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

A lgebraic & G eometric T opology XX (20XX) 1001– 999 1001 Algebraic K -theory ov er the inﬁnite dihedral group : an algebraic appr oach J AMES F D A VIS Q A YUM K HA N A NDREW R AN ICKI T wo types of Nil-grou ps arise in the codimen sion 1 splitting obstructio n theory for ho motopy equiv alence s of ﬁnite CW -com plexes: th e Farrell–Bass Nil-g roups in the no n-separating case when the fun damental group is an H NN extension and the W aldhausen Nil-gr oups in th e separating case when th e fun damental group is an amalgam ated fr ee product. W e obtain a general Nil-Nil theorem in algebraic K -theory relating the two types of Nil-group s. The inﬁnite dihedral group is a free produ ct and has an index 2 subg roup wh ich is an HNN extension, so both cases arise if the fundamental group s urjects on to the in ﬁnite dih edral group . The Nil-Nil theo rem implies th at the two typ es of th e reduced f Nil -grou ps arising from such a fund amental group are isomorp hic. Th ere is also a topolog ical app lication: in the ﬁnite-index case of an amalgamated free produ ct, a homoto py eq uiv alence of ﬁnite CW -complexes is semi-split along a separating subcomp lex. 19D35; 57R19 Introd uction The inﬁni te dihed ral group is both a free pr oduct and an e xtension of the inﬁnite cyclic group Z by the cyclic group Z 2 of order 2 D ∞ = Z 2 ∗ Z 2 = Z ⋊ Z 2 with Z 2 acting on Z by − 1 . A grou p G is said to be ov er D ∞ if it is equipp ed with an epimorphism p : G → D ∞ . W e study the algebrai c K -theory of R [ G ] , for an y ring R and any group G ov er D ∞ . Such a group G inherits from D ∞ an injecti ve amalgamate d free product struc ture G = G 1 ∗ H G 2 with H an inde x 2 subgrou p of G 1 and G 2 . Furthermore , there is a canonical ind ex 2 subgroup G ⊂ G with an injecti ve Published: XX Xxxe mber 20XX DOI: 10.2 140/agt.2 0XX.XX.1001 1002 J F Davis, Q Khan and A Ran ic ki HNN stru cture G = H ⋊ α Z for an automorp hism α : H → H . The v arious group s ﬁt into a commutati ve braid of short exact sequen ces: Z & & & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ G = H ⋊ α Z 9 9 9 9 r r r r r r r r r r % % θ % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ D ∞ = Z 2 ∗ Z 2 π " " " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ H = G 1 ∩ G 2 9 9 9 9 s s s s s s s s s s % % 9 9 G = G 1 ∗ H G 2 p 8 8 8 8 q q q q q q q q q q q π ◦ p : : : : Z 2 The algeb raic K -theory decompos ition theorems of W aldhaus en for injec tiv e amalga- mated free produc ts and HNN ex tensions giv e (1) K ∗ ( R [ G ]) = K ∗ ( R [ H ] → R [ G 1 ] × R [ G 2 ]) ⊕ f Nil ∗− 1 ( R [ H ]; R [ G 1 − H ] , R [ G 2 − H ]) and (2) K ∗ ( R [ G ]) = K ∗ (1 − α : R [ H ] → R [ H ]) ⊕ f Nil ∗− 1 ( R [ H ] , α ) ⊕ f Nil ∗− 1 ( R [ H ] , α − 1 ) . W e establ ish isomorp hisms f Nil ∗ ( R [ H ]; R [ G 1 − H ] , R [ G 2 − H ]) ∼ = f Nil ∗ ( R [ H ] , α ) ∼ = f Nil ∗ ( R [ H ] , α − 1 ) . A ho motopy equi v alence f : M → X of ﬁnite CW -comple xes is split a long a subco m- ple x Y ⊂ X if it is a cellul ar map and the restriction f | : N = f − 1 ( Y ) → Y is also a homotopy equi v alence. The f Nil ∗ -group s arise as the obstr uction groups to spli tting homotop y equi v alences of ﬁ nite CW -complex es for codimens ion 1 Y ⊂ X with injec- ti ve π 1 ( Y ) → π 1 ( X ) , so that π 1 ( X ) is either an HN N ext ension or an amalgamated free product (Farrell–Hsi ang, W aldhausen) — see Section 4 for a brief revie w of the codimen sion 1 splittin g obstruc tion theory in the separati ng case of an amalgamated free product. In this paper we introduce the consider ably weaker notion of a ho motopy equi valenc e in the separating case being semi-split ( Deﬁnition 4.4 ). By way of geo- metric applic ation we prov e in Theorem 4.5 that there is no ob struction to topolog ical semi-spl itting in the ﬁnite-in dex case. 0.1 Algebraic semi-splitting The fol lowing is a speci al case of our main algebraic resul t ( 1.1 , 2.7 ) which sho ws t hat there is no obstr uction to algebra ic semi-s plitting. A lgebraic & G eo metric T opology XX (20XX) Algebraic K -theory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1003 Theor em 0.1 Let G be a grou p ov er D ∞ , with H = G 1 ∩ G 2 ⊂ G = H ⋊ α Z ⊂ G = G 1 ∗ H G 2 . (1) For any ring R and n ∈ Z the corres ponding re duced Nil -group s are isomorph ic: f Nil n ( R [ H ]; R [ G 1 − F ] , R [ G 2 − H ]) ∼ = f Nil n ( R [ H ] , α ) ∼ = f Nil n ( R [ H ] , α − 1 ) . (2) The inclusion θ : R [ G ] → R [ G ] determin es indu ction and trans fer maps θ ! : K n ( R [ G ]) → K n ( R [ G ]) , θ ! : K n ( R [ G ]) → K n ( R [ G ]) . For all integ ers n 6 1 , the f Nil n ( R [ H ] , α ) - f Nil n ( R [ H ]; R [ G 1 − H ] , R [ G 2 − H ]) - compone nts of th e maps θ ! and θ ! in the deco mpositions ( 2 ) and ( 1 ) are isomor- phisms. Pro of Part (i) is a spe cial case of Theor em 0.4 . Part (ii) f ollo ws from Theorem 0.4 , Lemma 3.20 , and Proposit ion 3.23 . The n = 0 case will be discu ssed in more detai l in Section 0.2 and Section 3.1 . Remark 0.2 W e do not serious ly doubt that a more assiduou s applicat ion of higher K -theory would ex tend Theorem 0.1 ( 2 ) to all n ∈ Z (see also [ 5 ]). As a n applic ation of Theorem 0.1 , we shall prov e the f ollo wing theorem. It sho ws that the Farrell– Jones Is omorphism Conjecture in algebraic K -theory can be reduced (up to dimensio n one) to the famil y of ﬁnite-b y-cyc lic groups , so that virtuall y cycli c groups of inﬁnite dihedr al type need not be cons idered. Theor em 0.3 Let Γ be an y gr oup, and le t R be an y rin g. Then, for all inte gers n < 1 , the follo wing indu ced map of equiv ariant homolo gy gro ups, with coef ﬁ cients in the algebr aic K -theor y functo r K R , is an isomorp hism: H Γ n ( E fbc Γ ; K R ) − → H Γ n ( E vc Γ ; K R ) . Furthermor e, this map is an epimorph ism for n = 1 . In fact, this is a special case of a more general ﬁbered versio n ( 3.29 ). T heore m 0.3 has been proved for all deg rees n in [ 5 ]; ho wev er our proof here uses only algebra ic topolo gy , av oiding the use of controlled topology . A lgebraic & G eo metric T opology XX (20XX) 1004 J F Davis, Q Khan and A Ran ic ki The origi nal reduced Nil -groups f Nil ∗ ( R ) = f Nil ∗ ( R , id) featu re in the decomposit ions of Bass [ 2 ] and Quillen [ 9 ]: K ∗ ( R [ t ]) = K ∗ ( R ) ⊕ f Nil ∗− 1 ( R ) , K ∗ ( R [ Z ]) = K ∗ ( R ) ⊕ K ∗− 1 ( R ) ⊕ f Nil ∗− 1 ( R ) ⊕ f Nil ∗− 1 ( R ) . In Section 3 we shall compute se veral ex amples which require Theorem 0.1 : K ∗ ( R [ Z 2 ∗ Z 2 ]) = K ∗ ( R [ Z 2 ]) ⊕ K ∗ ( R [ Z 2 ]) K ∗ ( R ) ⊕ f Nil ∗− 1 ( R ) K ∗ ( R [ Z 2 ∗ Z 3 ]) = K ∗ ( R [ Z 2 ]) ⊕ K ∗ ( R [ Z 3 ]) K ∗ ( R ) ⊕ f Nil ∗− 1 ( R ) ∞ Wh( G 0 × Z 2 ∗ G 0 G 0 × Z 2 ) = Wh( G 0 × Z 2 ) ⊕ Wh( G 0 × Z 2 ) Wh( G 0 ) ⊕ f Nil 0 ( Z [ G 0 ]) where G 0 = Z 2 × Z 2 × Z . The point here is that f Nil 0 ( Z [ G 0 ]) is an inﬁnite torsion abelia n group. T his provi des the ﬁrst example ( 3.28 ) of a non-zero f Nil -group in the amalgamate d produc t case and hence the ﬁrst exa mple of a non-zero obstruc tion to splitti ng a homotop y equiv alence in the separa ting case ( A ) . 0.2 The Nil-Nil Theor em W e estab lish isomorphisms between two typ es of codimens ion 1 splitt ing obstru ction nilpot ent class groups, for any ring R . The ﬁrst type, for separ ated splitting, arises in the decompo sitions of the alg ebraic K -theory of the R -c oef ﬁcient group ring R [ G ] of a group G ov er D ∞ , with an epimorphis m p : G → D ∞ onto the inﬁnite dihedral group D ∞ . The second type, for non-separa ted splittin g, arises from the α -twisted polyn omial ring R [ H ] α [ t ] , with H = ker( p ) and α : F → F an automorphism such that G = ker( π ◦ p : G → Z 2 ) = H ⋊ α Z where π : D ∞ → Z 2 is the uniqu e epimorp hism with inﬁnite cycl ic kernel. Note: (A) D ∞ = Z 2 ∗ Z 2 is the free product of two cycli c groups of order 2, whose genera tors will be denoted t 1 , t 2 . (B) D ∞ = h t 1 , t 2 | t 2 1 = 1 = t 2 2 i contains the inﬁnite cycl ic group Z = h t i as a subgro up of inde x 2 with t = t 1 t 2 . In fact there is a short exac t sequence w ith a split epimorphi sm { 1 } / / Z / / D ∞ π / / Z 2 / / { 1 } . More gene rally , if G is a group ov er D ∞ , with an epimorp hism p : G → D ∞ , then: A lgebraic & G eo metric T opology XX (20XX) Algebraic K -theory over the inﬁnite dihedral gr o up: an algebraic appr o ach 1005 (A) G = G 1 ∗ H G 2 is a free product with amalgamation of two grou ps G 1 = ker( p 1 : G → Z 2 ) , G 2 = ker( p 2 : G → Z 2 ) ⊂ G amalgamate d over thei r common subgroup H = ker( p ) = G 1 ∩ G 2 of index 2 in both G 1 and G 2 . (B) G has a subgro up G = ker( π ◦ p : G → Z 2 ) of index 2 which is an HNN ext ension G = H ⋊ α Z where α : H → H is conjugatio n by an element t ∈ G with p ( t ) = t 1 t 2 ∈ D ∞ . The K -theory of type (A) For any ring S and S -bimodule s B 1 , B 2 , we write the S -bimodule B 1 ⊗ S B 2 as B 1 B 2 , and we suppre ss left-tensor products of maps with the identit ies id B 1 or id B 2 . The exa ct category NIL( S ; B 1 , B 2 ) has object s being quadruple s ( P 1 , P 2 , ρ 1 , ρ 2 ) consis ting of ﬁnitely g enerated (= ﬁnitely ge nerated) projecti ve S -modules P 1 , P 2 and S -module morphisms ρ 1 : P 1 − → B 1 P 2 , ρ 2 : P 2 − → B 2 P 1 such that ρ 2 ρ 1 : P 1 → B 1 B 2 P 1 is nilpotent in the sense that ( ρ 2 ◦ ρ 1 ) k = 0 : P 1 − → ( B 1 B 2 )( B 1 B 2 ) · · · ( B 1 B 2 ) P 1 for some k > 0 . The morphi sms are pairs ( f 1 : P 1 → P ′ 1 , f 2 : P 2 → P ′ 2 ) such that f 2 ◦ ρ 1 = ρ ′ 1 ◦ f 1 and f 1 ◦ ρ 2 = ρ ′ 2 ◦ f 2 . Recall the W aldhausen Nil -group s Nil ∗ ( S ; B 1 , B 2 ) : = K ∗ (NIL( S ; B 1 , B 2 )) , and the r educed Nil -groups f Nil ∗ satisfy Nil ∗ ( S ; B 1 , B 2 ) = K ∗ ( S ) ⊕ K ∗ ( S ) ⊕ f Nil ∗ ( S ; B 1 , B 2 ) . An object ( P 1 , P 2 , ρ 1 , ρ 2 ) i n NIL( S ; B 1 , B 2 ) i s semi-split if the S -module isomorphism ρ 2 : P 2 → B 2 P 1 is an isomorph ism. Let R be a ring which is an amalgamated free product R = R 1 ∗ S R 2 with R k = S ⊕ B k for S -bimodules B k which are free S -modules, k = 1 , 2 . The algebr aic K -group s were sho wn in [ 24 , 25 , 26 ] to ﬁt into a long ex act sequence · · · → K n ( S ) ⊕ f Nil n ( S ; B 1 , B 2 ) → K n ( R 1 ) ⊕ K n ( R 2 ) → K n ( R ) → K n − 1 ( S ) ⊕ f Nil n − 1 ( S ; B 1 , B 2 ) → . . . with K n ( R ) → f Nil n − 1 ( S ; B 1 , B 2 ) a split surjectio n. A lgebraic & G eo metric T opology XX (20XX) 1006 J F Davis, Q Khan and A Ran ic ki For any ring R a based ﬁnitely generated fre e R -module chain comple x C has a torsion τ ( C ) ∈ K 1 ( R ) . The torsion of a chain equi v alence f : C → D of bas ed ﬁnitely genera ted free R -module chain co mplex es is the torsion of the algebraic mapp ing con e τ ( f ) = τ ( C ( f )) ∈ K 1 ( R ) . By deﬁnition , the chai n equi valen ce is simple if τ ( f ) = 0 ∈ K 1 ( R ) . For R = R 1 ∗ S R 2 the algebraic analogue of codimens ion 1 manifold transv ersalit y sho ws that eve ry based ﬁnitely generate d free R -module chain comple x C admits a Mayer –V ietoris presentatio n C : 0 → R ⊗ S D → ( R ⊗ R 1 C 1 ) ⊕ ( R ⊗ R 2 C 2 ) → C → 0 with C k a based ﬁnitely generated free R k -module chain comp lex, D a based ﬁnitely genera ted free S -module chain comp lex with R k -module cha in maps R k ⊗ S D → C k , and τ ( C ) = 0 ∈ K 1 ( R ) . This was ﬁrst pro ved in [ 24 , 25 ]; see also [ 18 , Remark 8.7] and [ 19 ]. A contra ctible C splits if it is simple chain equi valent to a chain complex (also denoted by C ) with a Mayer –V ietoris presenta tion C with D contr actible, in which case C 1 , C 2 are also contra ctible and the torsion τ ( C ) ∈ K 1 ( R ) is such that τ ( C ) = τ ( R ⊗ R 1 C 1 ) + τ ( R ⊗ R 2 C 2 ) − τ ( R ⊗ S D ) ∈ im( K 1 ( R 1 ) ⊕ K 1 ( R 2 ) → K 1 ( R )) . By the alge braic obst ruction theo ry of [ 24 ] C splits if and only if τ ( C ) ∈ im( K 1 ( R 1 ) ⊕ K 1 ( R 2 ) → K 1 ( R )) = ker( K 1 ( R ) → K 0 ( S ) ⊕ f Nil 0 ( S ; B 1 , B 2 )) . For any ring R the group ring R [ G ] of an amalgamate d free produ ct of group s G = G 1 ∗ H G 2 is an amalgamate d free prod uct of rings R [ G ] = R [ G 1 ] ∗ R [ H ] R [ G 2 ] . If H → G 1 , H → G 2 are inj ecti ve then the R [ H ] -bimodules R [ G 1 − H ] , R [ G 2 − H ] are free, and W aldhaus en [ 26 ] decompo sed the algebr aic K -theory of R [ G ] as K ∗ ( R [ G ]) = K ∗ ( R [ H ] → R [ G 1 ] × R [ G 2 ]) ⊕ f Nil ∗− 1 ( R [ F ]; R [ G 1 − H ] , R [ G 2 − H ]) . In parti cular , there is deﬁned a split monomorphism σ A : f Nil ∗− 1 ( R [ H ]; R [ G 1 − H ] , R [ G 2 − H ]) − → K ∗ ( R [ G ]) , which for ∗ = 1 is gi ven by σ A : f Nil 0 ( R [ H ]; R [ G 1 − H ] , R [ G 2 − H ]) − → K 1 ( R [ G ]) ; [ P 1 , P 2 , ρ 1 , ρ 2 ] 7− →  R [ G ] ⊗ R [ H ] ( P 1 ⊕ P 2 ) ,  1 ρ 2 ρ 1 1  . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -theory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1007 The K -theory of type (B) Giv en a ring R and an R -bimodule B , consider the tensor algebra T R ( B ) of B over R : T R ( B ) : = R ⊕ B ⊕ B B ⊕ · · · . The Nil -groups Nil ∗ ( R ; B ) are deﬁned to be the algeb raic K -groups K ∗ (NIL( R ; B )) of the exact cate gory NIL( R ; B ) with object s pairs ( P , ρ ) with P a ﬁnitely gener ated projec tiv e R -mod ule a nd ρ : P → B P an R -module morphism, nilpot ent in th e se nse that for some k > 0 , we ha ve ρ k = 0 : P − → B P − → · · · − → B k P . The reduce d Nil -groups f Nil ∗ are such that Nil ∗ ( R ; B ) = K ∗ ( R ) ⊕ f Nil ∗ ( R ; B ) . W aldhausen [ 26 ] prove d that if B is ﬁnitely generated projecti ve as a left R -module and free as a right R -module, then K ∗ ( T R ( B )) = K ∗ ( R ) ⊕ f Nil ∗− 1 ( R ; B ) . There is a split monomorphi sm σ B : f Nil ∗− 1 ( R ; B ) − → K ∗ ( T R ( B )) which for ∗ = 1 is gi ven by σ B : f Nil 0 ( R ; B ) − → K 1 ( T R ( B )) ; [ P , ρ ] 7− → [ T R ( B ) P , 1 − ρ ] . In parti cular , for B = R , we ha ve: Nil ∗ ( R ; R ) = Nil ∗ ( R ) , f Nil ∗ ( R ; R ) = f Nil ∗ ( R ) T R ( B ) = R [ t ] , K ∗ ( R [ t ]) = K ∗ ( R ) ⊕ f Nil ∗− 1 ( R ) . Relating the K -theory of types (A) and (B) Recall that a small cate gory I is ﬁlter ed if: • fo r any pair of objects α, α ′ in I , there exist an object β and morphisms α → β and α ′ → β in I , and • fo r any pair of morph isms u , v : α → α ′ in I , there exists an object β and morphism w : α ′ → β such tha t w ◦ u = w ◦ v . A lgebraic & G eo metric T opology XX (20XX) 1008 J F Davis, Q Khan and A Ran ic ki Note that an y d irected pos et I is a ﬁltered categ ory . A ﬁltere d colimit is a colimit o ver a ﬁltered cate gory . Theor em 0.4 (The Nil-Nil Theorem) Let R be a ring . Let B 1 , B 2 be R -bimodu les. Suppose that B 2 = colim α ∈ I B α 2 is a ﬁltered colimit limit of R -bimodu les such that each B α 2 is a ﬁnitely generated projecti ve left R -module. Then, for all n ∈ Z , the Nil -group s of the triple ( R ; B 1 , B 2 ) are related to the Nil -group s of the pair ( R ; B 1 B 2 ) by isomor phisms Nil n ( R ; B 1 , B 2 ) ∼ = Nil n ( R ; B 1 B 2 ) ⊕ K n ( R ) , f Nil n ( R ; B 1 , B 2 ) ∼ = f Nil n ( R ; B 1 B 2 ) . In pa rticular , for n = 0 and B 2 a ﬁnitely gene rated proj ecti ve left R -module, the re are deﬁned in verse isomorp hisms i ∗ : Nil 0 ( R ; B 1 B 2 ) ⊕ K 0 ( R ) ∼ = − − → Nil 0 ( R ; B 1 , B 2 ) ; ([ P 1 , ρ 12 : P 1 → B 1 B 2 P 1 ] , [ P 2 ]) 7− → [ P 1 , B 2 P 1 ⊕ P 2 ,  ρ 12 0  , (1 0)] , j ∗ : Nil 0 ( R ; B 1 , B 2 ) ∼ = − − → Nil 0 ( R ; B 1 B 2 ) ⊕ K 0 ( R ) ; [ P 1 , P 2 , ρ 1 : P 1 → B 1 P 2 , ρ 2 : P 2 → B 2 P 1 ] 7− → ([ P 1 , ρ 2 ◦ ρ 1 ] , [ P 2 ] − [ B 2 P 1 ]) . The reduce d versi ons are the in vers e isomorph isms i ∗ : f Nil 0 ( R ; B 1 B 2 ) ∼ = − − → f Nil 0 ( R ; B 1 , B 2 ) ; [ P 1 , ρ 12 ] 7− → [ P 1 , B 2 P 1 , ρ 12 , 1] , j ∗ : f Nil 0 ( R ; B 1 , B 2 ) ∼ = − − → f Nil 0 ( R ; B 1 B 2 ) ; [ P 1 , P 2 , ρ 1 , ρ 2 ] 7− → [ P 1 , ρ 2 ◦ ρ 1 ] with i ∗ ( P 1 , ρ 12 ) = ( P 1 , B 2 P 1 , ρ 12 , 1) semi-spl it. Pro of This follo ws immediately from T heore m 1.1 and Theor em 2.7 . Remark 0.5 Theo rem 0.4 was alr eady known to Pierre V ogel in 1990—see [ 22 ]. 1 Higher Nil -gr oups In this secti on, we shall prov e Theorem 0.4 for non-n egati ve de grees. Quillen [ 17 ] deﬁned the K -theory sp ace K E : = Ω BQ ( E ) of an exac t cate gory E . T he space BQ ( E ) is the geometri c realizati on of the simplicial set N • Q ( E ) , which is the A lgebraic & G eo metric T opology XX (20XX) Algebraic K -theory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1009 nerv e of a certain catego ry Q ( E ) associated to E . The algebraic K -groups of E are deﬁned for ∗ ∈ Z K ∗ ( E ) : = π ∗ ( K E ) using a nonconnec tiv e deloo ping for ∗ 6 − 1 . In p articular , the alg ebraic K -groups of a ring R are the algebraic K -groups K ∗ ( R ) : = K ∗ (PR OJ( R )) of the e xact cate gory PR OJ( R ) of ﬁnitely generated projecti ve R -modules. The NIL - cate gories deﬁned in the I ntroducti on all hav e the structure of exact categ ories. Theor em 1.1 Let B 1 and B 2 be bimodu les ove r a ring R . Let j be th e exact fu nctor j : NIL( R ; B 1 , B 2 ) − → NIL( R ; B 1 B 2 ) ; ( P 1 , P 2 , ρ 1 , ρ 2 ) 7− → ( P 1 , ρ 2 ◦ ρ 1 ) . (1) If B 2 is ﬁnitely gene rated projecti ve as a left R -module, then there is an exact functo r i : NIL( R ; B 1 B 2 ) − → NIL( R ; B 1 , B 2 ) ; ( P , ρ ) 7− → ( P , B 2 P , ρ, 1) such that i ( P , ρ ) = ( P , B 2 P , ρ, 1) is semi-sp lit, j ◦ i = 1 , and i ∗ and j ∗ induce in verse isomorphisms on the reduced Nil-groups f Nil ∗ ( R ; B 1 B 2 ) ∼ = f Nil ∗ ( R ; B 1 , B 2 ) . (2) If B 2 = coli m α ∈ I B α 2 is a ﬁltered colimit of bimodules each of whic h is ﬁnitely genera ted projecti ve as a le ft R -module, the n th ere is a un ique exa ct fu nctor i so that the follo w ing diagram commutes for all α ∈ I NIL( R ; B 1 B 2 ) NIL( R ; B 1 , B 2 ) NIL( R ; B 1 B α 2 ) NIL( R ; B 1 , B α 2 ) . ✲ i ✲ i α ✻ ✻ Then j ◦ i = 1 and i ∗ and j ∗ induce in verse isomor phisms on the reduced Nil-grou ps f Nil ∗ ( R ; B 1 B 2 ) ∼ = f Nil ∗ ( R ; B 1 , B 2 ) . Pro of (1) Note that there are split injecti ons of exac t categorie s PR OJ( R ) × PR OJ( R ) → NIL( R ; B 1 , B 2 ) ; ( P 1 , P 2 ) 7− → ( P 1 , P 2 , 0 , 0) , PR OJ( R ) → NIL( R ; B 1 B 2 ) ; ( P ) 7− → ( P , 0) , A lgebraic & G eo metric T opology XX (20XX) 1010 J F Davis, Q Khan and A Ran ic ki which underl y the deﬁnition of the reduced Nil groups . Since both i and j take the image of the spli t injectio n to the image of the other split injection, they induce maps i ∗ and j ∗ on the reduc ed Nil gro ups. Since j ◦ i = 1 , it follo ws tha t j ∗ ◦ i ∗ = 1 . In prepa ration fo r the proof that i ∗ ◦ j ∗ = 1 , consider the follo wing objects of NIL( R ; B 1 , B 2 ) x : = ( P 1 , P 2 , ρ 1 , ρ 2 ) x ′ : = ( P 1 , B 2 P 1 ⊕ P 2 ,  0 ρ 1  ,  1 ρ 2  ) x ′′ : = ( P 1 , B 2 P 1 , ρ 2 ◦ ρ 1 , 1) a : = (0 , P 2 , 0 , 0) a ′ : = (0 , B 2 P 1 , 0 , 0) with x ′′ semi-spl it. Note that ( i ◦ j )( x ) = x ′′ . Deﬁne morp hisms f : = (1 ,  0 1  ) : x − → x ′ f ′ : = (1 ,  1 ρ 2  ) : x ′ − → x ′′ g : = (0 ,  − ρ 2 1  ) : a − → x ′ g ′ : = (0 ,  1 0  ) : x ′ − → a ′ h : = (0 , ρ 2 ) : a − → a ′ . There are ex act seq uences 0 − − − − → x ⊕ a   f g 0 1   − − − − − − → x ′ ⊕ a  g ′ h  − − − − − → a ′ − − − − → 0 0 − − − − → a g − − − − → x ′ f ′ − − − − → x ′′ − − − − → 0 . Deﬁne exa ct functors F ′ , F ′′ , G , G ′ : NIL( R ; B 1 , B 2 ) → NIL( R ; B 1 , B 2 ) by F ′ ( x ) = x ′ , F ′′ ( x ) = x ′′ , G ( x ) = a , G ′ ( x ) = a ′ . Thus we ha ve two ex act sequences of exact functo rs 0 − − − − → 1 ⊕ G − − − − → F ′ ⊕ G − − − − → G ′ − − − − → 0 0 − − − − → G − − − − → F ′ − − − − → F ′′ − − − − → 0 . Recall j ◦ i = 1 , and note i ◦ j = F ′′ . By Quillen’ s Additi vity Theorem [ 17 , page 98, Corollary 1], we obtain homotop ies K F ′ ≃ 1 + KG ′ and KF ′ ≃ KG + KF ′′ . Then Ki ◦ Kj = KF ′′ ≃ 1 + ( KG ′ − K G ) , A lgebraic & G eo metric T opology XX (20XX) Algebraic K -theory over the inﬁnite dihedral gr o up: an algebraic appr o ach 1011 where the subtrac tion uses the loop space stru cture. O bserv e tha t both G and G ′ send NIL( R ; B 1 , B 2 ) to the image of PROJ ( R ) × PR OJ( R ) . T hus i ∗ ◦ j ∗ = 1 as desire d. (2) It is straightfor ward to sho w that tens or produ ct commutes with colimits ov er a cate gory . Moreov er , for any obj ect x = ( P 1 , P 2 , ρ 1 : P 1 → B 1 P 2 , ρ 2 : P 2 → B 2 P 1 ) , since P 2 is ﬁnitely gene rated , there exists α ∈ I suc h that ρ 2 fact ors through a map P 2 → B α 2 P 1 , and similarly for short ex act sequences of nil-ob jects. W e thus obtain induce d isomor phisms of exa ct categori es: colim α ∈ I NIL( R ; B 1 B α 2 ) − → NIL( R ; B 1 B 2 ) colim α ∈ I NIL( R ; B 1 , B α 2 ) − → NIL( R ; B 1 , B 2 ) . This justiﬁes the exis tence and uniquenes s of the f unctor i . By Q uillen ’ s colimit ob serv ation [ 17 , S ection 2, Equation (9), page 20], we obtain induce d weak homotop y equiv alences of K -theory spaces: colim α ∈ I K NIL( R ; B 1 B α 2 ) − → K NIL( R ; B 1 B 2 ) colim α ∈ I K N IL( R ; B 1 , B α 2 ) − → K NIL( R ; B 1 , B 2 ) . The remainin g asser tions of part (2) then follo w from part (1). Remark 1.2 The proo f of Theorem 1.1 is best understoo d in terms of ﬁnite chain comple xes x = ( P 1 , P 2 , ρ 1 , ρ 2 ) in the cate gory NIL( R ; B 1 , B 2 ) , assuming that B 2 is a ﬁnitely gene rated proj ecti ve left R -module. Any such x represents a cla ss [ x ] = ∞ X r = 0 ( − 1) r [( P 1 ) r , ( P 2 ) r , ρ 1 , ρ 2 ] ∈ Nil 0 ( R ; B 1 , B 2 ) . The key observ ation is that x determines a ﬁnite chain complex x ′ = ( P ′ 1 , P ′ 2 , ρ ′ 1 , ρ ′ 2 ) in NIL( R ; B 1 , B 2 ) which is semi-spl it in the sense that ρ ′ 2 : P ′ 2 → B 2 P ′ 1 is a chain equi valenc e, and such that (3) [ x ] = [ x ′ ] ∈ f Nil 0 ( R ; B 1 , B 2 ) . Speciﬁcally , let P ′ 1 = P 1 , P ′ 2 = M ( ρ 2 ) , the algebra ic m appin g cylind er of the chain map ρ 2 : P 2 → B 2 P 1 , and let ρ ′ 1 =   0 0 ρ 1   : P ′ 1 = P 1 − → B 1 P ′ 2 = M (1 B 1 ⊗ ρ 2 ) , ρ ′ 2 =  1 0 ρ 2  : P ′ 2 = M ( ρ 2 ) − → B 2 P 1 , A lgebraic & G eo metric T opology XX (20XX) 1012 J F Davis, Q Khan and A Ran ic ki so that P ′ 2 / P 2 = C ( ρ 2 ) is the algebraic mapping cone of ρ 2 . Moreo ver , the proof of ( 3 ) is suf ﬁciently fu nctorial to e stablish not o nly that th e fol lowin g maps of the reduced nilpot ent class grou ps are in vers e isomorp hisms: i : f Nil 0 ( R ; B 1 B 2 ) − → f Nil 0 ( R ; B 1 , B 2 ) ; ( P , ρ ) 7− → ( P , B 2 P , ρ, 1) , j : f Nil 0 ( R ; B 1 , B 2 ) − → f Nil 0 ( R ; B 1 B 2 ) ; [ x ] 7− → [ x ′ ] , b ut also that there ex ist isomorp hisms of f Nil n for all highe r dimensions n > 0 , as sho wn abov e. In order to pro ve equatio n ( 3 ), note that x ﬁts into the sequence (4) 0 / / x (1 , u ) / / x ′ (0 , v ) / / y / / 0 with y = (0 , C ( ρ 2 ) , 0 , 0) , u =   0 0 1   : P 2 → P ′ 2 = M ( ρ 2 ) , v =  1 0 0 0 1 0  : P ′ 2 = M ( ρ 2 ) → C ( ρ 2 ) and [ y ] = ∞ X r = 0 ( − ) r [0 , ( B 2 P 1 ) r − 1 ⊕ ( P 2 ) r , 0 , 0] = 0 ∈ f Nil 0 ( R ; B 1 , B 2 ) . The projec tion M ( ρ 2 ) → B 2 P 1 deﬁnes a chain equi valenc e x ′ ≃ ( P 1 , B 2 P 1 , ρ 2 ◦ ρ 1 , 1) = ij ( x ) so that [ x ] = [ x ′ ] − [ y ] = [ P 1 , B 2 P 1 , ρ 2 ◦ ρ 1 , 1] = ij [ x ] ∈ f Nil 0 ( R ; B 1 , B 2 ) . No w suppos e that x is a 0-dimension al chain comple x in NIL( R ; B 1 , B 2 ) , that is, an object as in the proof of Theorem 1.1 . Let x ′ , x ′′ , a , a ′ , f , f ′ , g , g ′ , h be as deﬁned there. The exact sequen ce of ( 4 ) can be written as the short exact sequence of chain complex es a g   a − h   0 / / x f / / x ′ g ′ / / a ′ / / 0 . The ﬁrst ex act sequence of the pro of of Theorem 1.1 is no w immediat e: 0 / / x ⊕ a   f g 0 1   / / x ′ ⊕ a  g ′ h  / / a ′ / / 0 . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -theory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1013 The secon d ex act sequence is self-ev ident: 0 / / a g / / x ′ f ′ / / x ′′ / / 0 . 2 Lower Nil -groups 2.1 Cone and suspension rings Let us recal l some additio nal struct ures on the tensor pro duct of modu les. Originati ng from ideas of Kar oubi–V illamayo r [ 11 ], the fo llowin g con cept w as studied indepe ndently by S M Gersten [ 8 ] and J B W agoner [ 23 ] in the con struction of the non-co nnecti ve K -theory spectr um of a ring . Deﬁnition 2.1 (Gersten, W agoner) The cone ring Λ Z is the subrin g of ω × ω - matrices o ver Z such that eac h row an d column hav e only a ﬁnite numbe r of non-zero entries . T he suspension ring Σ Z is the quotient ring of Λ Z by the two-si ded ideal of matrices with only a ﬁnite number of non-ze ro entries. For each n ∈ N , deﬁne the rings Σ n Z : = Σ Z ⊗ Z · · · ⊗ Z Σ Z | {z } n copies with Σ 0 Z = Z . For a rin g R and for n ∈ N , deﬁne the ring Σ n R : = Σ n Z ⊗ Z R . Roughly spe aking, the suspension should be re garded as the ring of “ bounded modulo compact operators. ” Gersten and W agoner showed that K i ( Σ n R ) is naturally isomorphic to K i − n ( R ) for all i , n ∈ Z , in the sense of Quillen when the subscript is positi ve, in the sense of Grothendieck when the subscript is zero, and in the sense of Bass when the subscr ipt is ne gati ve. For a n R -bimodule B , deﬁne the Σ n R -bimodule Σ n B : = Σ n Z ⊗ Z B . Lemma 2.2 Let R be a ring. Let B 1 , B 2 be R -bimodu les. Then, for each n ∈ N , there is a natura l isomorph ism of Σ n R -bimodu les: t n : Σ n ( B 1 B 2 ) − → Σ n B 1 ⊗ Σ n R Σ n B 2 ; s ⊗ ( b 1 ⊗ b 2 ) 7− → ( s ⊗ b 1 ) ⊗ (1 Σ n Z ⊗ b 2 ) . Pro of By transpo sition of the middle two f actors, note tha t Σ n B 1 ⊗ Σ n R Σ n B 2 = ( Σ n Z ⊗ Z B 1 ) ⊗ ( Σ n Z ⊗ Z R ) ( Σ n Z ⊗ Z B 2 ) is isomorp hic to ( Σ n Z ⊗ Σ n Z Σ n Z ) ⊗ Z ( B 1 B 2 ) = Σ n Z ⊗ Z ( B 1 B 2 ) = Σ n ( B 1 B 2 ) . A lgebraic & G eo metric T opology XX (20XX) 1014 J F Davis, Q Khan and A Ran ic ki 2.2 Deﬁnition of lower Ni l -groups Deﬁnition 2.3 Let R be a ring. Let B be an R -bi module. For a ll n ∈ N , deﬁne Nil − n ( R ; B ) : = Nil 0 ( Σ n R ; Σ n B ) f Nil − n ( R ; B ) : = f Nil 0 ( Σ n R ; Σ n B ) . Deﬁnition 2.4 Let R be a ring. Let B 1 , B 2 be R -b imodules. For all n ∈ N , deﬁne Nil − n ( R ; B 1 , B 2 ) : = Nil 0 ( Σ n R ; Σ n B 1 , Σ n B 2 ) f Nil − n ( R ; B 1 , B 2 ) : = f Nil 0 ( Σ n R ; Σ n B 1 , Σ n B 2 ) . The nex t two theorems follo w from the deﬁnitions and [ 26 , Theorems 1,3]. Theor em 2 .5 (W aldhausen ) Let R be a ring and B be an R -bimodu le. Consider the tensor ring T R ( B ) : = R ⊕ B ⊕ B 2 ⊕ B 3 ⊕ · · · . Suppose B is ﬁnitely generate d projecti ve as a left R -module and free as a right R -module. Then, for all n ∈ N , th ere is a spl it monomorp hism σ B : f Nil − n ( R ; B ) − → K 1 − n ( T R ( B )) gi ven for n = 0 by the map σ B : Nil 0 ( R ; B ) − → K 1 ( T R ( B )) ; [ P , ρ ] 7− →  T R ( B ) P , 1 − b ρ  , where b ρ is deﬁned usin g ρ and multiplic ation in T R ( B ) . Furthermor e, there is a natural decompo sition K 1 − n ( T R ( B )) = K 1 − n ( R ) ⊕ f Nil − n ( R ; B ) . For e xample, the last asserti on of the theorem follo ws from the equations : K 1 − n ( T R ( B )) = K 1 ( Σ n T R ( B )) = K 1 ( T Σ n R ( Σ n B )) = K 1 ( Σ n R ) ⊕ f Nil 0 ( Σ n R ; Σ n B ) = K 1 − n ( R ) ⊕ f Nil − n ( R ; B ) . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1015 Theor em 2.6 (W aldhause n) Let R , A 1 , A 2 be rings. L et R → A i be ring monomor - phisms such that A i = R ⊕ B i for R -bimodu les B i . Consider the pushout of rings A = A 1 ∗ R A 2 = R ⊕ ( B 1 ⊕ B 2 ) ⊕ ( B 1 B 2 ⊕ B 2 B 1 ) ⊕ ( B 1 B 2 B 1 ⊕ B 2 B 1 B 2 ) ⊕ · · · . Suppose each B i is free as a right R -module. Then, for all n ∈ N , there is a split monomorph ism σ A : f Nil − n ( R ; B 1 , B 2 ) − → K 1 − n ( A ) , gi ven for n = 0 by the map Nil 0 ( R ; B 1 , B 2 ) − → K 1 ( A ) ; [ P 1 , P 2 , ρ 1 , ρ 2 ] 7− →  ( AP 1 ) ⊕ ( AP 2 ) ,  1 b ρ 2 b ρ 1 1  , where b ρ i is deﬁned using ρ i and multiplic ation in A i for i = 1 , 2 . Furthermor e, there is a natural Mayer –V ietoris type exact seque nce · · · ∂ − − − − → K 1 − n ( R ) − − − − → K 1 − n ( A 1 ) ⊕ K 1 − n ( A 2 ) − − − − → K 1 − n ( A ) f Nil − n ( R ; B 1 , B 2 ) ∂ − − − − → K − n ( R ) − − − − → · · · 2.3 The isomorphism f o r lower N il -group s Theor em 2.7 Let R be a ring. Let B 1 , B 2 be R -bimodu les. Suppose that B 2 = colim α ∈ I B α 2 is a ﬁltered colimit of R -bimodu les B α 2 , each of which is a ﬁnitely gen- erated projecti ve lef t R -module. Then, fo r all n ∈ N , the re is an in duced iso morphism: Nil − n ( R ; B 1 B 2 ) ⊕ K − n ( R ) − → Nil − n ( R ; B 1 , B 2 ) . Pro of Let n ∈ N . By Lemma 2.2 and Theorem 1.1 , there are induce d isomorph isms: Nil − n ( R ; B 1 B 2 ) ⊕ K − n ( R ) = Nil 0 ( Σ n R ; Σ n ( B 1 B 2 )) ⊕ K 0 Σ n ( R ) − → Nil 0 ( Σ n R ; Σ n B 1 ⊗ Σ n R Σ n B 2 ) ⊕ K 0 Σ n ( R ) − → Nil 0 ( Σ n R ; Σ n B 1 , Σ n B 2 ) = Nil − n ( R ; B 1 , B 2 ) . A lgebraic & G eo metric T opology XX (20XX) 1016 J F Davis, Q Khan and A Ran ic ki 3 A pplications W e indicate some applica tions of our main theo rem ( 0.4 ). In Sectio n 3.1 we prove Theorem 0.1 (ii), which descri bes the restrictio ns of the maps θ ! : K ∗ ( R [ G ]) → K ∗ ( R [ G ]) , θ ! : K ∗ ( R [ G ]) → K ∗ ( R [ G ]) to the f Nil -terms, with θ : G → G the inclusion of the canonic al inde x 2 subgrou p G for any group G over D ∞ . In Section 3.2 we gi ve the ﬁrst kno wn example of a non-zero Nil -group occurr ing in the K -theory of an integr al group ring of an amalgamated free produ ct. In Section 3.3 we sharpen the Farrell –Jones Conjectur e in K -theory , rep lacing the family of virtually cyclic groups by the smaller family of ﬁ nite-by -cycli c groups. In Section 3.4 we compute the K ∗ ( R [ Γ ]) for the modular group Γ = PSL 2 ( Z ) . 3.1 Algebraic K -theory over D ∞ The over all goal here is to s how that t he ab stract isomor phisms i ∗ and j ∗ coinci de with the restrict ions of the inductio n and transfer maps θ ! and θ ! in the group ring setting. 3.1.1 T wisting W e start by recalli ng the algebra ic K -theory of twisted polyno mial rings. Statement 3.1 Consider any (unital, associat iv e) ring R and any ring automorphis m α : R → R . Let t be an indetermin ate over R such that rt = t α ( r ) ( r ∈ R ) . For a ny R -module P , let tP : = { tx | x ∈ P } be the set with left R -mod ule structure tx + ty = t ( x + y ) , r ( tx ) = t ( α ( r ) x ) ∈ tP . Further endo w the left R -module tR with the R -bimodul e structure R × tR × R − → tR ; ( q , tr , s ) 7− → t α ( q ) r s . The Nil -categ ory of R with respect to α is the ex act categor y deﬁned by NIL( R , α ) : = NIL( R ; tR ) . The objects ( P , ρ ) consi st of any ﬁnitely genera ted projecti ve R -module P and any nilpot ent m orphi sm ρ : P → tP = tRP . The Nil -groups are w ritten Nil ∗ ( R , α ) : = Nil ∗ ( R ; tR ) , f Nil ∗ ( R , α ) : = f Nil ∗ ( R ; tR ) , A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1017 so that Nil ∗ ( R , α ) = K ∗ ( R ) ⊕ f Nil ∗ ( R , α ) . Statement 3.2 The ten sor algebra on tR is the α -twisted polynomial exte nsion of R T R ( tR ) = R α [ t ] = ∞ X k = 0 t k R . Giv en an R -module P there is induced an R α [ t ] -module R α [ t ] ⊗ R P = P α [ t ] whose elements are ﬁnit e lin ear c ombinations ∞ P j = 0 t j x j ( x j ∈ P ). Giv en R -modules P , Q and an R -module m orphis m ρ : P → tQ , deﬁne its exte nsion as the R α [ t ] -module morphism b ρ = t ρ : P α [ t ] − → Q α [ t ] ; ∞ X j = 0 t j x j 7− → ∞ X j = 0 t j ρ ( x j ) . Statement 3.3 Bass [ 2 ], Farrel l–Hsiang [ 6 ], and Quille n [ 9 ] giv e decomposit ions: K n ( R α [ t ]) = K n ( R ) ⊕ f Nil n − 1 ( R , α ) , K n ( R α − 1 [ t − 1 ]) = K n ( R ) ⊕ f Nil n − 1 ( R , α − 1 ) , K n ( R α [ t , t − 1 ]) = K n (1 − α : R → R ) ⊕ f Nil n − 1 ( R , α ) ⊕ f Nil n − 1 ( R , α − 1 ) . In parti cular for n = 1 , by Theorem 2.5 , there are deﬁned split monomorph isms: σ + B : f Nil 0 ( R , α ) − → K 1 ( R α [ t ]) ; [ P , ρ ] 7− → [ P α [ t ] , 1 − t ρ ] , σ − B : f Nil 0 ( R , α − 1 ) − → K 1 ( R α − 1 [ t − 1 ]) ; [ P , ρ ] 7− →  P α − 1 [ t − 1 ] , 1 − t − 1 ρ  , σ B =  ψ + σ + B ψ − σ − B  : f Nil 0 ( R , α ) ⊕ f Nil 0 ( R , α − 1 ) − → K 1 ( R α [ t , t − 1 ]) ; ([ P 1 , ρ 1 ] , [ P 2 , ρ 2 ]) 7− →  ( P 1 ⊕ P 2 ) α [ t , t − 1 ] ,  1 − t ρ 1 0 0 1 − t − 1 ρ 2  . These exte nd to all intege rs n 6 1 by the suspen sion isomorphi sms of Section 2 . 3.1.2 Scaling Next, con sider the effec t an inner automorphism on α . A lgebraic & G eo metric T opology XX (20XX) 1018 J F Davis, Q Khan and A Ran ic ki Statement 3.4 Supp ose α, α ′ : R → R are automorph isms satisfy ing α ′ ( r ) = u α ( r ) u − 1 ∈ R ( r ∈ R ) for some unit u ∈ R , and that t ′ is an ind eterminate over R satisfy ing rt ′ = t ′ α ′ ( r ) ( r ∈ R ) . Denote the canon ical inclusions ψ + : R α [ t ] − → R α [ t , t − 1 ] ψ − : R α − 1 [ t − 1 ] − → R α [ t , t − 1 ] ψ ′ + : R α ′ [ t ′ ] − → R α ′ [ t ′ , t ′ − 1 ] ψ ′− : R α ′ − 1 [ t ′ − 1 ] − → R α ′ [ t ′ , t ′ − 1 ] . Statement 3.5 The v arious polynomial rings are related by scaling isomorphisms β + u : R α [ t ] − → R α ′ [ t ′ ] ; t 7− → t ′ u , β − u : R α − 1 [ t − 1 ] − → R α ′ − 1 [ t ′ − 1 ] ; t − 1 7− → u − 1 t ′ − 1 , β u : R α [ t , t − 1 ] − → R α ′ [ t ′ , t ′ − 1 ] ; t 7− → t ′ u satisfy ing the equations β u ◦ ψ + = ψ ′ + ◦ β + u : R α [ t ] − → R α ′ [ t ′ , t ′ − 1 ] β u ◦ ψ − = ψ ′− ◦ β − u : R α − 1 [ t − 1 ] − → R α ′ [ t ′ , t ′ − 1 ] . Statement 3.6 There are corr espondin g scaling isomorphis ms of exact cate gories β + u : NIL( R , α ) − → NIL( R , α ′ ) ; ( P , ρ ) 7− → ( P , t ′ ut − 1 ρ : P → t ′ P ) β − u : NIL( R , α − 1 ) − → NIL( R , α ′ − 1 ) ; ( P , ρ ) 7− → ( P , t ′ − 1 ut ρ : P ′ → t ′ − 1 P ′ ) , where we mean ( t ′ ut − 1 ρ )( x ) : = t ′ ( uy ) with ρ ( x ) = ty , ( t ′ − 1 ut ρ )( x ) : = t ′ − 1 ( uy ) with ρ ( x ) = t − 1 y . Statement 3.7 For all n 6 1 , the vario us scaling isomorphisms are related by equa- tions ( β + u ) ∗ ◦ σ + B = σ ′ + B ◦ β + u : f Nil n − 1 ( R , α ) − → K n ( R α ′ [ t ′ ]) ( β − u ) ∗ ◦ σ − B = σ ′− B ◦ β − u : f Nil n − 1 ( R , α − 1 ) − → K n ( R α ′ − 1 [ t ′ − 1 ]) ( β u ) ∗ ◦ σ B = σ ′ B ◦  β + u 0 0 β − u  : f Nil n − 1 ( R , α ) ⊕ f Nil n − 1 ( R , α − 1 ) − → K n ( R α ′ [ t ′ , t ′ − 1 ]) . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1019 3.1.3 Gr oup rings W e no w adapt these isomorphisms to the case of group rings R [ G ] of groups G over the inﬁnite dihedral group D ∞ . In order to pro ve Lemma 3.20 and P roposi tion 3.23 , the overa ll idea is to trans form information about the product t 2 t 1 arising from the transp osition B 2 ⊗ B 1 into informat ion about the product t − 1 2 t − 1 1 arising in the sec- ond f Nil -summand of the twisted Bass decomposi tion. W e continue to discuss the ingred ients in a seque nce of statements. Statement 3.8 Let F be a group, and let α : F → F be an automorphism. R ecall that the injecti ve H NN ext ension F ⋊ α Z is the set F × Z with group multiplica tion ( x , n )( y , m ) : = ( α m ( x ) y , m + n ) ∈ F ⋊ α Z . Then, for any ring R , writing t = (1 F , 1) and ( x , n ) = t n x ∈ F ⋊ α Z , we ha ve R [ F ⋊ α Z ] = R [ F ] α [ t , t − 1 ] . Statement 3.9 Cons ider any group G = G 1 ∗ F G 2 ov er D ∞ , where F = G 1 ∩ G 2 ⊂ G = F ⋊ α Z = F ⋊ α ′ Z ⊂ G = G 1 ∗ F G 2 . Fix elements t 1 ∈ G 1 − F , t 2 ∈ G 2 − F , and deﬁne elements t : = t 1 t 2 ∈ G , t ′ : = t 2 t 1 ∈ G , u : = ( t ′ ) − 1 t − 1 ∈ F . Deﬁne the automorp hisms α 1 : F − → F ; x 7− → ( t 1 ) − 1 xt 1 , α 2 : F − → F ; x 7− → ( t 2 ) − 1 xt 2 , α : = α 2 ◦ α 1 : F − → F ; x 7− → t − 1 xt , α ′ : = α 1 ◦ α 2 : F − → F ; x 7− → t ′ − 1 xt ′ such that xt = t α ( x ) , xt ′ = t ′ α ′ ( x ) , α ′ ( x ) = u α − 1 ( x ) u − 1 ( x ∈ F ) . In particula r , n ote α ′ and α − 1 (not α ) are r elated by inner automorphism by u . Statement 3.10 Denote the cano nical inclusion s ψ + : R α [ t ] − → R α [ t , t − 1 ] ψ − : R α − 1 [ t − 1 ] − → R α [ t , t − 1 ] ψ ′ + : R α ′ [ t ′ ] − → R α ′ [ t ′ , t ′ − 1 ] ψ ′− : R α ′ − 1 [ t ′ − 1 ] − → R α ′ [ t ′ , t ′ − 1 ] . A lgebraic & G eo metric T opology XX (20XX) 1020 J F Davis, Q Khan and A Ran ic ki The inclus ion R [ F ] → R [ G ] exten ds to ring monomorphisms θ : R [ F ] α [ t , t − 1 ] − → R [ G ] θ ′ : R [ F ] α ′ [ t ′ , t ′ − 1 ] − → R [ G ] such that im( θ ) = im( θ ′ ) = R [ G ] ⊂ R [ G ] = R [ G 1 ] ∗ R [ F ] R [ G 2 ] . Furthermor e, the inclusion R [ F ] → R [ G ] exte nds to ring monomorphisms φ = θ ◦ ψ + : R [ F ] α [ t ] − → R [ G ] φ ′ = θ ′ ◦ ψ ′ + : R [ F ] α ′ [ t ′ ] − → R [ G ] . Statement 3.11 By 3.5 , there are deﬁned scaling isomorph isms of rings β + u : R [ F ] α − 1 [ t − 1 ] − → R [ F ] α ′ [ t ′ ] ; t − 1 7− → t ′ u , β − u : R [ F ] α [ t ] − → R [ F ] α ′ − 1 [ t ′ − 1 ] ; t 7− → u − 1 t ′ − 1 , β u : R [ F ] α [ t , t − 1 ] − → R [ F ] α ′ [ t ′ , t ′ − 1 ] ; t 7− → u − 1 t ′ − 1 which sati sfy the equations β u ◦ ψ − = ψ ′ + ◦ β + u : R [ F ] α − 1 [ t − 1 ] − → R [ F ] α ′ [ t ′ , t ′ − 1 ] β u ◦ ψ + = ψ ′− ◦ β − u : R [ F ] α [ t ] − → R [ F ] α ′ [ t ′ , t ′ − 1 ] θ = θ ′ ◦ β u : R [ F ] α [ t , t − 1 ] − → R [ G ] . Statement 3.12 By 3.6 , there are scaling isomorphi sms of exa ct categor ies β + u : NIL( R [ F ] , α − 1 ) − → NIL( R [ F ] , α ′ ) ; ( P , ρ ) 7− → ( P , t ′ ut ρ ) , β − u : NIL( R [ F ] , α ) − → NIL( R [ F ] , α ′ − 1 ) ; ( P , ρ ) 7− → ( P , t ′ − 1 ut − 1 ρ ) . Statement 3.13 By 3.7 , for all n 6 1 , the vario us scaling isomorphisms are related by: ( β + u ) ∗ ◦ σ − B = σ ′ + B ◦ β + u : f Nil ∗− 1 ( R [ F ] , α − 1 ) − → K ∗ ( R [ F ] α ′ [ t ′ ]) ( β − u ) ∗ ◦ σ + B = σ ′− B ◦ β − u : f Nil ∗− 1 ( R [ F ] , α ) − → K ∗ ( R [ F ] α ′ − 1 [ t ′ − 1 ]) ( β u ) ∗ ◦ σ B = σ ′ B ◦  0 β + u β − u 0  : f Nil ∗− 1 ( R [ F ] , α ) ⊕ f Nil ∗− 1 ( R [ F ] , α − 1 ) − → K ∗ ( R [ F ] α ′ [ t ′ , t ′ − 1 ]) . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -theory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1021 3.1.4 T ransposition Next, we study the ef fect of transpos ition of the bimodules B 1 and B 2 in order to relate α and α ′ . In pa rticular , there is no mention of α − 1 in this secti on. Statement 3.14 The R [ F ] -bimodule s B 1 = R [ G 1 − F ] = t 1 R [ F ] , B 2 = R [ G 2 − F ] = t 2 R [ F ] are free left and righ t R [ F ] -modules of ran k one. The R [ F ] -bimodule isomorphi sms B 1 ⊗ R [ F ] B 2 − → tR [ F ] ; t 1 x 1 ⊗ t 2 x 2 7− → t α 2 ( x 1 ) x 2 B 2 ⊗ R [ F ] B 1 − → t ′ R [ F ] ; t 2 x 2 ⊗ t 1 x 1 7− → t ′ α 1 ( x 2 ) x 1 shall be used to make th e identiﬁcations B 1 ⊗ R [ F ] B 2 = tR [ F ] , N IL( R [ F ]; B 1 ⊗ R [ F ] B 2 ) = NIL( R [ F ] , α ) , B 2 ⊗ R [ F ] B 1 = t ′ R [ F ] , NIL( R [ F ]; B 2 ⊗ R [ F ] B 1 ) = NIL( R [ F ] , α ′ ) . Statement 3.15 Theorem 0.4 gi ves in vers e isomorphisms i ∗ : f Nil ∗ ( R [ F ] , α ) − → f Nil ∗ ( R [ F ]; B 1 , B 2 ) , j ∗ : f Nil ∗ ( R [ F ]; B 1 , B 2 ) − → f Nil ∗ ( R [ F ] , α ) which for ∗ = 0 are giv en by i ∗ : f Nil 0 ( R [ F ] , α ) − → f Nil 0 ( R [ F ]; B 1 , B 2 ) ; [ P , ρ ] 7− → [ P , t 2 P , ρ, 1] , j ∗ : f Nil 0 ( R [ F ]; B 1 , B 2 ) − → f Nil 0 ( R [ F ] , α ) ; [ P 1 , P 2 , ρ 1 , ρ 2 ] 7− → [ P 1 , ρ 2 ◦ ρ 1 ] . Statement 3.16 Similarly , there are deﬁned in verse isomorphis ms i ′ ∗ : f Nil ∗ ( R [ F ] , α ′ ) − → f Nil ∗ ( R [ F ]; B 2 , B 1 ) , j ′ ∗ : f Nil ∗ ( R [ F ]; B 2 , B 1 ) − → f Nil ∗ ( R [ F ] , α ′ ) which for ∗ = 0 are giv en by i ′ ∗ : f Nil 0 ( R [ F ] , α ′ ) − → f Nil 0 ( R [ F ]; B 2 , B 1 ) ; [ P ′ , ρ ′ ] 7− → [ P ′ , t 1 P ′ , ρ ′ , 1] , j ′ ∗ : f Nil 0 ( R [ F ]; B 2 , B 1 ) − → f Nil 0 ( R [ F ] , α ′ ) ; [ P 2 , P 1 , ρ 2 , ρ 1 ] 7− → [ P 2 , ρ 1 ◦ ρ 2 ] . Statement 3.17 The tran sposition isomorphism of exact cate gories τ A : NIL( R [ F ]; B 1 , B 2 ) − → NIL( R [ F ]; B 2 , B 1 ) ; ( P 1 , P 2 , ρ 1 , ρ 2 ) 7− → ( P 2 , P 1 , ρ 2 , ρ 1 ) A lgebraic & G eo metric T opology XX (20XX) 1022 J F Davis, Q Khan and A Ran ic ki induce s isomorphisms τ A : Nil ∗ ( R [ F ]; B 1 , B 2 ) ∼ = Nil ∗ ( R [ F ]; B 2 , B 1 ) , τ A : f Nil ∗ ( R [ F ]; B 1 , B 2 ) ∼ = f Nil ∗ ( R [ F ]; B 2 , B 1 ) . Note, by Theorem 0.4 , the compo sites τ B : = j ′ ∗ ◦ τ A ◦ i ∗ : f Nil ∗ ( R [ F ] , α ) − → f Nil ∗ ( R [ F ] , α ′ ) , τ ′ B : = j ∗ ◦ τ − 1 A ◦ i ′ ∗ : f Nil ∗ ( R [ F ] , α ′ ) − → f Nil ∗ ( R [ F ] , α ) are in verse isomorphi sms, w hich for ∗ = 0 are giv en by τ B : f Nil 0 ( R [ F ] , α ) − → f Nil 0 ( R [ F ] , α ′ ) ; [ P , ρ ] 7− → [ t 2 P , t 2 ρ ] , τ ′ B : f Nil 0 ( R [ F ] , α ′ ) − → f Nil 0 ( R [ F ] , α ) ; [ P ′ , ρ ′ ] 7− → [ t 1 P ′ , t 1 ρ ′ ] . Furthermor e, note that the variou s transpositi ons are related by the equation τ A ◦ i ∗ = i ′ ∗ ◦ τ B : f Nil ∗ ( R [ F ] , α ) − → f Nil ∗ ( R [ F ]; B 2 , B 1 ) . Statement 3.18 Recall from Theor em 2.6 that there is a split monomorphism σ A : f Nil n − 1 ( R [ F ]; B 1 , B 2 ) − → K n ( R [ G ]) such that the n = 1 case is giv en by σ A : f Nil 0 ( R [ F ]; B 1 , B 2 ) − → K 1 ( R [ G ]) ; [ P 1 , P 2 , ρ 1 , ρ 2 ] 7− →  P 1 [ G ] ⊕ P 2 [ G ] ,  1 t 2 ρ 2 t 1 ρ 1 1  . Elementary ro w and column operation s produc e an equi v alent represent ativ e:  1 − t 2 ρ 2 0 1   1 t 2 ρ 2 t 1 ρ 1 1   1 0 − ρ 1 1  =  1 − t ρ 2 ρ 1 0 0 1  . Thus the n = 1 case satisﬁes the equa tions (similarly for the second equality): σ A [ P 1 , P 2 , ρ 1 , ρ 2 ] = [ P 1 [ G ] , 1 − t ρ 2 ρ 1 ] =  P 2 [ G ] , 1 − t ′ ρ 1 ρ 2  . Therefore for all n 6 1 , the split monomorphism σ ′ A , associated to the amalgamate d free produ ct G = G 2 ∗ F G 1 , satisﬁes the equati on σ A = σ ′ A ◦ τ A : f Nil n − 1 ( R [ F ]; B 1 , B 2 ) − → K n ( R [ G ]) . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1023 3.1.5 Induction W e analyz e the ef fect of inductio n maps on f Nil -summands . Statement 3.19 Recall from Theorem 0.4 the isomorp hism i ∗ : f Nil ∗− 1 ( R [ F ] , α ) = f Nil ∗− 1 ( R [ F ]; B 1 ⊗ R [ F ] B 2 ) − → f Nil ∗− 1 ( R [ F ]; B 1 , B 2 ) ; [ P , ρ ] 7− → [ P , t 2 P , ρ, 1] . Let ( P , ρ ) be an object in the exact cate gory NIL( R [ F ] , α ) . By 3.18 , note σ A i ∗ [ P , ρ ] = σ A [ P , t 2 P , ρ, 1] = [ P [ G ] , 1 − t ρ ] = φ ! σ + B [ P , ρ ] . Thus, for all n 6 1 , we obtain the key equ ality σ A ◦ i ∗ = φ ! ◦ σ + B : f Nil n − 1 ( R [ F ] , α ) − → K n ( R [ G ]) . Lemma 3.20 Let n 6 1 be an inte ger . The split monomorp hisms σ A , σ ′ A , σ + B , σ ′ + B are related by a commutati ve diagram f Nil n − 1 ( R [ F ] , α ) / / σ + B / / τ B ∼ =   i ∗ ∼ = & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ K n ( R [ F ] α [ t ]) φ !   ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ψ + !   f Nil n − 1 ( R [ F ]; B 1 , B 2 ) ) ) σ A ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ∼ = τ A   K n ( R [ F ] α [ t , t − 1 ]) θ ! u u ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ∼ = ( β u ) !   K n ( R [ G ]) f Nil n − 1 ( R [ F ]; B 2 , B 1 ) 5 5 σ ′ A 5 5 ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ K n ( R [ F ] α ′ [ t ′ , t ′ − 1 ]) θ ′ ! i i ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ f Nil n − 1 ( R [ F ] , α ′ ) / / σ ′ + B / / i ′ ∗ ∼ = 8 8 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ K n ( R [ F ] α ′ [ t ′ ]) φ ′ ! ^ ^ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ψ ′ + ! O O Pro of Commutati vity of the vario us parts follo w from the follo wing implications : • 3.10 gi ves φ ! = θ ! ◦ ψ + ! and φ ′ ! = θ ′ ! ◦ ψ ′ + ! • 3.11 gi ves θ ! = θ ′ ! ◦ ( β u ) ! • 3.17 gi ves τ A ◦ i ∗ = i ′ ∗ ◦ τ B A lgebraic & G eo metric T opology XX (20XX) 1024 J F Davis, Q Khan and A Ran ic ki • 3.18 gi ves σ A = σ ′ A ◦ τ A • 3.19 gi ves σ A ◦ i ∗ = φ ! ◦ σ + B and σ ′ A ◦ i ′ ∗ = φ ′ ! ◦ σ ′ + B . Observ e the action of G / G on K n ( R [ G ]) is inn er , hence is triv ial. Ho wev er , the action of C 2 = G / G on K n ( R [ G ]) is outer , indu ced by , say c 1 : G → G ; y 7→ t 1 y ( t 1 ) − 1 . (Note c 1 may not hav e order two.) This C 2 -action on K n ( R [ G ]) is non-tr ivia l, as follo w s. Pro position 3.21 Let n ≤ 1 be an int eger . The ind uced m ap θ ! is such that ther e is a commutati ve diagram f Nil n − 1 ( R [ F ] , α ) ⊕ f Nil n − 1 ( R [ F ] , α − 1 ) K n ( R [ G ]) f Nil n − 1 ( R [ F ]; B 1 , B 2 ) K n ( R [ G ]) ❄  i ∗ τ − 1 A i ′ ∗ β + u  ✲ σ B ❄ θ ! ✲ σ A Furthermor e, there is a C 2 -action on the upper lef t hand co rner which inte rchanges the two Nil -summands , and all maps are C 2 -equi varian t. Here, the actio n of C 2 = G / G on the uppe r right is gi ven by ( c 1 ) ! , and the C 2 -action on each lo wer corner is triv ial. Pro of First, we check commutati vity of the square on each Nil -summand: • Lemma 3.20 gi ves σ A ◦ i ∗ = φ ! ◦ σ + B = θ ! ◦ ψ + ! ◦ σ + B = θ ! ◦ σ B | f Nil n − 1 ( R [ F ] , α ) • 3.13 and 3.11 gi ve σ A ◦ τ − 1 A ◦ i ′ ∗ ◦ β + u = σ ′ A ◦ i ′ ∗ ◦ β + u = φ ′ ! ◦ σ ′ + B ◦ β + u = θ ! ◦ ( β u ) − 1 ! ◦ ψ ′ + ! ◦ ( β + u ) ! ◦ σ − B = θ ! ◦ ψ − ! ◦ σ − B = θ ! ◦ σ B | f Nil n − 1 ( R [ F ] , α − 1 ) . Next, deﬁne the i n v olution  0 ε ∗ ε − 1 ∗ 0  on f Nil n − 1 ( R [ F ] , α ) ⊕ f Nil n − 1 ( R [ F ] , α − 1 ) by ε : = ( α − 1 1 ) ! ◦ β + u : NIL( R [ F ] , α − 1 ) − → NIL( R [ F ] , α ) . Here, the automorphi sm α − 1 1 : F → F was deﬁned in 3.9 by x 7→ t 1 x ( t 1 ) − 1 and is the restric tion of c 1 . It remains to sho w σ B and ( i ∗ τ − 1 A ∗ i ′ ∗ β + u ) are C 2 -equi varian t, that is: ( c 1 ) ! ◦ ψ − σ − B = ψ + σ + B ◦ ε (5) τ − 1 A ∗ i ′ ∗ β + u = i ∗ ◦ ε. (6) Observ e that the induced ring automorph ism ( c 1 ) ! : R [ F ] α [ t , t − 1 ] → R [ F ] α [ t , t − 1 ] restric ts to a ring isomorphism ( c 1 ) + ! : R [ F ] α − 1 [ t − 1 ] − → R [ F ] α [ t ] ; x 7− → α − 1 1 ( x ) ; t − 1 7− → t α − 1 1 ( u ) . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1025 Then ( c 1 ) ! ◦ ψ − = ψ + ◦ ( c 1 ) + ! . So ( 5 ) follo ws from the commutati ve square ( c 1 ) + ! ◦ σ − B = σ + B ◦ ( α − 1 1 ) ! β + u : f Nil n − 1 ( R [ F ] , α − 1 ) − → K n ( R [ F ] α [ t ]) , which can be ve riﬁed by formulas for n = 1 and exten ds to n < 1 by v ariation of R . Observ e that ( 6 ) follo ws from the existen ce of an exact natural transfo rmation T : τ − 1 A ◦ i ′ → i ◦ ( α − 1 1 ) ! : NIL( R [ F ] , α ′ ) − → NIL( R [ F ]; t 1 R [ F ] , t 2 R [ F ]) deﬁned on object s ( P , ρ : P → t ′ P = t 2 t 1 P ) by the rule T ( P ,ρ ) : = (1 , ρ ) : ( t 1 P , P , 1 , ρ ) − → ( t 1 P , t ′ P , t 1 ρ, 1); a ke y observ ation f rom 3.9 is the isomorphi sm R [ F ] ⊗ c 1 P → t 1 P ; x ⊗ p 7→ α 1 ( x ) p . 3.1.6 T ransfer W e analyz e the ef fect of transfer maps on f Nil -summands . Statement 3.22 Give n an R [ G ] -module M , let M ! be the abelian gro up M with R [ G ] -action the restrictio n of th e R [ G ] -action. The transfer functor of exac t cate gories θ ! : PR OJ( R [ G ]) − → PR OJ( R [ G ]) ; M 7− → M ! induce s the transfer maps in algebraic K -theory θ ! : K ∗ ( R [ G ]) − → K ∗ ( R [ G ]) . The exa ct functors of Theorem 0.4 combine to gi ve an exa ct functor  j j ′  : NIL( R [ F ]; B 1 , B 2 ) − → N IL( R [ F ] , α ) × NIL( R [ F ] , α ′ ) ; [ P 1 , P 2 , ρ 1 , ρ 2 ] 7− →  [ P 1 , ρ 2 ◦ ρ 1 ] , [ P 2 , ρ 1 ◦ ρ 2 ]  induci ng a m ap betwee n reduced Nil -groups  j ∗ j ′ ∗  : f Nil ∗ ( R [ F ]; B 1 , B 2 ) − → f Nil ∗ ( R [ F ] , α ) ⊕ f Nil ∗ ( R [ F ] , α ′ ) . Pro position 3.23 Let n 6 1 be an integer . The transfer m ap θ ! restric ts to the isomorph ism j ∗ in a commutati ve diagram f Nil n − 1 ( R [ F ]; B 1 , B 2 )   j ∗ ( β + u ) − 1 j ′ ∗     / / σ A / / K n ( R [ G ]) θ !   f Nil n − 1 ( R [ F ] , α ) ⊕ f Nil n − 1 ( R [ F ] , α − 1 ) / /  ψ + σ + B β u ψ − σ − B  / / K n ( R [ G ]) . A lgebraic & G eo metric T opology XX (20XX) 1026 J F Davis, Q Khan and A Ran ic ki Pro of Using the suspens ion isomo rphisms of Section 2 , we may assume n = 1 . Let ( P 1 , P 2 , ρ 1 , ρ 2 ) be an object in NIL( R [ F ]; B 1 , B 2 ) . Deﬁne an R [ G ] -module automorp hism f : =  1 t 2 ρ 2 t 1 ρ 1 1  : P 1 [ G ] ⊕ P 2 [ G ] − → P 1 [ G ] ⊕ P 2 [ G ] . By Theorem 2.6 , we ha ve [ f ] = σ A [ P 1 , P 2 , ρ 1 , ρ 2 ] ∈ K 1 ( R [ G ]) . Note the trans fer is θ ! ( f ) =     1 t 2 ρ 2 0 0 t 1 ρ 1 1 0 0 0 0 1 t 1 ρ 1 0 0 t 2 ρ 2 1     as an R [ G ] -module automorp hism of P 1 [ G ] ⊕ t 1 P 2 [ G ] ⊕ P 2 [ G ] ⊕ t 1 P 1 [ G ] . Furthermore, elementa ry row and colu mn operations produce a diagonal representati on:     1 − t 2 ρ 2 0 0 0 1 0 0 0 0 1 − t 1 ρ 1 0 0 0 1     θ ! ( f )     1 0 0 0 − t 1 ρ 1 1 0 0 0 0 1 0 0 0 − t 2 ρ 2 1     =     1 − t ′ ρ 2 ρ 1 0 0 0 0 1 0 0 0 0 1 − t ρ 1 ρ 2 0 0 0 0 1     . So θ ! [ f ] = [1 − t ′ ρ 2 ρ 1 ] + [1 − t ρ 1 ρ 2 ] . Thus we obtai n a commutati ve diagram f Nil 0 ( R [ F ]; B 1 , B 2 )   j ∗ j ′ ∗     / / σ A / / K 1 ( R [ G ]) θ !   f Nil 0 ( R [ F ] , α ) ⊕ f Nil 0 ( R [ F ] , α ′ ) / /  ψ + σ + B ψ ′ + σ ′ + B  / / K 1 ( R [ G ]) Finally , by 3.13 and 3.11 , note ψ ′ + ◦ σ ′ + B ◦ β + u = ψ ′ + ◦ β + u ◦ σ − B = β u ◦ ψ − ◦ σ − B . 3.2 W aldhausen Nil Examples of bimod ules originate from group rings of amalgamat ed product of groups. Deﬁnition 3 .24 A s ubgroup H of a group G is almost- normal if | H : H ∩ xHx − 1 | < ∞ for ev ery x ∈ G . In other words, H is commensurate with all its conjugat es. Equi va lently , H is a n almos t-normal sub group of G if e very ( H , H ) -double c oset HxH is both a union of ﬁnitely many left c osets gH and a union of ﬁnitely many right co sets Hg . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1027 Remark 3.25 Almost-nor mal subgroups arise in the Shimura theory of automorphic functi ons, with ( G , H ) called a Hecke pair . Here are two suf ﬁcient conditio ns for a subgro up H ⊂ G to be almost-normal: if H is a ﬁnite-inde x subgr oup of G , or if H is a normal subgroup of G . Examples of almost-normal subgroups are giv en in [ 12 , page 9]. Here is our reduction for a certain class of group rings, sp ecializing the General Algebrai c S emi-spli tting of Theorem 0.4 . Cor ollary 3.26 Let R be a ring. Let G = G 1 ∗ F G 2 be an injecti ve amalgamated produ ct of groups over a subgro up F of G 1 and G 2 . Suppose F is an almost-normal subgro up of G 2 . Then, for all n ∈ Z , there is an isomo rphism of abelian groups: j ∗ : f Nil n ( R [ F ]; R [ G 1 − F ] , R [ G 2 − F ]) − → f Nil n ( R [ F ]; R [ G 1 − F ] ⊗ R [ F ] R [ G 2 − F ]) . Pro of Consider the se t J : = ( F \ G 2 / F ) − F of non-tri vial doub le coset s. Let I be the poset of all ﬁnite subsets of J , p artially ordered by inclusion. Note, as R [ F ] -bimodule s: R [ G 2 − F ] = colim I ∈I R [ I ] where R [ I ] : = M FgF ∈ I R [ FgF ] . Since F is an almost-no rmal subgroup of G 2 , each R [ F ] -bimodul e R [ I ] is a ﬁnitely genera ted free (hence projecti ve) left R [ F ] -module. Observ e that I is a ﬁltered poset: if I , I ′ ∈ I then I ∪ I ′ ∈ I . Therefo re we are done by Theorem 0.4 . The case of G = D ∞ = Z 2 ∗ Z 2 has a particu larly simple form. Cor ollary 3.27 Let R be a ring and n ∈ Z . There are natural isomorphis ms: (1) f Nil n ( R ; R , R ) ∼ = f Nil n ( R ) (2) K n ( R [ D ∞ ]) ∼ = ( K n ( R [ Z 2 ]) ⊕ K n ( R [ Z 2 ])) / K n ( R ) ⊕ f Nil n − 1 ( R ) . Pro of Part (i) follows from Corollary 3.26 with F = 1 and G i = Z 2 . Then Part (ii) follo w s from W aldhausen’ s exact sequenc e ( 2.6 ), where the group retractio n Z 2 → 1 induce s a splitting of the map K n ( R ) → K n ( R [ Z 2 ]) × K n ( R [ Z 2 ]) . Example 3.28 Consi der the gro up G = G 0 × D ∞ where G 0 = Z 2 × Z 2 × Z . Since G surjec ts onto the inﬁnite dihedral group, there is an amalgamated product decompo sition G = ( G 0 × Z 2 ) ∗ G 0 ( G 0 × Z 2 ) A lgebraic & G eo metric T opology XX (20XX) 1028 J F Davis, Q Khan and A Ran ic ki with the corres ponding index 2 subgro up G = G 0 × Z . Corollary 3.27 (1) gi ves an isomo rphism f Nil − 1 ( Z [ G 0 ]; Z [ G 0 ] , Z [ G 0 ]) ∼ = f Nil − 1 ( Z [ G 0 ]) . On the other hand, Bass showed that the latter group is an inﬁnitely generated abelian group of expo nent a po wer of two [ 2 , XII, 10.6]. Hence, by W aldhaus en’ s algebraic K -theory decompositio n resu lt, Wh( G ) is inﬁnitely generat ed due to Nil elemen ts. No w constru ct a codimension 1, ﬁnite CW -pair ( X , Y ) w ith π 1 X = G realizing the abo ve amalgamate d product decomposit ion – for ex ample, let Y → Z be a map of con nected CW -complex es induci ng the ﬁrst factor inclu sion G 0 → G 0 × Z 2 on the fundamenta l group and let X be the dou ble mapping c ylinder of Z ← Y → Z . Next construct a homotop y equi v alence f : M → X of ﬁnite C W -complex es whose torsio n τ ( f ) ∈ Wh( G ) is a non-zero Nil element. Then f is non-splittab le along Y by W aldhause n [ 24 ] (see Theorem 4.3 ). This is the ﬁrst explic it example of a non-zer o W aldhausen f Nil group and a n on-splittable homotopy equiv alence in the two-si ded case. 3.3 Farr ell–Jones Conjectur e The Farrell–Jon es Conject ure asserts the fa mily of virt ually cyclic subgroups is a “gener ating” family for K n ( R [ G ]) . In this section we apply our main theorem to sho w the Farrell–J ones Conjecture holds up to dimensio n o ne if and only if the smaller family of ﬁnite -by-cy clic subgroups is a ge nerating family fo r K n ( R [ G ]) up to di mension one. Let O r G be the orb it cate gory of a grou p G ; objects are G -sets G / H w here H is a subgrou p of G and morph isms are G -maps. Da vis–L ¨ uck [ 4 ] deﬁne d a functor K R : Or G → Spectra with the ke y pr operty π n K R ( G / H ) = K n ( R [ H ]) . The utility of such a fun ctor is that it allo ws the deﬁnition of an equi varia nt homology theo ry , indeed for a G -CW -comple x X , one deﬁnes H G n ( X ; K R ) : = π n (map G ( − , X ) + ∧ Or G K R ( − )) (see [ 4 , Sections 4, 7] for basic properti es). Note that the “coef ﬁcients” of the homology theory are gi ven by H G n ( G / H ; K R ) = K n ( R [ H ]) . A family F of subgroups of G is a nonempty set of subgroups closed under conjugati on and taking subgroups . For such a family , E F G is the classifying space for G -actions with isotropy in F . It is cha racterized up to G -h omotopy typ e as a G -CW -comple x so A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1029 that ( E F G ) H is co ntractible for subgro ups H ∈ F and is e mpty for subg roups H 6∈ F . Four rele vant families are fin ⊂ fbc ⊂ vc ⊂ all , the familie s of ﬁ nite subgroup s, ﬁnite-by -cyclic , virtually cyclic subgrou ps and all subgro ups respecti vely . Here fbc : = fin ∪ { H < G | H ∼ = F ⋊ Z with F ﬁ nite } vc : = { H < G | ∃ cyc lic C < H with ﬁnite index } . The Farrell– Jones C onjec ture in K -theo ry for the group G [ 7 , 4 ] assert s an isomor - phism: H G n ( E vc G ; K R ) − → H G n ( E all G ; K R ) = K n ( R [ G ]) . W e no w state a more general versio n, the Fibered Farrell– Jones Conjecture. Let ϕ : Γ → G be a group homomorphism. If F is a family of subg roups of G , d eﬁne the family of su bgroups ϕ ∗ F : = { H < Γ | ϕ ( H ) ∈ F } . The Fibered Farrell –Jones Conjecture in K -theory for the group G asserts, for eve ry ring R and homomorphism ϕ : Γ → G , that follo wing induced map is an isomorphism: H Γ n ( E ϕ ∗ vc ( G ) Γ ; K R ) − → H Γ n ( E ϕ ∗ all ( G ) Γ ; K R ) = K n ( R [ Γ ]) . The follo w ing theorem was prov ed for all n in [ 5 ] using controlle d topolog y . W e giv e a proof belo w up to dimension one using only algebraic topology . Theor em 3.29 Let ϕ : Γ → G be an homomorphism of groups. Let R be any ring. The inclus ion-indu ced map H Γ n ( E ϕ ∗ fbc ( G ) Γ ; K R ) − → H Γ n ( E ϕ ∗ vc ( G ) Γ ; K R ) . is an isomorph ism for all integers n < 1 and an epimorphi sm for n = 1 . Hence we propose a s harpening of the Farrell–Jon es Conjecture in al gebraic K -theor y . Conjectur e 3.30 Let G be a discrete group, and let R be a ring . Let n be an integ er . (1) T here is an isomorp hism: H G n ( E fbc G ; K R ) − → H G n ( E all G ; K R ) = K n ( R [ G ]) . (2) For any homomorp hism ϕ : Γ → G of groups, there is an isomorphism: H Γ n ( E ϕ ∗ fbc ( G ) Γ ; K R ) − → H Γ n ( E all Γ ; K R ) = K n ( R [ Γ ]) . A lgebraic & G eo metric T opology XX (20XX) 1030 J F Davis, Q Khan and A Ran ic ki The proof of Theorem 3.29 will require three auxiliary results, some of which w e quote from other s ources. The ﬁrst is a varian t of Theorem A.10 of F arrell–Jo nes [ 7 ], whose proof is identi cal to the proof of Theorem A.10. T ransitivit y Principle Let F ⊂ G be familie s of subgroups of a grou p Γ . Let E : Or Γ → Spectra be a funct or . Let N ∈ Z ∪ {∞} . If for all H ∈ G − F , the assembly map H H n ( E F | H H ; E ) − → H H n ( E all H ; E ) is an isomorph ism for n < N and an epimorphi sm if n = N , then the map H Γ n ( E F Γ ; E ) − → H Γ n ( E all H ; E ) is an isomorph ism for n < N and an epimorphi sm if n = N . Of course , we apply this principle to the f amilies fbc ⊂ vc . The second aux iliary result is a well-kno wn lemma (see [ 21 , Theor em 5.12]), b ut w e offer an alternati ve proof. Lemma 3.31 Let G be a virtuall y cyclic gro up. Then either (1) G is ﬁnite. (2) G maps onto Z ; hen ce G = F ⋊ α Z with F ﬁnite. (3) G maps onto D ∞ ; hence G = G 1 ∗ F G 2 with | G i : F | = 2 and F ﬁnite. Pro of Assume G is an inﬁnite virtually cyclic group. The intersec tion of the conju- gates of a ﬁ nite inde x, inﬁnite cyclic subgro up is a normal, ﬁnite index, inﬁnite cyclic subgro up C . Let Q be the ﬁnite quotien t group. Embed C as a subgro up of index | Q | in an inﬁnite cyclic group C ′ . There exists a un ique Z [ Q ] -module structur e on C ′ such that C is a Z [ Q ] -submodule. Observ e that the image of the obstruction cocycle under the map H 2 ( Q ; C ) → H 2 ( Q ; C ′ ) is tri vial. Hence G embeds as a ﬁnite index s ubgroup of a semidirect product G ′ = C ′ ⋊ Q . N ote G ′ maps epimorph ically to Z (if Q acts tri vially) or to D ∞ (if Q acts non-t rivi ally). In either case, G maps epimorph ically to a subgroup of ﬁnite inde x in D ∞ , which must be either inﬁnite cyclic or inﬁnite dihedr al. In order t o see ho w the reduce d Nil -groups rela te to equi var iant homology (and hence to the Farre ll–Jones Conjecture), w e need [ 5 , Lemma 3.1], the third auxili ary result. A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1031 Lemma 3.32 (Da vis–Quinn –Reich) Let G be a group of the form G 1 ∗ F G 2 with | G i : F | = 2 , and let fac be the smallest family of subgrou ps of G contai ning G 1 and G 2 . Let G be a group of the form F ⋊ α Z , and let fac be the smallest family of subgro ups of G contai ning F . Note that F need not be ﬁnite. (1) The follo wing exact sequen ces are split, and hence short exa ct: H G n ( E fac G ; K R ) f A − − → H G n ( E all G ; K R ) η A − − → H G n ( E all G , E fac G ; K R ) H G n ( E fac G ; K R ) f B − − → H G n ( E all G ; K R ) η B − − → H G n ( E all G , E fac G ; K R ) . Here f A , η A and f B , η B are inclusi on-induc ed maps. (2) The maps η A ◦ σ A : f Nil n − 1 ( R [ F ]; R [ G 1 − F ] , R [ G 2 − F ]) ∼ = − → H G n ( E all G , E fac G ; K R ) η B ◦ σ B : f Nil n − 1 ( R [ F ] , α ) ⊕ f Nil n − 1 ( R [ F ] , α − 1 ) ∼ = − → H G n ( E all G , E fac G ; K R ) are isomorph isms where σ A and σ B are W aldhausen ’ s split injections. The statement of Lemma 3.1 of [ 5 ] does not explici tly identify the isomorph isms in Part (2) a bov e, but the identiﬁcat ion follo ws from the last paragraph of the proof. It is not dif ﬁcult to compute H G n ( E fac G ; K R ) and H G n ( E fac G ; K R ) in terms of a W ang sequen ce and a M ayer –V ietoris sequence respecti vely . An example is in Section 3.4 . Next, we further assume G ⊂ G with | G : G | = 2 . Then C 2 = G / G acts on K n ( R G ) = H G n ( E all G ; K R ) by conjugation . By [ 5 , Remark 3.21], there is a C 2 -action on H G n ( E all G , E fac G ; K R ) so that η B and θ !! belo w are C 2 -equi varian t. Lemma 3.33 Let n ≤ 1 be an integ er . There is a commutati ve diagra m of C 2 - equi varian t homomorphis ms: f Nil n − 1 ( R [ F ] , α ) ⊕ f Nil n − 1 ( R [ F ] , α − 1 ) H G n ( E all G , E fac G ; K R ) f Nil n − 1 ( R [ F ]; B 1 , B 2 ) H G n ( E all G , E fac G ; K R ) ❄  i ∗ τ − 1 A i ′ ∗ β + u  ✲ η B ◦ σ B ❄ θ !! ✲ η A ◦ σ A Here, the C 2 = G / G -action on t he upper left-hand corner is gi ven in Proposit ion 3.21 , on th e upper r ight it is giv en by [ 5 , Remark 3.21 ], and on ea ch lo wer cor ner it is tri vial. A lgebraic & G eo metric T opology XX (20XX) 1032 J F Davis, Q Khan and A Ran ic ki Pro of This follo ws from Proposit ion 3.21 and the C 2 -equi varian ce of η B . Recall that if C 2 = { 1 , T } and if M is a Z [ C 2 ] -module then the coin variant group M C 2 = H 0 ( C 2 ; M ) is the quoti ent group of M modulo the subgroup { m − Tm | m ∈ M } . Lemma 3.34 Let n ≤ 1 be an inte ger . There is an inductio n-induced isomorphis m:  H G n ( E all G , E fac G ; K R )  C 2 − → H G n ( E fac G ∪ sub G G , E fac G ; K R ) . Pro of Recall G / G = C 2 . Since fac = fac ∩ G , by [ 5 , Lemma 4.1(i)] there is a identi ﬁcation of Z [ C 2 ] -modules H G n ( E all G , E fac G ; K ) = π n ( K / K fac )( G / G ) . The C 2 -coin varian ts can be inter preted as a C 2 -homolo gy group:  π n ( K / K fac )( C 2 )  C 2 = H C 2 0 ( EC 2 ; π n ( K / K fac )( C 2 )) . By Lemma 3.32 (2) and Lemm a 3.3 3 , the coefﬁcie nt Z [ C 2 ] -module is induce d from a Z -module. By the Ati yah–Hirzebr uch spectral sequ ence (which colla pses at E 2 ), n ote H C 2 0 ( EC 2 ; π n ( K / K fac )( C 2 )) = H C 2 n ( EC 2 ; ( K / K fac )( C 2 )) . Therefore , by [ 5 , Lemma 4.6, Lemma 4.4, Lemma 4.1], we concl ude: H C 2 n ( EC 2 ; ( K / K fac )( C 2 )) = H G n ( E sub G G ; K / K fac ) = H G n ( E fac G ∪ sub G G ; K / K fac ) = H G n ( E fac G ∪ sub G G , E fac G ; K ) . The ide ntiﬁcations in the abov e proof a re ex tracted from the proof of [ 5 , Theore m 1.5]. Pro of of Theor em 3.29 Let ϕ : Γ → G be a homomorphism of groups. Using the T ransiti vity Principle applied to the famili es ϕ ∗ fbc ⊂ ϕ ∗ vc , it sufﬁces to sho w that H H n ( E all H , E ϕ ∗ fbc | H H ; K R ) = 0 for all n ≤ 1 and for all H ∈ ϕ ∗ vc − ϕ ∗ fbc . T o identify the family ϕ ∗ fbc | H we will use two f acts, the proofs of which are left to the reader . • If q : A → B is a group epimorphis m with ﬁnite kernel, then fbc A = q ∗ fbc B . (The ke y step is to sho w that an epimorphic image of a ﬁnite-by-c yclic group is ﬁnite-by -cyclic .) A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1033 • If q : A → B = G 1 ∗ F G 2 is an group epimorp hism, then A = q − 1 G 1 ∗ q − 1 F q − 1 G 2 and fac A = q ∗ fac B . Let ϕ | : H → ϕ ( H ) denote the restriction of ϕ to H . By the deﬁnition of both sides, ϕ ∗ fbc | H = ϕ | ∗ ( fbc ϕ ( H )) . Since H ∈ ϕ ∗ vc − ϕ ∗ fbc , we ha ve ϕ ( H ) ∈ vc − fbc . S o, by Lemma 3.31 , the re is an epimorph ism p : ϕ ( H ) → D ∞ = Z 2 ∗ 1 Z 2 with ﬁnite ker nel. By the ﬁrst fact abo ve ϕ | ∗ ( fbc ϕ ( H )) = ϕ | ∗ ( p ∗ ( fbc D ∞ )) . Next, write H : = ( p ◦ ϕ | ) − 1 ( Z ) . Note ϕ | ∗ ( p ∗ ( fbc D ∞ )) = ( p ◦ ϕ | ) ∗ ( fbc D ∞ ) = ( p ◦ ϕ | ) ∗ ( fac D ∞ ∪ sub Z ) = ( p ◦ ϕ | ) ∗ ( fac D ∞ ) ∪ ( p ◦ ϕ | ) ∗ ( sub Z ) = fac H ∪ sub H , where the last equalit y uses the second fac t abo ve. Thus it sufﬁces to prov e, for any group H mappin g epimorphica lly to D ∞ and for all n ≤ 1 , that H H n ( E all H , E fac H ∪ sub H ; K R ) = 0 where H is th e in verse image of the maximal inﬁnite cycli c subgroup of D ∞ . Consider the follo w ing composite  H H n ( E all H , E fac H ; K R )  H / H α − − → H H n ( E fac H ∪ sub H H , E fac H ; K R ) β − − → H H n ( E all H , E fac H ; K R ) . The map α exists and is an isomorph ism by Lemma 3.34 . Apply C 2 -co vari ants to the commutat iv e diagram in the state ment of Lemma 3.33 . In this diagram of C 2 - coin v ariants, the to p and bottom are isomorphisms by Lemma 3.32 (2) an d the left map is an isomorphis m by Proposit ion 3.21 and Theorem 0.4 . Hence the right-hand map, which is β ◦ α , is an isomorphism for all n ≤ 1 . It follo ws that β is an isomorphism for all n ≤ 1 . So, by the e xact sequence of a triple, we obtain H H n ( E all H , E fac H ∪ sub H H ; K R ) = 0 for all n ≤ 1 as desired. A lgebraic & G eo metric T opology XX (20XX) 1034 J F Davis, Q Khan and A Ran ic ki 3.4 K -theory of the modular group Let Γ = Z 2 ∗ Z 3 = PSL 2 ( Z ) . The follo wing theorem follo ws from applyin g our main theore m and the recent proof [ 1 ] of the Farrell–Jo nes conjecture in K -theory for word hyperb olic groups. The Cayley graph for Z 2 ∗ Z 3 with respect to the generating set giv en by the nonzero elements of Z 2 and Z 3 has the quasi-isomet ry typ e of the usual Bass–Serre tree for the amalgamated product ( Figure 1 ). This is an inﬁnite tree with altern ating vertices • ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ • • ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ • ① ① ① ① ① ① ① ① ① • • • ① ① ① ① ① ① ① ① ① ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ • ① ① ① ① ① ① ① ① ① • ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ • ① ① ① ① ① ① ① ① ① • Figure 1: Bass–Serre tree for P SL 2 ( Z ) of va lence two and three. The group Γ acts on the tree, with the generator of order two acting by reﬂection thro ugh an v alence two verte x and th e ge nerator of or der thre e acting by rotat ion through an adjoining verte x of vale nce three. Any geodesic triangle in the B ass–Ser re tree has the propert y that the union of two sides is the union of all three sides. It follo ws that the Bass–Serre graph is δ -hyperb olic for any δ > 0 , the Cayley graph is δ -hy perbolic for some δ > 0 , and hence Γ is a hyperb olic group. Theor em 3.35 For a ny ring R and inte ger n , K n ( R [ Γ ]) = ( K n ( R [ Z 2 ]) ⊕ K n ( R [ Z 3 ])) / K n ( R ) ⊕ M M C f Nil n − 1 ( R ) ⊕ f Nil n − 1 ( R ) ⊕ M M D f Nil n − 1 ( R ) where M C and M D are the set of conjuga cy classes of maximal inﬁnite cyclic sub- group s and maximal inﬁnite dihedral subgrou ps, respe ctiv ely . Moreove r , all virtually cyc lic subgroups of Γ are cycl ic or inﬁnite dihedral. A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1035 Pro of By Lemma 3.32 , the exac t sequence of ( E all Γ , E fin Γ ) is short exac t and split: H Γ n ( E fin Γ ; K R ) → H Γ n ( E all Γ ; K R ) → H Γ n ( E all Γ , E fin Γ ; K R ) . Then, by the Farrell –Jones C onject ure [ 1 ] for word hyp erbolic groups, w e obt ain K n ( R [ Γ ]) = H Γ n ( E fin Γ ; K R ) ⊕ H Γ n ( E all Γ , E fin Γ ; K R ) = H Γ n ( E fin Γ ; K R ) ⊕ H Γ n ( E vc Γ , E fin Γ ; K R ) . Observ e E fin Γ is construct ed as a pushout of Γ -spaces Γ ⊔ Γ − − − − → Γ / Z 2 ⊔ Γ / Z 3   y   y Γ × D 1 − − − − → E fin Γ . Then E fin Γ is the Bass–Serre tree for Γ = Z 2 ∗ Z 3 . Note that H Γ ∗ ( Γ / H ; K R ) = K ∗ ( R [ H ]) . The pushout giv es, after ca nceling a K n ( R ) term, a split long exact sequenc e · · · → K n ( R ) → K n ( R [ Z 2 ]) ⊕ K n ( R [ Z 3 ]) → H Γ n ( E fin Γ ; K R ) → K n − 1 ( R ) → · · · . Hence H Γ n ( E fin Γ ; K R ) = ( K n ( R [ Z 2 ]) ⊕ K n ( R [ Z 3 ])) / K n ( R ) . Next, for a w ord hyperbo lic group G , H G n ( E vc G , E fin G ; K ) ∼ = M [ V ] ∈M ( G ) H V n ( E vc V , E fin V ; K ) where M ( G ) is the set of conjugac y classes of m aximal virtually cyclic subgrou ps of G (see [ 16 , Theorem 8.11 ] and [ 10 ]). T he g eometric interpr etation of this re sult is that E vc G is obtaine d by coning of f each geodesic in the tree E fin G ; then apply ex cision. The Kurosh subgro up theor em implies that a subgro up of Z 2 ∗ Z 3 is a free product of Z 2 ’ s, Z 3 ’ s, and Z ’ s. Note that Z 2 ∗ Z 3 = h a , b | a 2 = 1 = b 3 i , Z 3 ∗ Z 3 = h c , d | c 3 = 1 = d 3 i , and Z 2 ∗ Z 2 ∗ Z 2 = h e , f , g | e 2 = f 2 = g 2 = 1 i hav e free subgro ups of ra nk 2 , for example h ab , ab 2 i , h cd , cd 2 i , and h ef , fg i . On the other han d, the free grou p F 2 rank 2 is not a virtually c yclic gro up since its ﬁrst B etti number β 1 ( F 2 ) = rank H 1 ( F 2 ) = 2 , while for a virtually c yclic group V , transfe rring to the cyclic subgroup C ⊂ V of ﬁ nite inde x sho ws that β 1 ( V ) is 0 or 1. Subg roups of virtua lly cyclic groups are also virtually cycli c. T herefo re all virtua lly cyclic subgroups of Γ are cyclic or inﬁnite dihed ral. By the fund amental theore m of K -theory and W aldhausen’ s Theorem ( 3.32 ): H Z n ( E vc Z , E fin Z ; K R ) = f Nil n − 1 ( R ) ⊕ f Nil n − 1 ( R ) H D ∞ n ( E vc D ∞ , E fin D ∞ ; K R ) = f Nil n − 1 ( R ; R , R ) A lgebraic & G eo metric T opology XX (20XX) 1036 J F Davis, Q Khan and A Ran ic ki Finally , by Corollary 3.27 (1), we obtai n exactly one type of Nil-group: f Nil n − 1 ( R ; R , R ) ∼ = f Nil n − 1 ( R ) . Remark 3.36 The sets M C and M D are countably inﬁnite. This can be sho w n by parameterizi ng these subsets either: combinato rially (using that elements in Γ are words in a , b , b 2 ), geometrica lly (maximal virtually cy clic sub groups corr espond to stabili zers of geodesics in the B ass–Serre tree E fin Γ , where the geodesic m ay or may not be in va riant under an element of order 2), or numbe r theoreticall y (using solution s to Pell’ s equation and Gauss’ theory of binary quadratic forms [ 20 ]). Let us gi ve an over vie w and history of some related work. The Farrell–J ones Conjecture and the classiﬁcation of virtual ly cyclic groups ( 3.31 ) focused attention on the algebraic K -theory of groups mapping to the inﬁnite dihedra l group. Sev eral years ago James Dav is and Bogdan V ajiac outlined a unpublished proof of Theorem 0.1 when n 6 0 using controll ed top ology and hyper bolic geometry . L afont and Ortiz [ 13 ] prov ed that f Nil n ( Z [ F ]; Z [ V 1 − F ] , Z [ V 2 − F ]) = 0 if and only if f Nil n ( Z [ F ] , α ) = 0 for any virtua lly cyc lic group V with an epimorphis m V → D ∞ and n = 0 , 1 . More recently , Lafont–Orti z [ 15 ] hav e s tudied t he more g eneral c ase o f t he K -theory K n ( R [ G 1 ∗ F G 2 ]) of an injecti ve amalgam, where F , G 1 , G 2 are ﬁnite groups. F inally , we mentioned the paper [ 5 ], which was written in parallel w ith this one; it an altern ate pro of of Theorem 0.1 . Also, [ 5 ] pro vides se veral auxiliary result s used in Section 3.3 of this paper . T he Nil-Nil isomorp hism of Theorem 0.1 has been used in a geometric ally moti va ted computatio n of Lafont–Orti z [ 14 , Section 6.4]. 4 Codimension 1 splitting and semi-splitting W e shal l now giv e a topologic al interpretat ion of the Nil-Nil Theorem 1.1 , proving in Theorem 4.5 that e very homotop y equi vale nce of ﬁnite CW -comple xes f : M → X = X 1 ∪ Y X 2 with X 1 , X 2 , Y connecte d and π 1 ( Y ) → π 1 ( X ) inje ctiv e is “semi-spli t” along Y ⊂ X , assuming that π 1 ( Y ) is of ﬁ nite index in π 1 ( X 2 ) . Indeed, the proof of Theorem 1.1 is motiv ated by the codimens ion 1 splitting obstruc tion theory of W ald- hausen [ 24 ], and the su bsequent algebraic K -theory decompos ition theorems of W ald- hausen [ 25 , 26 ]. The papers [ 24 , 25 ] de veloped both an algebra ic splitting obstructio n theory for chain comple xes over injecti ve general ized free produ cts, and a geometri c codimen sion 1 splitting obstruction theory; the geometric splitting obstruction is the algebr aic splitting obstructio n of the cellul ar ch ain complex. There are parallel theories A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1037 for the separating t ype (A) (amalgamated fr ee product) a nd th e no n-separat ing type ( B) (HNN exte nsion). W e ﬁrst brieﬂy outline the theory , mainly for type (A). The cellul ar chain complex of the uni versal cov er e X of a connecte d CW -complex X is a based free Z [ π 1 ( X )] -module chain co mplex C ( e X ) such that H ∗ ( e X ) = H ∗ ( C ( e X )) . T he ker nel Z [ π 1 ( X )] -modules of a map f : M → X are deﬁned by K ∗ ( M ) : = H ∗ + 1 ( e f : e M → e X ) with e M : = f ∗ e X the pullback cove r of M and e f : e M → e X a π 1 ( X ) -equi var iant lift of f . For a cel lular map f of CW -complex es let K ( M ) : = C ( e f : C ( e M ) → C ( e X )) ∗ + 1 be the algebraic mapping cone of the induced Z [ π 1 ( X )] -module cha in map e f , with homolog y Z [ π 1 ( X )] -modules H ∗ ( K ( M )) = K ∗ ( M ) = H ∗ + 1 ( e f : e M → e X ) . For n > 1 the m ap f : M → X is n -connect ed if and only if f ∗ : π 1 ( M ) ∼ = π 1 ( X ) and K r ( M ) = 0 for r < n , in which case the Hurewicz map is a n isomorphism: π n + 1 ( f ) = π n + 1 ( e f ) → K n ( M ) = H n + 1 ( e f ) . By t he th eorem of J H C Wh itehead, f : M → X is a homotop y equi vale nce if an d only if f ∗ : π 1 ( M ) ∼ = π 1 ( X ) and K ∗ ( M ) = 0 (if and only if K ( M ) is chain contrac tible). Decompose the bounda ry of the ( n + 1) -disk as a union of upper and lower n -disks: ∂ D n + 1 = S n = D n + ∪ S n − 1 D n − . Giv en a CW -complex M and a cellu lar map φ : D n + → M deﬁne a ne w CW -complex M ′ = ( M ∪ ∂ φ D n − ) ∪ φ ∪ 1 D n + 1 by attac hing an n -cell and an ( n + 1) -cell, with ∂ φ = φ | : S n − 1 → M , φ ∪ 1 : S n = D n + ∪ S n − 1 D n − → M ∪ ∂ φ D n − . The inclus ion M ⊂ M ′ is a homoto py equi v alence called an elementary expansion . The cellular based free Z [ π 1 ( M )] -module chain complex es ﬁ t into a short e xact se- quenc e 0 → C ( e M ) → C ( e M ′ ) → C ( e M ′ , e M ) → 0 with C ( e M ′ , e M ) : · · · / / 0 / / Z [ π 1 ( M )] 1 / / Z [ π 1 ( M )] / / 0 / / · · · A lgebraic & G eo metric T opology XX (20XX) 1038 J F Davis, Q Khan and A Ran ic ki concen trated in dimensions n , n + 1 . For a co mmutati ve diagram of cellular maps D n + φ / /   M f   D n + 1 δ φ / / X f e xtends to a cellular map f ′ = ( f ∪ δ φ | D n − ) ∪ δ φ : M ′ = ( M ∪ ∂ φ D n − ) ∪ φ ∪ 1 D n + 1 → X which is also called an elementary expansion , and there is de ﬁned a short exa ct sequen ce of based free Z [ π 1 ( X )] -module chain complex es 0 → K ( M ) → K ( M ′ ) → C ( e M ′ , e M ) → 0 . Recall the Whitehead gr oup of a group G is deﬁned by Wh( G ) : = K 1 ( Z [ G ]) / {± g | g ∈ G } . Suppose the CW -complex es M , M ′ , X are ﬁ nite. The Whitehead torsion of a homotopy equi valenc e f : M → X is τ ( f ) = τ ( K ( M )) ∈ Wh( π 1 ( X )) . Homotop y equi vale nces f : M → X and f ′ : M ′ → X are simple-homotopic if τ ( f ) = τ ( f ′ ) ∈ Wh( π 1 ( X )) . This is equiv alent to being able to obtain f ′ from f by a ﬁnite sequenc e of elemen- tary exp ansions and subdi visions and their formal in verses . For details , see Cohen’ s book [ 3 ]. A 2-sided codimension 1 pair ( X , Y ⊂ X ) is a pair of spaces such that the inclusi on Y = Y × { 0 } ⊂ X ex tends to an o pen embe dding Y × R ⊂ X . W e sa y tha t a homotopy equi valenc e f : M → X splits along Y ⊂ X if the restric tions f | : N = f − 1 ( Y ) → Y , f | : M − N → X − Y are also homotop y equiv alences. In dealing w ith maps f : M → X and 2-sided codimension 1 pair s ( X , Y ) we shall assume that f is cellula r and that both ( X , Y ) and ( M , N = f − 1 ( Y )) are a 2-sid ed codimen sion 1 CW -pair . A 2-sided codimensi on 1 CW -pair ( X , Y ) is π 1 -injecti ve if X , Y are connected and π 1 ( Y ) → π 1 ( X ) is injecti ve. As usual, there are two cases, accordin g as to whether Y separa tes X or not: A lgebraic & G eo metric T opology XX (20XX) Algebraic K -theory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1039 (A) The separating case: X − Y is disconnec ted, so X = X 1 ∪ Y X 2 with X 1 , X 2 conne cted. By the Seifert-v an Kampen theorem π 1 ( X ) = π 1 ( X 1 ) ∗ π 1 ( Y ) π 1 ( X 2 ) is the amalgamated free product determin ed by the injections i k : π 1 ( Y ) → π 1 ( X k ) ( k = 1 , 2 ). (B) The non-separating case: X − Y is connected , so X = X 1 / { y ∼ ty | y ∈ Y } for a connect ed space X 1 (a deformation retract of X − Y ) which contains two disjoi nt copies Y ⊔ tY ⊂ X 1 of Y . By the Seifert-v an Kampen theorem π 1 ( X ) = π 1 ( X 1 ) ∗ i 1 , i 2 { t } is the HN N exte nsion dete rmined by the i njections i 1 , i 2 : π 1 ( Y ) → π 1 ( X 1 ) , with i 1 ( y ) t = ti 2 ( y ) ( y ∈ π 1 ( Y ) ). Remark 4.1 Let e X be the uni versal cov er of X , and let X : = e X /π 1 ( Y ) , so that for both types (A) and (B), ( X , Y ) is a π 1 -inject iv e 2-sid ed codimens ion 1 pair of the separa ting type (A), with X = X − ∪ Y X + for connec ted subspaces X − , X + ⊂ X such that π 1 ( X ) = π 1 ( X − ) = π 1 ( X + ) = π 1 ( Y ) . Moreo ver f or ty pe (B), wh en i 1 , i 2 are isomorphisms, the HN N extension simpl iﬁes to: 1 − − − − → π 1 ( Y ) − − − − → π 1 ( X ) = π 1 ( Y ) ⋊ α Z − − − − → Z − − − − → 1 with automor phism α = ( i 1 ) − 1 i 2 of π 1 ( Y ) , studied originally by Farrell and H siang [ 6 ]. F r om now on, we shall only consid er the separ ating case (A) of X = X 1 ∪ Y X 2 . Write π 1 ( X ) = G , π 1 ( X 1 ) = G 1 , π 1 ( X 2 ) = G 2 , π 1 ( Y ) = H , i k : Z [ H ] → Z [ G k ] = Z [ H ] ⊕ B k , B k = Z [ G k − H ] with B k free as both a right and a left Z [ H ] -module, and Z [ G ] = Z [ G 1 ] ∗ Z [ H ] Z [ G 2 ] = Z [ H ] ⊕ B 1 ⊕ B 2 ⊕ B 1 B 2 ⊕ B 2 B 1 ⊕ . . . . Use the injecti ons i k : H → G k to deﬁne cov ers X 1 = e X 1 / H ⊂ X − , X 2 = e X 2 / H ⊂ X + A lgebraic & G eo metric T opology XX (20XX) 1040 J F Davis, Q Khan and A Ran ic ki such that X 1 ∩ X 2 = Y and e X =   [ g 1 G 1 ∈ G / G 1 g 1 e X 1   ∪ S hH ∈ G / H h e Y !   [ g 2 G 2 ∈ G / G 2 g 2 e X 2   X =   [ g 1 G 1 ∈ G / G 1 g 1 X 1   ∪ S hH ∈ G / H hY !   [ g 2 G 2 ∈ G / G 2 g 2 X 2   with e X k the uni vers al cove r of X k , and e Y the uni versal cove r of Y . Let ( f , g ) : ( M , N ) → ( X , Y ) be a m ap of separating π 1 -inject iv e codimensio n 1 ﬁnite CW -pairs. This giv es an exact sequ ence of based free Z [ H ] -module chain complex es (7) 0 − − − − → K ( N ) − − − − → K ( M ) − − − − → K ( M − , N ) ⊕ K ( M + , N ) − − − − → 0 induci ng a long exact sequen ce of homology modules · · · / / K r ( N ) / / K r ( M ) / / K r ( M + , N ) ⊕ K r ( M − , N ) / / K r − 1 ( N ) / / · · · . Note that f : M → X is a homotop y equi vale nce if and only if f ∗ : π 1 ( M ) → π 1 ( X ) is an isomor phism and K ( M ) is contractibl e. The map of pair s ( f , g ) : ( M , N ) → ( X , Y ) is a spl it homo topy eq uiv alence if and on ly if any two of the chain complex es in ( 7 ) are contra ctible, in w hich case the third cha in complex is also contra ctible. Suppose f : M → X is a homotop y equiv alence. Then K ∗ ( M ) = K ∗ ( M ) = 0 , K ∗ ( N ) = K ∗ + 1 ( M − , N ) ⊕ K ∗ + 1 ( M + , N ) . W e obtain an exac t sequence of Z [ H ] -module chain complex es 0 → K ( M 1 , N ) → K ( M − , N ) ρ 1 − − → K ( M − , M 1 ) = B 1 ⊗ Z [ H ] K ( M + , N ) → 0 0 → K ( M 2 , N ) → K ( M + , N ) ρ 2 − − → K ( M + , M 2 ) = B 2 ⊗ Z [ H ] K ( M − , N ) → 0 . The pair ( ρ 1 , ρ 2 ) of intertwined chain maps is chain homotopy nilpote nt , in the sense that the follo w ing chain map is a Z [ G ] -module chain equi vale nce:  1 ρ 2 ρ 1 1  : Z [ G ] ⊗ Z [ H ] ( K ( M − , N ) ⊕ K ( M + , N )) − → Z [ G ] ⊗ Z [ H ] ( K ( M − , N ) ⊕ K ( M + , N )) . Deﬁnition 4.2 Let x = ( P 1 , P 2 , ρ 1 , ρ 2 ) be an object of NIL( Z [ H ]; B 1 , B 2 ) . A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1041 (1) L et x ′ = ( P ′ 1 , P ′ 2 , ρ ′ 1 , ρ ′ 2 ) be another object. W e say x and x ′ are equiv alent if [ P 1 ] = [ P ′ 1 ] , [ P 2 ] = [ P ′ 2 ] ∈ e K 0 ( Z [ H ]) , [ x ] = [ x ′ ] ∈ f Nil 0 ( Z [ H ]; B 1 , B 2 ) , or equi v alently [ x ′ ] − [ x ] ∈ K 0 ( Z ) ⊕ K 0 ( Z ) ⊆ Nil 0 ( Z [ H ]; B 1 , B 2 ) = K 0 ( Z [ H ]) ⊕ K 0 ( Z [ H ]) ⊕ f Nil 0 ( Z [ H ]; B 1 , B 2 ) with K 0 ( Z ) ⊕ K 0 ( Z ) the subg roup generated by ( Z [ H ] , 0 , 0) and (0 , Z [ H ] , 0 , 0) . (2) L et k = 1 or 2 . L et y k ∈ ker( ρ k ) genera te a direct su mmand h y k i ⊆ P k . Deﬁne an object x ′ in NIL( Z [ H ]; B 1 , B 2 ) by x ′ = ( P ′ 1 , P ′ 2 , ρ ′ 1 , ρ ′ 2 ) = ( ( P 1 / h y 1 i , P 2 , [ ρ 1 ] , [ ρ 2 ]) if k = 1 ( P 1 , P 2 / h y 2 i , [ ρ 1 ] , [ ρ 2 ]) if k = 2 with an exa ct sequence in NIL( Z [ H ]; B 1 , B 2 )        0 / / ( Z [ H ] , 0 , 0 , 0) ( y 1 , 0) / / x / / x ′ / / 0 0 / / (0 , Z [ H ] , 0 , 0) (0 , y 2 ) / / x / / x ′ / / 0 . Thus x ′ is equi valen t to x , obtaine d by the algebraic cell-ex change which kills y k ∈ P 1 ⊕ P 2 . It can be shown that two obj ects x and x ′ in NIL( Z [ H ]; B 1 , B 2 ) are equi vale nt if and only if x ′ can be obtained from x by a ﬁnite seque nce of isomorp hisms, algebr aic cell-e xchang es, and their formal in verse s. Geometric cell-exch anges (called surgeries in [ 24 ]) determine algebraic cell-exchanges. In the high ly-conne cted case, al gebraic and geometric c ell-ex changes occur in tandem: Theor em 4.3 ([ 24 ]) Let ( f , g ) : ( M , N ) → ( X , Y ) be a map of separating π 1 -inject iv e codimen sion 1 ﬁ nite CW -pairs, with f : M → X a homotop y equiv alence. Write X = X 1 ∪ Y X 2 with induc ed amalgam π 1 ( X ) = G = G 1 ∗ H G 2 of fund amental groups. (i) Let k = 1 , 2 . Suppose for some n > 0 that we are gi ven a map ( φ, ∂ φ ) : ( D n + 1 , S n ) − → ( M k , N ) and a null-h omotopy of pairs ( θ , ∂ θ ) : ( f | M k ◦ φ, g ◦ ∂ φ ) ≃ ( ∗ , ∗ ) : ( D n + 1 , S n ) − → ( X k , Y ) . A lgebraic & G eo metric T opology XX (20XX) 1042 J F Davis, Q Khan and A Ran ic ki Assume they repr esent an element in ker( ρ k ) (with ǫ = − if k = 1 ; ǫ = + if k = 2 ): y k = [ φ, θ ] ∈ im( K n + 1 ( M k , N ) → K n + 1 ( M ǫ , N )) = ker( ρ k : K n + 1 ( M ǫ , N ) → B k ⊗ Z [ H ] K n + 1 ( M − ǫ , N )) ⊆ K n ( N ) . The map ( f , g ) ext ends to the map of codimension 1 pairs ( f ′ , g ′ ) : = (( f ∪ f | M k ◦ φ ) ∪ θ, g ∪ ∂ θ ) : ( M ′ , N ′ ) : = (( M ∪ ∂ φ D n + 1 ) ∪ φ ∪ 1 D n + 2 , N ∪ ∂ φ D n + 1 ) − → ( X , Y ) where the ne w ( n + 2) -cell has attach ing map φ ∪ 1 : ∂ D n + 2 = D n + 1 ∪ S n D n + 1 − → M ∪ ∂ φ D n + 1 . The homolog ical eff ect on ( f , g ) of this geometri c cell-exc hange is no chang e in    K r ( M ′ ǫ , N ′ ) = K r ( M ǫ , N ) for r 6 = n + 1 , n + 2 , K r ( M ′− ǫ , N ′ ) = K r ( M − ǫ , N ) for all r ∈ Z exc ept there is a ﬁve-term exac t sequence 0 − − − − → K n + 2 ( M ǫ , N ) − − − − → K n + 2 ( M ′ ǫ , N ′ ) − − − − → Z [ H ] y k − − − − → K n + 1 ( M ǫ , N ) − − − − → K n + 1 ( M ′ ǫ , N ′ ) − − − − → 0 . The incl usion h : M ⊂ M ′ is a simple homot opy equ iv alence with ( f , g ) ≃ ( f ′ h , g ′ h | N ) . (ii) Suppose for some n > 2 that K r ( N ) = 0 for all r 6 = n . Then K r ( M − , N ) = 0 = K r ( M + , N ) for all r 6 = n + 1 , and K n ( N ) is a stably ﬁ nitely gener ated free Z [ H ] -module. Moreov er , we may deﬁne an obje ct x in NIL( Z [ H ]; B 1 , B 2 ) by x : = ( K n + 1 ( M − , N ) , K n + 1 ( M + , N ) , ρ 1 , ρ 2 ) whose underl ying modules satisfy [ K n + 1 ( M − , N )] + [ K n + 1 ( M + , N )] = [ K n ( N )] = 0 ∈ e K 0 ( Z [ H ]) , [ Z [ G k ] ⊗ Z [ H ] K n + 1 ( M ǫ , N )] = 0 ∈ e K 0 ( Z [ G k ]) ( k = 1 , 2) . If ( f ′ , g ′ ) : ( M ′ , N ′ ) → ( X , Y ) is obtained from ( f , g ) by a geometri c cell-e xchange killing an element y k ∈ K n + 1 ( M ǫ , N ) (( k , ǫ ) = (1 , − ) or (2 , + )) which generates a di- rect summand h y k i ⊆ K n + 1 ( M ǫ , N ) , then the corres ponding object in NIL( Z [ H ]; B 1 , B 2 ) x ′ : = ( K n + 1 ( M ′− , N ′ ) , K n + 1 ( M ′ + , N ′ ) , ρ ′ 1 , ρ ′ 2 ) A lgebraic & G eo metric T opology XX (20XX) Algebraic K -th eory over the inﬁnite dihedral gr o up: an algebraic appr oa ch 1043 is obtained from x by an algebraic cell-exch ange. Since π n + 1 ( M k , N ) = K n + 1 ( M k , N ) by the relati ve Hurewic z theorem, there is a one-one corresp ondence b etween algebraic and geometri c cell-exch anges killing elements y k genera ting direct summands h y k i . (iii) For any n > 2 it is possible to modify the giv en ( f , g ) by a ﬁnite sequence of geometri c cell-ex changes and their formal in vers es to obtain a pair (also denoted by ( f , g ) ) such that K r ( N ) = 0 for all r 6 = n as in (ii) , and hence a canonica l equiv alence class of nilpot ent objects x = ( P 1 , P 2 , ρ 1 , ρ 2 ) in NIL( Z [ H ]; B 1 , B 2 ) such that [ P 1 ] + [ P 2 ] = 0 ∈ e K 0 ( Z [ H ]) , [ Z [ G k ] ⊗ Z [ H ] P k ] = 0 ∈ e K 0 ( Z [ G k ]) with P 1 : = K n + 1 ( M − , N ) , P 2 : = K n + 1 ( M + , N ) . Any x ′ in the equi va lence class of x is realized by a map ( f ′ , g ′ ) : ( M ′ , N ′ ) → ( X , Y ) with f ′ simple-h omotopic to f . The splitti ng obst ruction of f is the image of the W hiteh ead t orsion τ ( f ) ∈ Wh( G ) , namely: ∂ ( τ ( f )) = ([ P 1 ] , [ x ]) = ([ P 1 ] , [ P 1 , P 2 , ρ 1 , ρ 2 ]) ∈ ker( e K 0 ( Z [ H ]) → e K 0 ( Z [ G 1 ]) ⊕ e K 0 ( Z [ G 2 ])) ⊕ f Nil 0 ( Z [ H ]; B 1 , B 2 ) . Thus f is simpl y ho motopic to a sp lit ho motopy e qui vale nce if and only if ∂ ( τ ( f )) = 0 , if and only if x is equi v alent to 0 . (iv) The Whitehe ad group of G = G 1 ∗ H G 2 ﬁts into an ex act sequence Wh( H ) − − − − → Wh( G 1 ) ⊕ Wh( G 2 ) − − − − → Wh( G ) ∂ − − − − → e K 0 ( Z [ H ]) ⊕ f Nil 0 ( Z [ H ]; B 1 , B 2 ) − − − − → e K 0 ( Z [ G 1 ]) ⊕ e K 0 ( Z [ G 2 ]) . Furthermor e, the homomor phism ∂ : Wh( G ) − → e K 0 ( Z [ H ]) ⊕ f Nil 0 ( Z [ H ]; B 1 , B 2 ) ; τ ( f ) 7− → ([ P 1 ] , [ P 1 , P 2 , ρ 1 , ρ 2 ]) satisﬁes that proj 2 ◦ ∂ : Wh( G ) → f Nil 0 ( Z [ H ]; B 1 , B 2 ) is an epimorph ism split by ι : f Nil 0 ( Z [ H ]; B 1 , B 2 ) − → Wh( G ) ; [ P 1 , P 2 , ρ 1 , ρ 2 ] 7− →  1 ρ 2 ρ 1 1  . Deﬁnition 4.4 Let ( X , Y ) be a separating π 1 -inject iv e codimensio n 1 ﬁnite CW -pair . A homotopy equi v alence f : M → X from a ﬁnite CW -complex M is semi-split along Y ⊂ X if f is simple homoto pic to a map (also den oted by f ) such that for the correspon ding map of pairs ( f , g ) : ( M , N ) → ( X , Y ) the relati ve homolog y kernel Z [ H ] -modules K ∗ ( M 2 , N ) = H ∗ + 1 (( e M 2 , e N ) → ( e X 2 , e Y )) v anish, which is equi v alent to the induced Z [ H ] -module morphisms ρ 2 : K ∗ ( M + , N ) − → K ∗ ( M + , M 2 ) = Z [ G 2 − H ] ⊗ Z [ H ] K ∗ ( M − , N ) , A lgebraic & G eo metric T opology XX (20XX) 1044 J F Davis, Q Khan and A Ran ic ki being isomorphisms. Equi vale ntly , f is semi-split along Y if there is a semi-split object x = ( P 1 , P 2 , ρ 1 , ρ 2 ) in the canonic al equi va lence class of 4.3 , that is, with ρ 2 : P 2 → B 1 P 1 a Z [ H ] -module isomorp hism. In parti cular , a split homotop y equi va lence f of separatin g pairs is semi-split. Theor em 4.5 Let ( X , Y ) be a separating π 1 -inject iv e codimens ion 1 ﬁnite CW -pair , with X = X 1 ∪ Y X 2 . S uppos e that H = π 1 ( Y ) is a ﬁnite-ind ex subgroup of G 2 = π 1 ( X 2 ) . Every homo topy equi vale nce f : M → X with M a ﬁnite CW -complex is simple- homotop ic to a homotop y equi va lence which is semi-split along Y . Pro of Let x = ( P 1 , P 2 , ρ 1 , ρ 2 ) represent the canonic al equi val ence class of objects in NIL( Z [ H ]; B 1 , B 2 ) asso ciated to f in Theorem 4.3 (ii). S ince H is of ﬁ nite index in G 2 , as in the proof of Theorem 1.1 , we can deﬁne a semi-split object x ′′ : = ( P 1 , B 2 P 1 , ρ 2 ◦ ρ 1 , 1) satisfy ing [ x ′′ ] − [ x ] = [0 , B 2 P 1 , 0 , 0] − [0 , P 2 , 0 , 0] ∈ N il 0 ( Z [ H ]; B 1 , B 2 ) . By Theorem 4.3 (iii), the direct sum B 2 P 1 ⊕ P 1 = ( Z [ G 2 − H ] ⊗ Z [ H ] P 1 ) ⊕ P 1 = Z [ G 2 ] ⊗ Z [ H ] P 1 is a stably ﬁnitely generate d free Z [ G 2 ] -module. S ince Z [ G 2 ] is a ﬁ nitely generated free Z [ H ] -module, B 2 P 1 ⊕ P 1 is a stably ﬁnitely generat ed free Z [ H ] -module. S o [ B 2 P 1 ] − [ P 2 ] = [ B 2 P 1 ] + [ P 1 ] = [ Z [ G 2 ] ⊗ Z [ H ] P 1 ] = 0 ∈ e K 0 ( Z [ H ]) . Therefore x is equi valen t to x ′′ . Thus, by Theorem 4.3 (iii), there is a homotopy equi valenc e f ′′ : M ′′ → X simple-homoto pic to f realizing x ′′ ; note it is semi-split. Acknowledgeme nts W e would like to thank the participan ts of the workshop Nil Phenomena in T opolo gy (14–1 5 April 2007, V anderb ilt Univ ersity), where our intere sts intersected and moti- v ated the dev elopment of this paper . W e are grateful to the referee for making helpful comments and as king percepti ve question s. Moreo ver , Chuck W eibel helped us to fo r- mulate the ﬁltered c olimit hyp othesis in Theorem 0.4 , a nd Dan Ramras commun icated the con cept of almost-no rmal subgroup in Deﬁnition 3.24 to the se cond-named author . 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School of Mathematics, Uni versity of Edinburgh, Ed inburgh EH9 3JZ, SCO TLAND, U.K. jfdavi s@ind iana.edu , qkhan @nd.ed u , a.ra nicki@ ed.ac. uk Receiv ed: 17 Aug ust 2010 Re vised: 28 June 2011 A lgebraic & G eo metric T opology XX (20XX)

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

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