! 1 ˛2 F F ; x .t2/ xt2 W ! 7! 1 ˛ ˛2 ˛1 F F ; x t xt WD ı W ! 7! 1 ˛ ˛1 ˛2 F F ; x t xt 0 WD ı W ! 7! 0 0 such that

xt t˛.x/; xt t ˛ .x/ ; ˛ .x/ u˛ 1.x/u 1 .x F/: D 0 D 0 0 0 D 2 1 In particular, note ˛0 and ˛ (not ˛) are related by inner automorphism by u.

Statement 3.10 Denote the canonical inclusions 1 1 1 C R˛Œt R˛Œt; t  ; R˛ 1 Œt  R˛Œt; t  W ! W ! 1 1 1 0C R˛ Œt 0 R˛ Œt 0; t 0  ; 0 R˛ 1 Œt 0  R˛ Œt 0; t 0  : W 0 ! 0 W 0 ! 0 The inclusion RŒF RŒG extends to ring monomorphisms ! 1 1  RŒF˛Œt; t  RŒG ;  0 RŒF˛ Œt 0; t 0  RŒG W ! W 0 ! such that

im./ im. / RŒG RŒG RŒG1 RŒG2 : D 0 D x  D RŒF  Furthermore, the inclusion RŒF RŒG extends to ring monomorphisms !   C RŒF˛Œt RŒG ; 0  0 0C RŒF˛ Œt 0 RŒG : D ı W ! D ı W 0 !

Algebraic & Geometric Topology, Volume 11 (2011) 2410 James F Davis, Qayum Khan and Andrew Ranicki

Statement 3.11 By Statement 3.5, there are defined scaling isomorphisms of rings 1 1 ˇuC RŒF˛ 1 Œt  RŒF˛ Œt 0 ; t t 0u W ! 0 7! 1 1 1 ˇu RŒF˛Œt RŒF˛ 1 Œt 0  ; t u t 0 W ! 0 7! 1 1 1 1 ˇu RŒF˛Œt; t  RŒF˛ Œt 0; t 0  ; t u t 0 W ! 0 7! which satisfy the equations 1 1 ˇu 0C ˇuC RŒF˛ 1 Œt  RŒF˛ Œt 0; t 0  ı D ı W ! 0 1 ˇu C 0 ˇ RŒF˛Œt RŒF˛ Œt 0; t 0  ı D ı u W ! 0 1   ˇu RŒF˛Œt; t  RŒG : D 0 ı W ! Statement 3.12 By Statement 3.6, there are scaling isomorphisms of exact categories ˇ NIL.RŒF; ˛ 1/ NIL.RŒF; ˛ /.P; / .P; t ut/ ; uCW ! 0 7! 0 ˇ NIL.RŒF; ˛/ NIL.RŒF; ˛ 1/;.P; / .P; t 1ut 1/ : uW ! 0 7! 0 Statement 3.13 By Statement 3.7, for all n 6 1, the various scaling isomorphisms are related by 1 .ˇuC/ B B0C ˇuC Nilf 1.RŒF; ˛ / K .RŒF˛ Œt 0/  ı D ı W  !  0 1 .ˇu/ BC B0 ˇu Nilf 1.RŒF; ˛/ K .RŒF˛ 1 Œt 0 /  ı D ı W  !  0   0 ˇuC 1 .ˇu/ B B0 Nilf 1.RŒF; ˛/ Nilf 1.RŒF; ˛ /  ı D ı ˇu 0 W  ˚  1 K .RŒF˛ Œt 0; t 0 / : !  0

3.1.4 Transposition Next, we study the effect of transposition of the bimodules Ꮾ1 1 and Ꮾ2 in order to relate ˛ and ˛0 . In particular, there is no mention of ˛ in this section.

Statement 3.14 The RŒF–bimodules

Ꮾ1 RŒG1 F t1RŒF ; Ꮾ2 RŒG2 F t2RŒF D D D D are free left and right RŒF–modules of rank one. The RŒF–bimodule isomorphisms

Ꮾ1 Ꮾ2 tRŒF ; t1x1 t2x2 t˛2.x1/x2 ˝RŒF  ! ˝ 7! Ꮾ2 Ꮾ1 t RŒF ; t2x2 t1x1 t ˛1.x2/x1 ˝RŒF  ! 0 ˝ 7! 0 shall be used to make the identifications

Ꮾ1 Ꮾ2 tRŒF ; NIL.RŒF Ꮾ1 Ꮾ2/ NIL.RŒF; ˛/ ; ˝RŒF  D I ˝RŒF  D Ꮾ2 Ꮾ1 t RŒF ; NIL.RŒF Ꮾ2 Ꮾ1/ NIL.RŒF; ˛ /: ˝RŒF  D 0 I ˝RŒF  D 0

Algebraic & Geometric Topology, Volume 11 (2011) Algebraic K–theory over the infinite dihedral group: an algebraic approach 2411

Statement 3.15 Theorem 0.4 gives inverse isomorphisms

i Nilf .RŒF; ˛/ Nilf .RŒF Ꮾ1; Ꮾ2/ W  !  I j Nilf .RŒF Ꮾ1; Ꮾ2/ Nilf .RŒF; ˛/ W  I !  which for 0 are given by  D i Nilf 0.RŒF; ˛/ Nilf 0.RŒF Ꮾ1; Ꮾ2/;ŒP;  ŒP; t2P; ; 1 W ! I 7! j Nilf 0.RŒF Ꮾ1; Ꮾ2/ Nilf 0.RŒF; ˛/ ; ŒP1; P2; 1; 2 ŒP1; 2 1 : W I ! 7! ı Statement 3.16 Similarly, there are defined inverse isomorphisms

i 0 Nilf .RŒF; ˛0/ Nilf .RŒF Ꮾ2; Ꮾ1/ W  !  I j 0 Nilf .RŒF Ꮾ2; Ꮾ1/ Nilf .RŒF; ˛0/ W  I !  which for 0 are given by  D i 0 Nilf 0.RŒF; ˛0/ Nilf 0.RŒF Ꮾ2; Ꮾ1/;ŒP 0; 0 ŒP 0; t1P 0; 0; 1 W ! I 7! j 0 Nilf 0.RŒF Ꮾ2; Ꮾ1/ Nilf 0.RŒF; ˛0/;ŒP2; P1; 2; 1 ŒP2; 1 2 : W I ! 7! ı Statement 3.17 The transposition isomorphism of exact categories

A NIL.RŒF Ꮾ1; Ꮾ2/ NIL.RŒF Ꮾ2; Ꮾ1/; W I ! I .P1; P2; 1; 2/ .P2; P1; 2; 1/ 7! induces isomorphisms

A Nil .RŒF Ꮾ1; Ꮾ2/ Nil .RŒF Ꮾ2; Ꮾ1/ W  I Š  I A Nilf .RŒF Ꮾ1; Ꮾ2/ Nilf .RŒF Ꮾ2; Ꮾ1/: W  I Š  I Note, by Theorem 0.4, the composites

B j 0 A i Nilf .RŒF; ˛/ Nilf .RŒF; ˛0/ WD ı ı W  !   1 B0 j A i 0 Nilf .RŒF; ˛0/ Nilf .RŒF; ˛/ WD  ı ı W  !  are inverse isomorphisms, which for 0 are given by  D B Nilf 0.RŒF; ˛/ Nilf 0.RŒF; ˛0/;ŒP;  Œt2P; t2 W ! 7!  0 Nilf 0.RŒF; ˛0/ Nilf 0.RŒF; ˛/ ; ŒP 0; 0 Œt1P 0; t10 : BW ! 7! Furthermore, note that the various transpositions are related by the equation

A i i 0 B Nil .RŒF; ˛/ Nil .RŒF Ꮾ2; Ꮾ1/: ı  D  ı W  !  I

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Statement 3.18 Recall from Theorem 2.6 that there is a split monomorphism

A Niln 1.RŒF Ꮾ1; Ꮾ2/ Kn.RŒG/ W I ! such that the n 1 case is given by D

A Nilf 0.RŒF Ꮾ1; Ꮾ2/ K1.RŒG/ ; W I !    e 1 t22 ŒP1; P2; 1; 2 P1ŒG P2ŒG; : 7! ˚ t11 1

Elementary row and column operations produce an equivalent representative:         1 t22 1 t22 1 0 1 t21 0 : 0 1 t11 1 1 1 D 0 1 Thus the n 1 case satisfies the equations (similarly for the second equality) D   AŒP1; P2; 1; 2 ŒP1ŒG; 1 t21 P2ŒG; 1 t 12 : D D 0 6 Therefore for all n 1, the split monomorphism A0 , associated to the amalgamated free product G G2 F G1 , satisfies the equation D 

A A0 A Niln 1.RŒF Ꮾ1; Ꮾ2/ Kn.RŒG/ : D ı W I !

3.1.5 Induction We analyze the effect of induction maps on Nil –summands.

Statement 3.19 Recall from Theoreme 0.4 the isomorphism

i Nilf 1.RŒF; ˛/ Nilf 1.RŒF Ꮾ1 RŒF  Ꮾ2/ Nilf 1.RŒF Ꮾ1; Ꮾ2/; W  D  I ˝ ! e I ŒP;  ŒP; t2P; ; 1 : 7! Let .P; / be an object in the exact category NIL.RŒF; ˛/. By Statement 3.18, note

Ai ŒP;  AŒP; t2P; ; 1 ŒPŒG; 1 t !BCŒP;  :  D D D Thus, for all n 6 1, we obtain the key equality

A i ! BC Niln 1.RŒF; ˛/ Kn.RŒG/ : ı  D ı W !

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6 Lemma 3.20 Let n 1 be an integer. The split monomorphisms A; A0 ; BC; B0C are related by a commutative diagram

BC Nilf n 1.RŒF; ˛/ / / Kn.RŒF˛Œt/

i  !C Š ! $  1 Nilf n 1.RŒF Ꮾ1; Ꮾ2/ Kn.RŒF˛Œt; t / I ( A !

( Ó w B A Kn.RŒG/ .ˇu/! Š Š 6 [ g Š A0 !0

 6  1 Nilf n 1.RŒF Ꮾ2; Ꮾ1/ Kn.RŒF˛ Œt 0; t 0 / : I O 0

!0 i0  !0C Š

 B0C / / K .RŒF Œt / Nilf n 1.RŒF; ˛0/ n ˛0 0

Proof Commutativity of the various parts follow from the following implications:

Statement 3.10 gives   and   .  ! D ! ı !C !0 D !0 ı !0C Statement 3.11 gives   .ˇu/ .  ! D !0 ı !

Statement 3.17 gives A i i 0 B .  ı  D  ı Statement 3.18 gives A  A .  D A0 ı

Statement 3.19 gives A i ! BC and A0 i 0 !0 B0C .  ı  D ı ı  D ı

Observe the action of G=G on Kn.RŒG/ is inner, hence is trivial. However, the action x 1 of C2 G=G on Kn.RŒG/ is outer, induced by, say c1 G G , y t1y.t1/ . D x x W x ! x 7! (Note that c may not have order two.) This C –action on K .RŒG/ is nontrivial, as 1 2 n x follows.

Algebraic & Geometric Topology, Volume 11 (2011) 2414 James F Davis, Qayum Khan and Andrew Ranicki

Proposition 3.21 Let n 1 be an integer. The induced map  is such that there is a Ä ! commutative diagram

 1 B- Niln 1.RŒF; ˛/ Niln 1.RŒF; ˛ / Kn.RŒG/ ˚ x

1 . i A i0 ˇuC / !   ? ? A - e Niln 1.RŒFe Ꮾ1; Ꮾ2/ Kn.RŒG/ : I Furthermore, there is a C2 –action on the upper left hand corner which interchanges the two Nil–summands, and all maps are C2 –equivariant. Here, the action of C2 G=G D x on the upper right is given by .c1/! , and the C2 –action on each lower corner is trivial. Proof First, we checke commutativity of the square on each Nil–summand:

Lemma 3.20 gives A i ! BC ! !C BC ! B Niln 1.RŒF; ˛/.  ı  D ı D ı ı D ı j Statements 3.13 and 3.11 give   1 i ˇ  i ˇ   ˇ  A A 0 uC A0 0 uC !0 B0C uC 1 ı ı ı D ı  ı D ı ı 1 D ! .ˇu/! !0C .ˇuC/! B ! ! B ! B Nilf n 1.RŒF; ˛ /. ı ı ı ı D ı ı D ı j Next, define the involution  0 "  1  e " 0  1 on Niln 1.RŒF; ˛/ Niln 1.RŒF; ˛ / by ˚ " .˛ 1/ ˇ NIL.RŒF; ˛ 1/ NIL.RŒF; ˛/ : WD 1 ! ı uCW ! 1 1 Here, the automorphism ˛1 F F was defined in Statement 3.9 by x t1x.t1/ W ! 1 7! and is the restriction of c1 . It remains to show B and . i A i0 ˇuC / are C2 –equivariant,   thate is, e 

(5) .c1/   " ! ı B D C BC ı 1 (6) A i 0 ˇuC i ":   D  ı 1 1 Observe that the induced ring automorphism .c1/ RŒF˛Œt; t  RŒF˛Œt; t  !W ! restricts to a ring isomorphism 1 1 1 1 .c1/C RŒF˛ 1 Œt  RŒF˛Œt ; x ˛ .x/; t t ˛ .u/: ! W ! 7! 1 7! 1 Then .c1/ .c1/ . So (5) follows from the commutative square ! ı D C ı !C 1 1 .c1/!C B BC .˛1 /!ˇuC Niln 1.RŒF; ˛ / Kn.RŒF˛Œt/ ; ı D ı W ! which can be verified by formulas for n 1 and extends to n < 1 by variation of R. D

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Observe that (6) follows from the existence of an exact natural transformation

1 1 T  i i .˛ / NIL.RŒF; ˛ / NIL.RŒF t1RŒF; t2RŒF/ W A ı 0 ! ı 1 !W 0 ! I defined on objects .P;  P t P t2t1P/ by the rule W ! 0 D

T .1; / .t1P; P; 1; / .t1P; t P; t1; 1/: .P;/ WD W ! 0

A key observation from Statement 3.9 is the isomorphism RŒF c P t1P sending ˝ 1 ! x p ˛1.x/p. ˝ 7!

3.1.6 Transfer We analyze the effect of transfer maps on Nil –summands.

Statement 3.22 Given an RŒG–module M , let M ! be the abelian group M with RŒG–action the restriction of the RŒG–action. The transfer functor of exact categories x  ! PROJ.RŒG/ PROJ.RŒG/ ; M M ! W ! x 7!e induces the transfer maps in algebraic K–theory

 ! K .RŒG/ K .RŒG/ : W  !  x The exact functors of Theorem 0.4 combine to give an exact functor

 j  NIL.RŒF Ꮾ1; Ꮾ2/ NIL.RŒF; ˛/ NIL.RŒF; ˛0/; j 0 W I ! 
ŒP1; P2; 1; 2 ŒP1; 2 1; ŒP2; 1 2 7! ı ı inducing a map between reduced Nil–groups

j   Nil .RŒF Ꮾ1; Ꮾ2/ Nil .RŒF; ˛/ Nil .RŒF; ˛0/: j 0 W  I !  ˚  

Proposition 3.23 Let n 6 1 be an integer. The transfer map  ! restricts to the isomorphism j in a commutative diagram  e e e A Nilf n 1.RŒF Ꮾ1; Ꮾ2/ / / Kn.RŒG/ I  j  !  1  .ˇuC/ j 0   . CBC ˇu B /  1 Nilf n 1.RŒF; ˛/ Nilf n 1.RŒF; ˛ / / / Kn.RŒG/ : ˚ x

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Proof Using the suspension isomorphisms of Section 2, we may assume n 1. D Let .P1; P2; 1; 2/ be an object in NIL.RŒF Ꮾ1; Ꮾ2/. Define an RŒG–module I automorphism   1 t22 f P1ŒG P2ŒG P1ŒG P2ŒG : WD t11 1 W ˚ ! ˚

By Theorem 2.6, we have Œf  AŒP1; P2; 1; 2 K1.RŒG/. Note the transfer is D 2 0 1 1 t22 0 0 ! Bt11 1 0 0 C  .f / B C D @ 0 0 1 t11A 0 0 t22 1 as an RŒG–module automorphism of P1ŒG t1P2ŒG P2ŒG t1P1ŒG. Further- x x ˚ x ˚ x ˚ x more, elementary row and column operations produce a diagonal representation: 0 1 0 1 0 1 1 t22 0 0 1 0 0 0 1 t 21 0 0 0 0 B0 1 0 0 C ! B t11 1 0 0C B 0 1 0 0C B C  .f / B C B C : @0 0 1 t11A @ 0 0 1 0A D @ 0 0 1 t12 0A 0 0 0 1 0 0 t22 1 0 0 0 1 ! So  Œf  Œ1 t 21 Œ1 t12. Thus we obtain a commutative diagram D 0 C A Nilf 0.RŒF Ꮾ1; Ꮾ2/ / / K1.RŒG/ I   j !   j 0   . CBC 0CB0C /  Nilf 0.RŒF; ˛/ Nilf 0.RŒF; ˛ / / / K1.RŒG/ : ˚ 0 x Finally, by Statement 3.13 and Statement 3.11, note

 ˇ ˇ  ˇu  : 0C ı B0C ı uC D 0C ı uC ı B D ı ı B 3.2 Waldhausen Nil

Examples of bimodules originate from group rings of amalgamated product of groups.

Definition 3.24 A subgroup H of a group G is almost-normal if H H xHx 1 < j W \ j 1 for every x G . In other words, H is commensurate with all its conjugates. Equiva- 2 lently, H is an almost-normal subgroup of G if every .H; H/–double coset H xH is both a union of finitely many left cosets gH and a union of finitely many right cosets Hg.

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Remark 3.25 Almost-normal subgroups arise in the Shimura theory of automorphic functions, with .G; H / called a Hecke pair. Here are two sufficient conditions for a subgroup H G to be almost-normal: if H is a finite-index subgroup of G , or if  H is a normal subgroup of G . Examples of almost-normal subgroups are given by Kreig[12, page 9].

Here is our reduction for a certain class of group rings, specializing the General Algebraic Semi-splitting of Theorem 0.4.

Corollary 3.26 Let R be a ring. Let G G1 F G2 be an injective amalgamated D  product of groups over a subgroup F of G1 and G2 . Suppose F is an almost-normal subgroup of G2 . Then, for all n Z, there is an isomorphism of abelian groups 2 j Niln.RŒF RŒG1 F; RŒG2 F/ Niln.RŒF RŒG1 F RŒF  RŒG2 F/ : W I ! I ˝

Proof Consider the set J .F G2=F/ F of nontrivial double cosets. Let I be WD n the poset of all finite subsets of J , partially ordered by inclusion. Note, as RŒF– bimodules, e e M RŒG2 F colim RŒI where RŒI RŒFgF : D I I WD 2 FgF I 2 Since F is an almost-normal subgroup of G2 , each RŒF–bimodule RŒI is a finitely generated free (hence projective) left RŒF–module. Observe that I is a filtered poset: if I; I I then I I I . Therefore we are done by Theorem 0.4. 0 2 [ 0 2

The case of G D Z2 Z2 has a particularly simple form. D 1 D  Corollary 3.27 Let R be a ring and n Z. There are natural isomorphisms: 2 (1) Niln.R R; R/ Niln.R/. I Š (2) Kn.RŒD / .Kn.RŒZ2/ Kn.RŒZ2//=Kn.R/ Niln 1.R/. 1 Š ˚ ˚

Proof Part (i) follows from Corollary 3.26 with F 1 and Gi Z2 . Then Part (ii) D D follows from Waldhausen’s exact sequence 2.6, where the group retraction Z2 1 ! inducese a splitting of thee map Kn.R/ Kn.RŒZ2/ Kn.RŒZ2/. !  e Example 3.28 Consider the group G G0 D where G0 Z2 Z2 Z. Since G D  1 D   surjects onto the infinite dihedral group, there is an amalgamated product decomposition

G .G0 Z2/ G .G0 Z2/ D   0 

Algebraic & Geometric Topology, Volume 11 (2011) 2418 James F Davis, Qayum Khan and Andrew Ranicki with the corresponding index 2 subgroup

G G0 Z : x D  Corollary 3.27(1) gives an isomorphism

Nil 1.ZŒG0 ZŒG0; ZŒG0/ Nil 1.ZŒG0/ : I Š On the other hand, Bass showed that the latter group is an infinitely generated abelian group of exponent a power of two[2, XII, 10.6]. Hence, by Waldhausen’s algebraic K–theory decomposition result, Wh.G/ is infinitely generated due to Nil elements. Now construct a codimension 1, finite CW–pair .X; Y / with 1X G realizing D the above amalgamatede product decomposition – fore example, let Y Z be a map ! of connected CW–complexes inducing the first factor inclusion G0 G0 Z2 on !  the fundamental group and let X be the double mapping cylinder of Z Y Z. ! Next construct a homotopy equivalence f M X of finite CW–complexes whose W ! torsion .f / Wh.G/ is a nonzero Nil element. Then f is nonsplittable along Y 2 by Waldhausen[24] (see Theorem 4.3). This is the first explicit example of a nonzero Waldhausen Nil group and a nonsplittable homotopy equivalence in the two-sided case.

3.3 Farrell–Jones Conjecture

The Farrell–Jones Conjecture asserts the family of virtually cyclic subgroups is a “generating”e family for Kn.RŒG/. In this section we apply our main theorem to show the Farrell–Jones Conjecture holds up to dimension one if and only if the smaller family of finite-by-cyclic subgroups is a generating family for Kn.RŒG/ up to dimension one. Let Or G be the orbit category of a group G ; objects are G –sets G=H where H is a subgroup of G and morphisms are G –maps. Davis and Lück[4] defined a functor KR Or G Spectra with the key property nKR.G=H / Kn.RŒH /. The utility W ! D of such a functor is that it allows the definition of an equivariant homology theory, indeed for a G –CW–complex X , one defines

G Hn .X KR/ n.mapG. ; X / Or G KR. // I WD C ^ (see[4, Sections 4, 7] for basic properties). Note that the “coefficients” of the homology G theory are given by H .G=H KR/ Kn.RŒH /. n I D A family F of subgroups of G is a nonempty set of subgroups closed under conjugation and taking subgroups. For such a family, EF G is the classifying space for G –actions with isotropy in F . It is characterized up to G –homotopy type as a G –CW–complex so H that .EF G/ is contractible for subgroups H F and is empty for subgroups H F . 2 62

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Four relevant families are fin fbc vc all, the families of finite subgroups,    finite-by-cyclic, virtually cyclic subgroups and all subgroups respectively. Here fbc fin H < G H F Ì Z with F finite WD [ f j Š g vc H < G cyclic C < H with finite index : WD f j 9 g The Farrell–Jones Conjecture in K–theory for the group G [7;4] asserts an isomor- phism G G H .EvcG KR/ H .EallG KR/ Kn.RŒG/ : n I ! n I D We now state a more general version, the Fibered Farrell–Jones Conjecture. Let ' € G be a group homomorphism. If F is a family of subgroups of G , define the W ! family of subgroups ' F H < € '.H/ F :  WD f j 2 g The Fibered Farrell–Jones Conjecture in K–theory for the group G asserts, for every ring R and homomorphism ' € G , the following induced map is an isomorphism: W ! € € H .E' vc.G/€ KR/ H .E' all.G/€ KR/ Kn.RŒ€/ : n  I ! n  I D The following theorem was proved for all n in[5] using controlled topology. We give a proof below up to dimension one using only algebraic topology.

Theorem 3.29 Let ' € G be an homomorphism of groups. Let R be any ring. W ! The inclusion-induced map € € H .E' fbc.G/€ KR/ H .E' vc.G/€ KR/ n  I ! n  I is an isomorphism for all integers n < 1 and an epimorphism for n 1. D Hence we propose a sharpening of the Farrell–Jones Conjecture in algebraic K–theory.

Conjecture 3.30 Let G be a discrete group, and let R be a ring. Let n be an integer. (1) There is an isomorphism G G H .EfbcG KR/ H .EallG KR/ Kn.RŒG/ : n I ! n I D (2) For any homomorphism ' € G of groups, there is an isomorphism W ! € € H .E' fbc.G/€ KR/ H .Eall€ KR/ Kn.RŒ€/ : n  I ! n I D The proof of Theorem 3.29 will require three auxiliary results, some of which we quote from other sources. The first is a variant of Theorem A:10 of Farrell–Jones[7], whose proof is identical to the proof of Theorem A:10.

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Transitivity Principle Let F G be families of subgroups of a group € . Let  E Or € Spectra be a functor. Let N Z . If for all H G F , the W ! 2 [ f1g 2 assembly map H H Hn .EF H H E/ Hn .EallH E/ j I ! I is an isomorphism for n < N and an epimorphism if n N , then the map D € € H .EF € E/ H .EallH E/ n I ! n I is an isomorphism for n < N and an epimorphism if n N . D

Of course, we apply this principle to the families fbc vc. The second auxiliary  result is a well-known lemma (see Scott and Wall[21, Theorem 5.12]), but we offer an alternative proof.

Lemma 3.31 Let G be a virtually cyclic group. Then either

(1) G is finite.

(2) G maps onto Z; hence G F Ì˛ Z with F finite. D (3) G maps onto D ; hence G G1 F G2 with Gi F 2 and F finite. 1 D  j W j D

Proof Assume G is an infinite virtually cyclic group. The intersection of the con- jugates of a finite index, infinite cyclic subgroup is a normal, finite index, infinite cyclic subgroup C . Let Q be the finite quotient group. Embed C as a subgroup of index Q in an infinite cyclic group C . There exists a unique ZŒQ–module structure j j 0 on C 0 such that C is a ZŒQ–submodule. Observe that the image of the obstruction cocycle under the map H 2.Q C / H 2.Q C / is trivial. Hence G embeds as a finite I ! I 0 index subgroup of a semidirect product G C Ì Q. Note G maps epimorphically 0 D 0 0 to Z (if Q acts trivially) or to D (if Q acts nontrivially). In either case, G maps epimorphically to a subgroup of1 finite index in D , which must be either infinite cyclic or infinite dihedral. 1

In order to see how the reduced Nil–groups relate to equivariant homology (and hence to the Farrell–Jones Conjecture), we need[5, Lemma 3.1], the third auxiliary result.

Lemma 3.32 (Davis–Quinn–Reich) Let G be a group of the form G1 F G2 with  Gi F 2, and let fac be the smallest family of subgroups of G containing G1 j W j D and G . Let G be a group of the form F Ì Z, and let fac be the smallest family of 2 x ˛ subgroups of G containing F . (Note that F need not be finite.) x

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(1) The following exact sequences are split, and hence short exact:

G fA G A G H .EfacG KR/ H .EallG KR/ H .EallG; EfacG KR/ n I ! n I ! n I G fB G B G H x .E G KR/ H x .EallG KR/ H x .EallG; E G KR/: n fac xI ! n xI ! n x fac xI Here fA; A and fB; B are inclusion-induced maps. (2) The maps

G A A Nilf n 1.RŒF RŒG1 F; RŒG2 F/ Š Hn .EallG; EfacG KR/ ı W I ! I 1 Š G B B Nilf n 1.RŒF; ˛/ Nilf n 1.RŒF; ˛ / Hnx .EallG; EfacG KR/ ı W ˚ ! x xI are isomorphisms where A and B are Waldhausen’s split injections.

The statement of Lemma 3.1 of[5] does not explicitly identify the isomorphisms in Part (2) above, but the identification follows from the last paragraph of the proof.

G G It is not difficult to compute H .EfacG KR/ and H x .E G KR/ in terms of n I n fac xI a Wang sequence and a Mayer–Vietoris sequence respectively. An example is in Section 3.4.

Next, we further assume G G with G G 2. In this case, C2 G=G acts G x  j W xj D D x on Kn.RG/ Hnx .EallG KR/ by conjugation. By[5, Remark 3.21], there is a x D G xI C2 –action on H x .EallG; E G KR/ so that B and  below are C2 –equivariant. n x fac xI !!

Lemma 3.33 Let n 1 be an integer. There is a commutative diagram of C2 – Ä equivariant homomorphisms

B B 1 ı - G Niln 1.RŒF; ˛/ Niln 1.RŒF; ˛ / Hnx .EallG; EfacG KR/ ˚ x xI

1 . i A i0 ˇuC / !!  

e ?e ? A A ı - G Niln 1.RŒF Ꮾ1; Ꮾ2/ Hn .EallG; EfacG KR/: I I

Here, the C2 G=G –action on the upper left-hand corner is given in Proposition 3.21, D x on the upper right it is given by [5, Remark 3.21], and on each lower corner it is trivial.

Proof This followse from Proposition 3.21 and the C2 –equivariance of B .

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Recall that if C2 1; T and if M is a ZŒC2–module then the coinvariant group D f g MC H0.C2 M / is the quotient group of M modulo the subgroup m T m m M . 2 D I f j 2 g Lemma 3.34 Let n 1 be an integer. There is an induction-induced isomorphism Ä G
G H x .EallG; E G KR/ H .E G; EfacG KR/: n fac C2 n fac G sub G x xI ! [ x I

Proof Recall G=G C2 . Since fac fac G , by[5, Lemma 4.1(i)] there is a x D D \ x identification of ZŒC2–modules

G H x .EallG; E G K/ n.K=Kfac/.G=G/: n x fac xI D x The C2 –coinvariants can be interpreted as a C2 –homology group:

C2 .n.K=Kfac/.C2// H .EC2 n.K=Kfac/.C2// : C2 D 0 I By Lemma 3.32(2) and Lemma 3.33, the coefficient ZŒC2–module is induced from a Z–module. By the Atiyah–Hirzebruch spectral sequence (which collapses at E2 ), note

C2 C2 H .EC2 n.K=Kfac/.C2// H .EC2 .K=Kfac/.C2// : 0 I D n I Therefore, by[5, Lemma 4.6, Lemma 4.4, Lemma 4.1], we conclude

C2 G Hn .EC2 .K=Kfac/.C2// Hn .EsubGG K=Kfac/ I D x I G Hn .Efac G sub GG K=Kfac/ D [ x I G Hn .Efac G sub GG; EfacG K/: D [ x I The identifications in the above proof are extracted from the proof of[5, Theorem 1.5].

Proof of Theorem 3.29 Let ' € G be a homomorphism of groups. Using the W ! Transitivity Principle applied to the families ' fbc ' vc, it suffices to show that    H Hn .EallH; E' fbc H H KR/ 0  j I D for all n 1 and for all H ' vc ' fbc. To identify the family ' fbc H we will Ä 2    j use two facts, the proofs of which are left to the reader.

If q A B is a group epimorphism with finite kernel, then fbc A q fbc B .  W ! D  (The key step is to show that an epimorphic image of a finite-by-cyclic group is finite-by-cyclic.) 1 1 If q A B G1 F G2 is a group epimorphism, then A q G1 q 1F q G2  W ! D  D  and fac A q fac B . D 

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Let ' H '.H / denote the restriction of ' to H . By the definition of both sides, jW ! ' fbc H ' .fbc '.H // :  j D j Since H ' vc ' fbc, we have '.H / vc fbc. So, by Lemma 3.31, there is an 2   2 epimorphism p '.H / D Z2 1 Z2 with finite kernel. By the first fact above W ! 1 D  ' .fbc '.H // ' .p.fbc D // : j D j 1 Next, write H q .p ' / 1.Z/. Note x WD ı j ' .p.fbc D // .p ' /.fbc D / j 1 D ı j 1 .p ' /.fac D sub Z/ D ı j 1 [ .p ' /.fac D / .p ' /.sub Z/ D ı j 1 [ ı j fac H sub H ; D [ x where the last equality uses the second fact above. Thus it suffices to prove, for any group H mapping epimorphically to D and for all n 1, that 1 Ä H Hn .EallH; Efac H sub H KR/ 0 [ x I D where H is the inverse image of the maximal infinite cyclic subgroup of D . x 1 Consider the composite

H ˛ H x
Hn .EallH ; EfacH KR/ H =H Hn .Efac H sub H H; EfacH KR/ x xI x ! [ x I ˇ H H .EallH; EfacH KR/: ! n I The map ˛ exists and is an isomorphism by Lemma 3.34. Apply C2 –covariants to the commutative diagram in the statement of Lemma 3.33. In this diagram of C2 – coinvariants, the top and bottom are isomorphisms by Lemma 3.32(2) and the left map is an isomorphism by Proposition 3.21 and Theorem 0.4. Hence the right-hand map, which is ˇ ˛, is an isomorphism for all n 1. It follows that ˇ is an isomorphism ı Ä for all n 1. So, by the exact sequence of a triple, we obtain Ä H Hn .EallH; Efac H sub H H KR/ 0 [ x I D for all n 1 as desired. Ä 3.4 K –theory of the modular group

Let € Z2 Z3 PSL2.Z/. The following theorem follows from applying our D  D main theorem and the recent proof by Bartels, Lück and Reich[1] of the Farrell–Jones conjecture in K–theory for word hyperbolic groups.

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The Cayley graph for Z2 Z3 with respect to the generating set given by the nonzero  elements of Z2 and Z3 has the quasi-isometry type of the usual Bass–Serre tree for the amalgamated product (Figure 1). This is an infinite tree with alternating vertices

 

 

  

 

 

Figure 1: Bass–Serre tree for PSL2.Z/ of valence two and three. The group € acts on the tree, with the generator of order two acting by reflection through an valence two vertex and the generator of order three acting by rotation through an adjoining vertex of valence three. Any geodesic triangle in the Bass–Serre tree has the property that the union of two sides is the union of all three sides. It follows that the Bass–Serre graph is ı–hyperbolic for any ı > 0, the Cayley graph is ı–hyperbolic for some ı > 0, and hence € is a hyperbolic group. Theorem 3.35 For any ring R and integer n,

Kn.RŒ€/ .Kn.RŒZ2/ Kn.RŒZ3//=Kn.R/ D ˚ M M Nilf n 1.R/ Nilf n 1.R/ Nilf n 1.R/; ˚ ˚ ˚ MC MD where MC and MD are the set of conjugacy classes of maximal infinite cyclic subgroups and maximal infinite dihedral subgroups, respectively. Moreover, all virtually cyclic subgroups of € are cyclic or infinite dihedral.

Proof By Lemma 3.32, the exact sequence of .Eall€; Efin€/ is short exact and split: € € € H .Efin€ KR/ H .Eall€ KR/ H .Eall€; Efin€ KR/: n I ! n I ! n I Then, by the Farrell–Jones Conjecture[1] for word hyperbolic groups, we obtain € € Kn.RŒ€/ H .Efin€ KR/ H .Eall€; Efin€ KR/ D n I ˚ n I € € H .Efin€ KR/ H .Evc€; Efin€ KR/: D n I ˚ n I

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Observe Efin€ is constructed as a pushout of € –spaces

€ € €=Z2 €=Z3 t? ! t? ? ? y y 1 € D Efin€ :  ! € Then Efin€ is the Bass–Serre tree for € Z2 Z3 . Note that H .€=H KR/ D   I D K .RŒH /. The pushout gives, after canceling a Kn.R/ term, a split long exact sequence

€ Kn.R/ Kn.RŒZ2/ Kn.RŒZ3/ Hn .Efin€ KR/ Kn 1.R/ :


! ! ˚ ! I ! !


€ Hence H .Efin€ KR/ .Kn.RŒZ2/ Kn.RŒZ3//=Kn.R/: n I D ˚ Next, for a word hyperbolic group G ,

G M V H .EvcG; EfinG K/ H .EvcV; EfinV K/; n I Š n I ŒV  M.G/ 2 where M.G/ is the set of conjugacy classes of maximal virtually cyclic subgroups of G (see Lück 16, Theorem 8.11 and Juan-Pineda and Leary[10]). The geometric interpretation of this result is that EvcG is obtained by coning off each geodesic in the tree EfinG ; then apply excision.

The Kurosh subgroup theorem implies that a subgroup of Z2 Z3 is a free product 2  3 of Z2 ’s, Z3 ’s, and Z’s. Note that Z2 Z3 a; b a 1 b , Z3 Z3 3 3  D h 2j D2 D 2 i  D c; d c 1 d , and Z2 Z2 Z2 e; f; g e f g 1 have free h j D D i   D h j D D D i subgroups of rank 2, for example ab; ab2 , cd; cd 2 , and ef; fg . On the other h i h i h i hand, the free group F2 rank 2 is not a virtually cyclic group since its first Betti number ˇ1.F2/ rank H1.F2/ 2, while for a virtually cyclic group V , transferring to the D D cyclic subgroup C V of finite index shows that ˇ1.V / is 0 or 1. Subgroups of  virtually cyclic groups are also virtually cyclic. Therefore all virtually cyclic subgroups of € are cyclic or infinite dihedral.

By the fundamental theorem of K–theory and Waldhausen’s Theorem 3.32,

Z Hn .EvcZ; EfinZ KR/ Nilf n 1.R/ Nilf n 1.R/ I D ˚ D Hn 1 .EvcD ; EfinD KR/ Nilf n 1.R R; R/: 1 1I D I Finally, by Corollary 3.27(1), we obtain exactly one type of Nil-group:

Niln 1.R R; R/ Niln 1.R/: I Š

Algebraic & Geometric Topology, Volume 11 (2011) e e 2426 James F Davis, Qayum Khan and Andrew Ranicki

Remark 3.36 The sets MC and MD are countably infinite. This can be shown by parameterizing these subsets either: combinatorially (using that elements in € are words in a; b; b2 ), geometrically (maximal virtually cyclic subgroups correspond to stabilizers of geodesics in the Bass–Serre tree Efin€ , where the geodesic may or may not be invariant under an element of order 2), or number theoretically (using solutions to Pell’s equation and Gauss’ theory of binary quadratic forms[20]).

Let us give an overview and history of some related work. The Farrell–Jones Conjecture and the classification of virtually cyclic groups 3.31 focused attention on the algebraic K–theory of groups mapping to the infinite dihedral group. Several years ago James Davis and Bogdan Vajiac outlined a unpublished proof of Theorem 0.1 when n 6 0 using controlled topology and hyperbolic geometry. Lafont and Ortiz[13] proved that Niln.ZŒF ZŒV1 F; ZŒV2 F/ 0 if and only if Niln.ZŒF; ˛/ 0 for any virtually I D D cyclic group V with an epimorphism V D and n 0; 1. More recently, Lafont ! 1 D and Ortiz[15] have studied the more general case of the K–theory Kn.RŒG1 F G2/  of an injective amalgam, where F; G1; G2 are finite groups. Finally, we mentioned the paper[5], which was written in parallel with this one; it an alternate proof of Theoreme 0.1. Also,[5] provides several auxiliary resultse used in Section 3.3 of this paper. The Nil-Nil isomorphism of Theorem 0.1 has been used in a geometrically motivated computation of Lafont and Ortiz[14, Section 6.4].

4 Codimension 1 splitting and semisplitting

We shall now give a topological interpretation of the Nil-Nil Theorem 1.1, proving in Theorem 4.5 that every homotopy equivalence of finite CW–complexes f M X W ! D X1 Y X2 with X1; X2; Y connected and 1.Y / 1.X / injective is “semisplit” [ ! along Y X , assuming that 1.Y / is of finite index in 1.X2/. Indeed, the proof  of Theorem 1.1 is motivated by the codimension 1 splitting obstruction theory of Waldhausen[24], and the subsequent algebraic K–theory decomposition theorems of Waldhausen[25; 26]. The papers[24; 25] developed both an algebraic splitting obstruction theory for chain complexes over injective generalized free products, and a geometric codimension 1 splitting obstruction theory; the geometric splitting ob- struction is the algebraic splitting obstruction of the cellular chain complex. There are parallel theories for the separating type (A) (amalgamated free product) and the nonseparating type (B) (HNN extension). We first briefly outline the theory, mainly for type (A). The cellular chain complex of the universal cover X of a connected CW–complex X is z a based free ZŒ1.X /–module chain complex C.X / such that H .X / H .C.X //. z  z D  z

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The kernel ZŒ1.X /–modules of a map f M X are defined by W ! K .M / H 1.f M X /  WD C zW ! z with M f X the pullback cover of M and f M X a 1.X /–equivariant lift WD  z zW ! z of f . For a cellular map f of CW–complexes let

᏷.M / Ꮿ.f C.M/ C.X // 1 WD zW ! z C be the algebraic mapping cone of the induced ZŒ1.X /–module chain map fz, with homology ZŒ1.X /–modules

H .᏷.M // K .M / H 1.f M X /:  D  D C zW ! z For n > 1 the map f M X is n–connected if and only if f 1.M / 1.X / W ! W Š and Kr .M / 0 for r < n, in which case the Hurewicz map is an isomorphism: D n 1.f / n 1.f / Kn.M / Hn 1.f / : C D C z ! D C z By the theorem of J H C Whitehead, f M X is a homotopy equivalence if and only W ! if f 1.M / 1.X / and K .M / 0 (if and only if ᏷.M / is chain contractible). W Š  D Decompose the boundary of the .n 1/–disk as a union of upper and lower n–disks: C n 1 n n n @D C S D S n 1 D : D D C [ Given a CW–complex M and a cellular map  Dn M define a new CW–complex W C ! n n 1 M 0 .M @ D /  1 D C D [ [ [ by attaching an n–cell and an .n 1/–cell, with C n 1 n n n n @  S M ;  1 S D S n 1 D M @ D : D jW ! [ W D C [ ! [ The inclusion M M is a homotopy equivalence called an elementary expansion. The  0 cellular based free ZŒ1.M /–module chain complexes fit into a short exact sequence

0 C.M/ C.M 0/ C.M 0; M/ 0 ! ! ! ! 1 with C.M ; M/ / 0 / ZŒ1.M / / ZŒ1.M / / 0 / 0 W





concentrated in dimensions n; n 1. For a commutative diagram of cellular maps C  Dn / M C f  ı  n 1 D C / X ;

Algebraic & Geometric Topology, Volume 11 (2011) 2428 James F Davis, Qayum Khan and Andrew Ranicki the map f extends to a cellular map n n 1 f 0 .f ı Dn / ı M 0 .M @ D /  1 D C X D [ j [ W D [ [ [ ! which is also called an elementary expansion, and there is defined a short exact sequence of based free ZŒ1.X /–module chain complexes

0 ᏷.M / ᏷.M 0/ C.M 0; M/ 0 : ! ! ! ! Recall the Whitehead group of a group G is defined by

Wh.G/ K1.ZŒG/= g g G : WD f˙ j 2 g Suppose the CW–complexes M; M 0; X are finite. The Whitehead torsion of a homo- topy equivalence f M X is W ! .f / .᏷.M // Wh.1.X // : D 2 Homotopy equivalences f M X and f M X are simple-homotopic if W ! 0W 0 ! .f / .f / Wh.1.X // : D 0 2 This is equivalent to being able to obtain f 0 from f by a finite sequence of elementary expansions and subdivisions and their formal inverses. For details, see Cohen’s book[3]. A 2–sided codimension 1 pair .X; Y X / is a pair of spaces such that the inclusion  Y Y 0 X extends to an open embedding Y R X . We say that a homotopy D f g    equivalence f M X splits along Y X if the restrictions f N f 1.Y / Y , W !  jW D ! f M N X Y are also homotopy equivalences. jW ! In dealing with maps f M X and 2–sided codimension 1 pairs .X; Y / we shall W ! assume that f is cellular and that both .X; Y / and .M; N f 1.Y // are a 2–sided D codimension 1 CW–pair.

A 2–sided codimension 1 CW–pair .X; Y / is 1 –injective if X; Y are connected and 1.Y / 1.X / is injective. As usual, there are two cases, according as to whether ! Y separates X or not: (A) The separating case: X Y is disconnected, so X X1 Y X2 D [ with X1; X2 connected. By the Seifert–van Kampen theorem

1.X / 1.X1/ 1.X2/ D 1.Y / is the amalgamated free product determined by the injections i 1.Y / k W ! 1.X / (k 1; 2). k D

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(B) The nonseparating case: X Y is connected, so X X1= y ty y Y D f  j 2 g for a connected space X1 (a deformation retract of X Y ) which contains two disjoint copies Y tY X1 of Y . By the Seifert–van Kampen theorem, t  1.X / 1.X1/ i ;i t D  1 2 f g is the HNN extension determined by the injections i1; i2 1.Y / 1.X1/, W ! with i1.y/t ti2.y/ (y 1.Y /). D 2

Remark 4.1 Let X be the universal cover of X , and let X X =1.Y /, so that for z x WD z both types (A) and (B), .X ; Y / is a  –injective 2–sided codimension 1 pair of the x 1 separating type (A), with X X Y X for connected subspaces X ; X X x D x [ xC x xC  x such that 1.X / 1.X / 1.X / 1.Y /: x D x D xC D Moreover for type (B), when i1; i2 are isomorphisms, the HNN extension simplifies to

1 1.Y / 1.X / 1.Y / Ì˛ Z Z 1 ! ! D ! ! 1 with automorphism ˛ .i1/ i2 of 1.Y /, studied originally by Farrell and Hsiang[6]. D

From now on, we shall only consider the separating case (A) of X X1 Y X2 . Write D [ 1.X / G ; 1.X1/ G1 ; 1.X2/ G2 ; 1.Y / H ; D D D D i ZŒH  ZŒG  ZŒH Ꮾ ; Ꮾ ZŒG H  ; k W ! k D ˚ k k D k with Ꮾk free as both a right and a left ZŒH –module, and

ZŒG ZŒG1 ZŒG2 ZŒH  Ꮾ1 Ꮾ2 Ꮾ1Ꮾ2 Ꮾ2Ꮾ1 : D ZŒH  D ˚ ˚ ˚ ˚ ˚


Use the injections i H G to define covers k W ! k X1 X1=H X ; X2 X2=H X x D z  x x D z  xC such that X1 X2 Y and x \ x D  [   [  X g X g X 1 1 S hY
2 2 z D z [ hH G=H z z g1G1 G=G1 2 g2G2 G=G2 2 2  [   [  X g1X1 S
g2X2 x D x [ hH G=H hY x g1G1 G=G1 2 g2G2 G=G2 2 2 with X the universal cover of X , and Y the universal cover of Y . zk k z

Algebraic & Geometric Topology, Volume 11 (2011) 2430 James F Davis, Qayum Khan and Andrew Ranicki

Let .f; g/ .M; N / .X; Y / be a map of separating 1 –injective codimension 1 W ! finite CW–pairs. This gives an exact sequence of based free ZŒH –module chain complexes

(7) 0 ᏷.N / ᏷.M / ᏷.M ; N / ᏷.M ; N / 0 ! ! S ! S ˚ S C ! inducing a long exact sequence of homology modules

/ Kr .N / / Kr .M / / Kr .M ; N / Kr .M ; N /


S S C ˚ S / Kr 1.N / / :


Note that f M X is a homotopy equivalence if and only if f 1.M / 1.X / is W ! W ! an isomorphism and ᏷.M / is contractible. The map of pairs .f; g/ .M; N / .X; Y / S W ! is a split homotopy equivalence if and only if any two of the chain complexes in (7) are contractible, in which case the third chain complex is also contractible. Suppose f M X is a homotopy equivalence. Then W ! K .M / K .M / 0 ; K .N / K 1.M ; N / K 1.M C; N /:  S D  D  D C S ˚ C S We obtain an exact sequence of ZŒH –module chain complexes

1 0 ᏷.M1; N / ᏷.M ; N / ᏷.M ; M1/ Ꮾ1 ᏷.M ; N / 0 ! S ! S ! S S D ˝ZŒH  S C ! 2 0 ᏷.M2; N / ᏷.M ; N / ᏷.M ; M2/ Ꮾ2 ᏷.M ; N / 0 : ! S ! S C ! S C S D ˝ZŒH  S ! The pair .1; 2/ of intertwined chain maps is chain homotopy nilpotent, in the sense that the following chain map is a ZŒG–module chain equivalence:   1 2 ZŒG ZŒH  .᏷.M ; N / ᏷.M C; N // 1 1 W ˝ S ˚ S ZŒG .᏷.M ; N / ᏷.M ; N // : ! ˝ZŒH  S ˚ S C

Definition 4.2 Let x .P1; P2; 1; 2/ be an object of NIL.ZŒH  Ꮾ1; Ꮾ2/. D I (1) Let x .P ; P ;  ;  / be another object. We say x and x are equivalent if 0 D 10 20 10 20 0 ŒP1 ŒP  ; ŒP2 ŒP  K0.ZŒH / ; Œx Œx  Nil0.ZŒH  Ꮾ1; Ꮾ2/; D 10 D 20 2 z D 0 2 I or equivalently,

Œx  Œx K0.Z/ K0.Z/ Nil0.ZŒH  Ꮾ1; Ꮾ2/ 0 2 ˚  I K0.ZŒH / K0.ZŒH / Nilf 0.ZŒH  Ꮾ1; Ꮾ2/ D ˚ e˚ I with K0.Z/ K0.Z/ the subgroup generated by .ZŒH ; 0; 0/ and .0; ZŒH ; 0; 0/. ˚

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(2) Let k 1 or 2. Let y ker. / generate a direct summand y P . Define D k 2 k h k i  k an object x in NIL.ZŒH  Ꮾ1; Ꮾ2/ by 0 I ( .P1= y1 ; P2; Œ1; Œ2/ if k 1; x0 .P10 ; P20 ; 10 ; 20 / h i D D D .P1; P2= y2 ; Œ1; Œ2/ if k 2; h i D with an exact sequence in NIL.ZŒH  Ꮾ1; Ꮾ2/ I 8 .y1;0/ < 0 / .ZŒH ; 0; 0; 0/ / x / x0 / 0

.0;y2/ : 0 / .0; ZŒH ; 0; 0/ / x / x0 / 0 :

Thus x0 is equivalent to x, obtained by the algebraic cell-exchange which kills y P1 P2 . k 2 ˚

It can be shown that two objects x and x in NIL.ZŒH Ꮾ1; Ꮾ2/ are equivalent if and 0 I only if x0 can be obtained from x by a finite sequence of isomorphisms, algebraic cell-exchanges, and their formal inverses. Geometric cell-exchanges (called surgeries in[24]) determine algebraic cell-exchanges. In the highly connected case, algebraic and geometric cell-exchanges occur in tandem:

Theorem 4.3 [24] Let .f; g/ .M; N / .X; Y / be a map of separating 1–injective W ! codimension 1 finite CW–pairs, with f M X a homotopy equivalence. Write W ! X X1 Y X2 with induced amalgam 1.X / G G1 H G2 of fundamental D [ D D  groups. (i) Let k 1; 2. Suppose for some n > 0 that we are given a map D .; @/ .Dn 1; S n/ .M ; N / W C ! k and a null-homotopy of pairs

n 1 n .; @/ .f M ; g @/ . ; / .D ; S / .X ; Y /: W j k ı ı '   W C ! k Assume they represent an element in ker. / (with  if k 1;  if k 2): k D D DC D  yk Œ;  im.Kn 1.Mk ; N / Kn 1.M ; N // D 2 C S ! C S   ker.k Kn 1.M ; N / Ꮾk ZŒH  Kn 1.M ; N // Kn.N /: D W C S ! ˝ C S  The map .f; g/ extends to the map of codimension 1 pairs

.f ; g / ..f f M / ; g @/ 0 0 WD [ j k ı [ [ W n 1 n 2 n 1 .M 0; N 0/ ..M @ D C /  1 D C ; N @ D C / .X; Y / WD [ [ [ [ !

Algebraic & Geometric Topology, Volume 11 (2011) 2432 James F Davis, Qayum Khan and Andrew Ranicki where the new .n 2/–cell has attaching map C n 2 n 1 n 1 n 1  1 @D D S n D M D : [ W C D C [ C ! [@ C The homological effect on .f; g/ of this geometric cell-exchange is no change in ( K .M ; N / K .M ; N / for r n 1; n 2 ; r S 0 0 r  D  ¤ C C Kr .M ; N / Kr .M ; N / for all r Z; S 0 0 D 2 except there is a five-term exact sequence

  0 Kn 2.M ; N / Kn 2.M 0 ; N 0/ ZŒH  ! C S ! C S ! yk   Kn 1.M ; N / Kn 1.M 0 ; N 0/ 0 : ! C S ! C S ! The inclusion h M M is a simple homotopy equivalence with .f;g/ .f h;g h N /. W  0 ' 0 0 j (ii) Suppose for some n > 2 that Kr .N / 0 for all r n. Then D ¤

Kr .M ; N / 0 Kr .M ; N / for all r n 1 ; S D D S C ¤ C and Kn.N / is a stably finitely generated free ZŒH –module. Moreover, we may define an object x in NIL.ZŒH  Ꮾ1; Ꮾ2/ by I

x .Kn 1.M ; N /; Kn 1.M C; N /; 1; 2/ WD C S C S whose underlying modules satisfy

ŒKn 1.M ; N / ŒKn 1.M C; N / ŒKn.N / 0 K0.ZŒH / ; C S C C S D D 2 z  ŒZŒGk  ZŒH  Kn 1.M ; N / 0 K0.ZŒGk / .k 1; 2/: ˝ C S D 2 z D

If .f 0; g0/ .M 0; N 0/ .X; Y / is obtained from .f; g/ by a geometric cell-exchange W !  killing an element yk Kn 1.M ; N / ..k; / .1; / or .2; // which gener- 2 C S  D C ates a direct summand yk Kn 1.M ; N /, then the corresponding object in h i  C S NIL.ZŒH  Ꮾ1; Ꮾ2/ I

x0 .Kn 1.M 0; N 0/; Kn 1.M 0C; N 0/; 10 ; 20 / WD C S C S is obtained from x by an algebraic cell-exchange. Since n 1.Mk ;N/ Kn 1.Mk ;N/ C S D C S by the relative Hurewicz theorem, there is a one-one correspondence between algebraic and geometric cell-exchanges killing elements y generating direct summands y . k h k i (iii) For any n > 2 it is possible to modify the given .f; g/ by a finite sequence of geometric cell-exchanges and their formal inverses to obtain a pair (also denoted by

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.f; g/) such that Kr .N / 0 for all r n as in (ii), and hence a canonical equivalence D ¤ class of nilpotent objects x .P1; P2; 1; 2/ in NIL.ZŒH  Ꮾ1; Ꮾ2/ such that D I ŒP1 ŒP2 0 K0.ZŒH / ; ŒZŒG  P  0 K0.ZŒG / C D 2 z k ˝ZŒH  k D 2 z k with P1 Kn 1.M ; N /, P2 Kn 1.M C; N /. Any x0 in the equivalence class WD C S WD C S of x is realized by a map .f ; g / .M ; N / .X; Y / with f simple-homotopic to f . 0 0 W 0 0 ! 0 The splitting obstruction of f is the image of the Whitehead torsion .f / Wh.G/, 2 namely,

@..f // .ŒP1; Œx/ .ŒP1; ŒP1; P2; 1; 2/ D D ker.K0.ZŒH / K0.ZŒG1/ K0.ZŒG2// Nilf 0.ZŒH  Ꮾ1; Ꮾ2/: 2 z ! z ˚ z ˚ I Thus f is simply homotopic to a split homotopy equivalence if and only if @..f // 0, D if and only if x is equivalent to 0.

(iv) The Whitehead group of G G1 H G2 fits into an exact sequence D  Wh.H / Wh.G1/ Wh.G2/ Wh.G/ ! ˚ ! @ K0.ZŒH / Nilf 0.ZŒH  Ꮾ1; Ꮾ2/ K0.ZŒG1/ K0.ZŒG2/ : ! z ˚ I ! z ˚ z Furthermore, the homomorphism

@ Wh.G/ K0.ZŒH/ Nil0.ZŒH  Ꮾ1; Ꮾ2/ ; .f / .ŒP1; ŒP1; P2; 1; 2/ W ! z ˚ I 7! satisfies that proj @ Wh.G/ Nil0.ZŒH  Ꮾ1; Ꮾ2/ is an epimorphism split by 2 ı W ! I   1 2  Nil0.ZŒH  Ꮾ1; Ꮾ2/ Wh.G/;ŒP1; P2; 1; 2 : W I ! 7! 1 1 e Definition 4.4 Let .X; Y / be a separating 1 –injective codimension 1 finite CW–pair. A homotopy equivalence f M X from a finite CW–complex M is semisplit along W !e Y X if f is simple homotopic to a map (also denoted by f ) such that for the  correspondinge map of pairs .f; g/ .M; N / .X; Y / the relative homology kernel W ! ZŒH –modules K .M2; N / H 1..M2; N / .X2; Y //  S D C z ! z z vanish, which is equivalent to the induced ZŒH –module morphisms

2 K .M C; N / K .M C; M2/ ZŒG2 H  ZŒH  K .M ; N /; W  S !  S S D ˝  S being isomorphisms. Equivalently, f is semisplit along Y if there is a semisplit object x .P1; P2; 1; 2/ in the canonical equivalence class of Theorem 4.3, that is, with D 2 P2 Ꮾ1P1 a ZŒH –module isomorphism. W ! In particular, a split homotopy equivalence f of separating pairs is semisplit.

Algebraic & Geometric Topology, Volume 11 (2011) 2434 James F Davis, Qayum Khan and Andrew Ranicki

Theorem 4.5 Let .X; Y / be a separating 1 –injective codimension 1 finite CW– pair, with X X1 Y X2 . Suppose that H 1.Y / is a finite-index subgroup of D [ D G2 1.X2/. Every homotopy equivalence f M X with M a finite CW–complex D W ! is simple-homotopic to a homotopy equivalence which is semisplit along Y .

Proof Let x .P1; P2; 1; 2/ represent the canonical equivalence class of objects D in NIL.ZŒH  Ꮾ1; Ꮾ2/ associated to f in Theorem 4.3(ii). Since H is of finite index I in G2 , as in the proof of Theorem 1.1, we can define a semisplit object

x .P1; Ꮾ2P1; 2 1; 1/ 00 WD ı satisfying

Œx  Œx Œ0; Ꮾ2P1; 0; 0 Œ0; P2; 0; 0 Nil0.ZŒH  Ꮾ1; Ꮾ2/: 00 D 2 I By Theorem 4.3(iii), the direct sum

Ꮾ2P1 P1 .ZŒG2 H  P1/ P1 ZŒG2 P1 ˚ D ˝ZŒH  ˚ D ˝ZŒH  is a stably finitely generated free ZŒG2–module. Since ZŒG2 is a finitely generated free ZŒH –module, Ꮾ2P1 P1 is a stably finitely generated free ZŒH –module. So ˚ ŒᏮ2P1 ŒP2 ŒᏮ2P1 ŒP1 ŒZŒG2 P1 0 K0.ZŒH / : D C D ˝ZŒH  D 2 z So x is equivalent to x00 . Thus, by Theorem 4.3(iii), there is a homotopy equivalence f M X simple-homotopic to f realizing x ; note it is semisplit. 00 W 00 ! 00 Acknowledgements We would like to thank the participants of the workshop Nil Phenomena in Topology (14–15 April 2007, Vanderbilt University), where our interests intersected and motivated the development of this paper. We are grateful to the referee for making helpful comments and asking perceptive questions. Moreover, Chuck Weibel helped us to formulate the filtered colimit hypothesis in Theorem 0.4, and Dan Ramras communicated the concept of almost-normal subgroup in Definition 3.24 to the second author.

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Department of Mathematics, Indiana University Bloomington IN 47405, USA Department of Mathematics, University of Notre Dame Notre Dame IN 46556, USA School of Mathematics, University of Edinburgh Edinburgh, EH9 3JZ, UK [email protected], [email protected], [email protected]

Received: 17 August 2010 Revised: 28 June 2011

Algebraic & Geometric Topology, Volume 11 (2011)