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On the Whitehead groups of semi-direct products of free groups Choo, Koo-Guan 1972

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• ON THE WHITEHEAD GROUPS OF SEMI-DIRECT PRODUCTS OF FREE GROUPS BY KOO-GUAN CHOO B.Sc. Nanyang University, Singapore, 1964 M.Sc. University of Ottawa, Ottawa, Ontario, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA . September, 1972 In present ing th i s thes is in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the Un ivers i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying o f t h i s t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representat i ves . It i s understood that copying or p u b l i c a t i o n o f th i s thes is fo r f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion . \ Department of The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada Supervisor : Dr. E. Luft 1 ABSTRACT Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring Z(G) of G by K 0Z(G). Then Wh G = 0 i f G i s free abelian (Bass-Heller-Swan), free (Gersten-Stailings) or a semi-direct product of a free group and an i n f i n i t e cyclic group (Farrell-Hsiang) and KDZ(G) = 0 i f G i s free abelian (Bass-Heller-Swan), free (Bass) or a direct product of a free abelian group and a free group (Gersten). In this thesis, we extend these results to a wider class of groups. Let a be an automorphism of G and F a free group. We denote the semi-direct product of G and F with respect to a by- G x F. New, l e t D be a direct product of n free groups and a an automorphism of D which leaves a l l but one of the noncyclic factors i n D pointwise fixed. F i r s t , by using techniques of Bass-Heller-Swan on Whitehead groups of certain direct products, together with techniques of Stallings on Whitehead groups of free products, we prove Wh D = 0 and KQZ(D) = 0. Next, we establish a fundamental theorem for coherent rings : If R is a (right) Noetherian ring and i f G1 and G 2 are groups such that the group rings R(Gj) and R(G2) are (right) coherent, then R(GX * G2) i s (right) coherent, where Gl * G 2 is the free product of Gj and G 2. A similar theorem has been anounced by Fi Waldhausen. From this fundamental theorem, we deduce that i f A is a free abelian group and F is a free group, the integral group ring Z(A x F) of A x F is (right) i i coherent. If A is of f i n i t e rank, then Z(A x F) has f i n i t e right global dimension. Combining these facts with techniques of Farrell-Hsiang on Whitehead groups of certain semi-direct products of groups and using the t r i v i a l i t y of Wh D and K 0Z(D), we show that Wh(D x a T) = 0 and K QZ(D * a T) = 0. The f i r s t result generalizes that of Farrell-Hsiang on semi-direct product F * a T, and the second result implies, in particular, that for the fundamental group TT^M) of a closed surface M (other than the real projective plane), the projective class group of Z(ir 1(M)) i s t r i v i a l . If M i s a closed surface (other than the real projective plane) l k i k and (S ) Is the k-dimensional torus, the fundamental group of M x (sl) Then the t r i v i a l i t y of Wh(3 x a T) impli es tue following result in topology : If N i s a differentiable or PL manifold l k of dim >_ 5 which i s h-cobordant to M * (S ) , then N is actually i k diffeomorphic or PL-homeomorphic to M x (S 1) respectively. Finally, by adapting Gersten's discussion on Whitehead group of free associative algebra to the case of a twisted free associative algebra, and by using the facts that Wh(D x Q T) =0 and KQZ(D x a T) = 0» we prove Wh((D * a T) x F) = 0. The factor T can presumably be dropped, although this i s not entirely obvious. There i s also a separate chapter on combinatorial group theory in-which we give certain necessary-and sufficient conditions for a given one relator group to be of the form F x T." i i i ACKNOWLEDGEMENT I am deeply indebted to my s u p e r v i s o r Dr. E. L u f t f o r suggesting the t o p i c and h i s generous a s s i s t a n c e and i n v a l u a b l e guidance throughout the research and p r e p a r a t i o n of t h i s t h e s i s . My s p e c i a l thanks i s due t o Dr. K.Y. Lam f o r h i s h e l p f u l suggestions and c r i t i c i s m s d u r ing the p r e p a r a t i o n of t h i s work. I would a l s o l i k e to thank Dr. S. Page and Dr. L.G. Roberts who c a r e f u l l y read the d r a f t of t h i s t h e s i s . The f i n a n c i a l support of the N a t i o n a l Research C o u n c i l of Canada a u u u u c u u j . v c i o x i , j f \J±. x ^ j . j _ i — i o n u u x u u i u x a i s ^ r c t L - c L u _ L X y cicK*now.i-eQged • I dedicate t h i s work to my w i f e f o r her constant encouragements during my research and f o r her e x c e l l e n t t y p i n g of t h i s t h e s i s . TABLE OF CONTENTS INTRODUCTION •CHAPTER 1 : PRELIMINARIES §1.1. D e f i n i t i o n s and Terminology §1.2. The F a r r e l l - H s i a n g Decomposition Formula f o r Wh(G * a T) §1.3. S t a l l i n g s ' Decomposition Formula f o r Free Products CHAPTER .2 : COHERENT RINGS §2.1. I n t r o d u c t i o n §2.2. The Group Ring of a Free Group over a Noetherian Ring CHAPTER 3 : WHITEHEAD GROUPS OF SOME SEMI-DIRECT PRODUCTS OF FREE GROUPS n §3.1. The Whitehead Group of a D i r e c t Product II F. of Free Groups F. f n §3.2. The Whitehead Group of II F^ i = l x xa T §3.3. of Twisted Free A s s o c i a t i v e Algebras §3.4. The Whitehead Group of §3.5. Concluding Remarks f n 1 n F. 1 i - i 1 J x a T x . , F a x i d T CHAPTER 4 : GROUPS F * a T WITH ONE DEFINING RELATOR §4.1. I n t r o d u c t i o n §4.2. Groups w i t h Two Generators and One D e f i n i n g R e l a t o r §4.3. Groups w i t h n (n > 2) Generators and One D e f i n i n g R e l a t o r BIBLIOGRAPHY V INTRODUCTION This t h e s i s deals w i t h the Whitehead groups of c e r t a i n s e m i - d i r e c t products of f r e e groups. Let G be a group. We denote the Whitehead group of G by Wh G and the p r o j e c t i v e c l a s s group of the i n t e g r a l group r i n g Z(G) of G by K QZ(G). Wh and K Q are c o v a r i a n t f u n c t o r s from groups to a b e l i a n groups. The problem of computing Wh G i s d i f f i c u l t but important i n A l g e b r a i c K-theory. I t i s not even easy to decide when Wh G i s t r i v i a l . I t i s known th a t Wh G = 0 i f G i s one of the f o l l o w i n g forms : (a) i n f i n i t e c y c l i c or f i n i t e c y c l i c of order 2, 3 or 4 .([12]) ; (b) f r e e a b e l i a n (.[4]) ; (c) f r e e ([18]) ; (d) a s e m i - d i r e c t product F x Q T of a f r e e group F and an i n f i n i t e c y c l i c group T w i t h respect to an automorphism a of F ([8]) ; (e) group of type n, that i s s e m i - d i r e c t product of n i n f i n i t e c y c l i c groups ([8]) ; ( f ) f r e e products of groups as given i n (a) - (e) '([18]).. In t h i s t h e s i s , we extend ( b ) , ( c ) , (d) to a wider c l a s s of groups. A l s o , i t i s known that K QZ(G) = 0 i f G i s one of the f o l l o w i n g forms : (a) f r e e a b e l i a n ([4]) ; (b) f r e e ( [3]) ; (c) d i r e c t product of a f r e e a b e l i a n group and a f r e e group.([11]). v i We a l s o extend these r e s u l t s i n the t h e s i s . Our work i s d i v i d e d i n t o four chapters. In s e c t i o n 1 of Chapter 1 , we s t a t e those d e f i n i t i o n s and terminology which are used throughout the t h e s i s . We w i l l make use of two techniques i n computing Wh G. One of them i s due to S t a l l i n g s ( [ 1 8 ] ) on f r e e products and the other i s due to F a r r e l l and Hsiang ( [ 8 ] ) on s e m i - d i r e c t products G x a T of groups, or B a s s - H e l l e r -Swan ([4]) when a i s the i d e n t i t y . In f a c t , F a r r e l l and Hsiang obtained a decomposition formula f o r Wh(G x a T ) , which i s due to Bass-Heller-Swan when a i s the i d e n t i t y . We r e c a l l , i n S e c t i o n 2 and S e c t i o n 3 of Chapter 1 , these techniques and those formulae which w i l l be subsequently used. Chapter 2 i s devoted to the study of a s p e c i a l c l a s s o f r i n g s , c a l l e d coherent r i n g s , which i s of some importance i n A l g e b r a i c K-theory, e s p e c i a l l y i n computing Wh G. A r i n g R i s c a l l e d r i g h t coherent i f any f i n i t e l y generated submodule of a f r e e r i g h t R-module i s f i n i t e l y presented. The importance of the coherent property can be expl a i n e d as f o l l o w s : I f the i n t e g r a l group r i n g Z(G) of G i s r i g h t coherent and has f i n i t e r i g h t g l o b a l dimension, the e x o t i c summand C(Z(G),a) i n the F a r r e l l - H s i a n g (or Bass-Heller-Swan when a = i d e n t i t y ) decomposition formula f o r Wh(G x a T ) becomes zero and t h i s g r e a t l y s i m p l i f i e s the determination of Wh(G• x- T). We e s t a b l i s h i n t h i s chapter the f o l l o w i n g fundamental theorem f o r coherent r i n g s : Let R be a r i g h t Noetherian r i n g and l e t Gj and G 2 be groups such that the group r i n g s R(Gj) and R(G 2) are r i g h t coherent. Then R(G. * G„) i s r i g h t coherent where G. * G 2 i s the f r e e product of G, v i i and G 2 . A s i m i l a r theorem has been anounced i n ( [ 1 9 ] ) . From t h i s fundamental theorem, we deduce that the r i n g R(F) of a f r e e group F over a r i g h t Noetherian r i n g R i s r i g h t coherent and that the i n t e g r a l group r i n g Z(A x F) i s r i g h t coherent f o r any a b e l i a n group A. In Chapter 3, we show the t r i v i a l i t y of Wh G and K 0Z(G) f o r . c e r t a i n s e m i - d i r e c t products G of f r e e groups. Let D be a d i r e c t product of n f r e e groups. By usi n g the Bass-Heller-Swan decomposition formula together w i t h S t a i l i n g s ' technique, we show i n S e c t i o n 1 of Chapter 3 that Wh D = 0. This extends the r e s u l t s of Bass-Heller-Swan ([4]) f o r f r e e a b e l i a n groups and of G e r s t e n - S t a i l i n g s ([18]) f o r f r e e groups. In a d d i t i o n to the t r i v i a l i t y of Wh D, we have K QZ(D) = 0. This g e n e r a l i z e s ..those r e s u l t s f o r K Q p r e v i o u s l y mentioned. Next, l e t a be an automorphism of D which leaves a l l but one of the n o n c y c l i c f a c t o r s i n D po i n t w i s e f i x e d and D x a x the s e m i - d i r e c t product of D and T w i t h respect to a. In S e c t i o n 2 of Chapter 3, we show that Wh(D x a x) = 0. I n the proof, we need to use the coherence property o f Z(A x F ) , which we have e s t a b l i s h e d i n Chapter 2. This g e n e r a l i z e s t h a t of F a r r e l l - H s i a n g ([8]) on F x a T. As a consequence, we have K QZ(D x a T) = 0 . This i m p l i e s , i n p a r t i c u l a r , that f o r the fundamental group i r ^ M ) of a cl o s e d s u r f a c e M (other than the r e a l p r o j e c t i v e p l a n e ) , the p r o j e c t i v e c l a s s group of Z ( T T J ( M ) ) i s t r i v i a l . I f M i s a cl o s e d s u r f a c e (other than the r e a l p r o j e c t i v e plane) 1 k 1 k and (S ) i s the k-dimensional t o r u s , the fundamental group of M x (S ) v i i i i s of the form D x a T . Then the t r i v i a l i t y of Wh(D x a T) i m p l i e s the f o l l o w i n g r e s u l t i n topology : I f N i s a d i f f e r e n t i a b l e or PL m a n i f o l d 1 k of dim >_ 5 which i s h-cobordant to M x (S ) , then N i s a c t u a l l y 1 k diffeomorphic or PL-homeomorphic to M x (S ) r e s p e c t i v e l y . In the l a s t s e c t i o n of Chapter 3, we e s t a b l i s h the r e s u l t Wh((D x a T) x F) = 0, where F i s a f r e e group. This g e n e r a l i z e s a x i a ^ the r e s u l t i n §3.2. I n pr o v i n g t h i s a s s e r t i o n , we come across the s o - c a l l e d " t w i s t e d f r e e a s s o c i a t i v e algebras". We adapt Gersten's d i s c u s s i o n on f r e e a s s o c i a t i v e algebras ([10]) to t h i s case of t w i s t e d f r e e a s s o c i a t i v e algebras i n S e c t i o n 3 of Chapter 3. Chapter 4 i s a separate chapter d e a l i n g w i t h groups w i t h one d e f i n i n g r e l a t o r . We o b t a i n c e r t a i n necessary and s u f f i c i e n t c o n d i t i o n s f o r such a one r e l a t o r group to be of the form F x a T , w i t h F f r e e . CHAPTER I PRELIMINARIES §1.1. D e f i n i t i o n s and Terminology Throughout the t h e s i s , a r i n g R always mean an a s s o c i a t i v e r i n g w i t h i d e n t i t y , and r i n g homomorphisms are assumed to map the i d e n t i t y i n t o the i d e n t i t y . Trie r i n g of i n t e g e r s i s denoted by Z. I f G i s a group, the group r i n g of G over R i s denoted by R(G). The purpose of t h i s s e c t i o n i s to r e c a l l those d e f i n i t i o n s and terminology from A l g e b r a i c K-theory that w i l l be used i n the t h e s i s . A general r e f e r e n c e f o r these w i l l be [ 2 ] , [8] and [14]. The Whitehead group K^R of a r i n g Let R be a r i n g . Denote the group of a l l n o n s i n g u l a r n x n matrices over R by GL(n,R). We have a n a t u r a l i n c l u s i o n GL(n,R) c G L ( n + l , R ) . The commutator subgroup of GL(n,R) i s denoted by [GL(n,R), GL(n,R)] and the Whitehead group K^R of R i s defined by KjR = d i r e c t l i m i t GL(n,R)/[GL(n,R), GL(n,R)]. n -> °° I f a e GL(n,R), denote the corresponding element i n KjR by [ a ] . C l e a r l y , i s a c o v a r i a n t f u n c t o r from r i n g s to a b e l i a n groups. That i s , a r i n g homomorphism f : Rj —> R 2 induces a homomorphism f ^ : K.R, — > K,R„. - 2 -The whitehead group Wh G of a group Let G be a group. Let J(G) be the subgroup of KjZ(G) generated by the elements [(±g)] f o r g e G where (±g) i s the l x l ma t r i x w i t h s i n g l e entry g or -g. The q u o t i e n t group K 1Z(G)/J(G) i s c a l l e d the whitehead group of G, denoted by Wh G. C l e a r l y , Wh i s a co v a r i a n t f u n c t o r from groups to a b e l i a n groups. In other words, any group homomorphism f : Gj — > G 2 induces a homomorphism f * : Wh Gj — > Wh G 2. In general, i t i s a d i f f i c u l t problem to compute Wh G ; i t i s even not easy to determine when i s Wh G t r i v i a l . I n [12], Higman proved that Wh G = 0 i f G i s i n f i n i t e c y c l i c , or i s f i n i t e of order 2, 3 or 4. Bass, H e l l e r and Swan ([4]) have shown that the whitehead group of any f r e e a b e l i a n group i s zero, w h i l e S t a i l i n g s ([18]) and Gersten ([10]) have proved that the whitehead group of any f r e e group i s zero. In Chapter 3, we w i l l determine some c l a s s e s of groups G i n which Wh G = 0, and t h i s w i l l g e n e r a l i z e the r e s u l t s mentioned above. The Grothendieck group K QR of a r i n g C l o s e l y r e l a t e d to i s the f u n c t o r K Q which i s d e f i n e d as f o l l o w s . L e t I(R) be the category whose o b j e c t s are f i n i t e l y generated p r o j e c t i v e r i g h t R-modules and whose morphisms are R l i n e a r homomorphisms. Then K QR i s the Grothendieck group of E(R) ; i . e . K QR i s the a b e l i a n group generated by the isomorphism c l a s s e s of ob j e c t s i n 1(R) modulo the r e l a t i o n s ( P 2 - P j - P 3) f o r s h o r t exact sequences 0 Pj ->• P 2 -> P 3 0 - 3 -i n I ( R ) . The c l a s s of P e I(R) i n K QR i s denoted by [ P ] . The c l a s s of the f r e e r i g h t R-module of rank 1 generates a c y c l i c subgroup of K QR. The q u o t i e n t K QR/(subgroup generated by f r e e r i g h t R-modules) i s c a l l e d the p r o j e c t i v e c l a s s group K QR of R. C l e a r l y , K Q i s a l s o a c o v a r i a n t f u n c t o r from r i n g s to a b e l i a n groups. Bass, H e l l e r and Swan ([4]) proved that K QZ(A) = 0 f o r any f r e e a b e l i a n group A, w h i l e Bass ([3]) has shown that K QZ(F) = 0 f o r any f r e e group F. Moreover, Gersten ([11]) proved that K QZ(A x F) = 0 where A .is fzee ..abelian .and J? .is f r e e . We,.w.ill also, o b t a i n .some g e n e r a l i z a t i o n of these r e s u l t s i n Chapter 3. The group C(R,ct) There i s another c l a s s of group (introduced by F a r r e l l i n [9]) a s s o c i a t e d to a given r i n g w i t h an automorphism, which i s a l s o c l o s e l y r e l a t e d to and K D, and i s defined as f o l l o w s . Let R be a r i n g and a an automorphism of R. F i r s t , r e c a l l t hat an a d d i t i v e map <f> from a r i g h t R-module M^^ to a r i g h t R-module M 2 i s a - l i n e a r i f cj)(mr) = <Km)a(r) f o r m e Mj and r e R. Let (5(R,a) be the category whose o b j e c t s are p a i r s (P,<j>) where P e I(R) and cf) i s an a - l i n e a r n i l p o t e n t endomorphism of P, and whose morphisms - 4 -g : ( P ^ , ^ ) — > (P 2,(j) 2) are R l i n e a r homomorphisms g : P j — > P 2 such that the f o l l o w i n g diagram -> P. -> P, i s commutative. We have the f o r g e t t i n g f u n c t o r F : £(R,ot) — > 1(R) defined by F(P,<}>) = P f o r (P,<(>) e C(R,a), and the zero f u n c t o r J : E(R) — > C(R,a) defined by J(P) = (P,0) f o r P e I ( R ) . Both F and J are c o v a r i a n t f u n c t o r s and F°J i s the i d e n t i t y f u n c t o r of 1(R). L e t C'-C-RjCt) be the-Grothendieck group of -the category C(R,a). The c l a s s of an element (P><j>) e C(R,a) i n C'(R,a) i s denoted by [P,<j>]. The c l a s s [R,0] generates a c y c l i c subgroup 3F(R) of C'(R,a). Let C(R,a) = C'(R,a)/F(R) and l e t C(R,a) be the subgroup of C(R,a) generated by [Rn,<J>] f o r (Rn,(f>) e 6(R,a). The f o l l o w i n g r e s u l t gives us more p r e c i s e l y the r e l a t i o n s between K c, C'(R,a), C(R,a) and C(R,a) : Theorem 1.1.1. ([9]) The f o l l o w i n g sequences are s p l i t exact : 0 > C(R,a) > C'(R,a) ~T±> K^R > 0, F* ~ 0 > C(R,a) > C(R,a) ~ ~ > K QR > 0, J * - 5 -where F* and J * are homomorphisms induced by F and J r e s p e c t i v e l y , and I[Rn,(f»] = [Rn,<j>] - [ R n , 0 ] . Semi-direct product of groups Let G be a group and a an automorphism of G. L e t F be a f r e e group generated by { x^) • I f w i s a word i n x^ d e f i n i n g an element i n F, we denote by |w| the t o t a l exponent sum of the x^ appearing i n w. The s e m i - d i r e c t product G * a F of G and F w i t h  respect to a i s defined as f o l l o w s : G x a F = G x F as s e t s and m u l t i p l i c a t i o n i n G x a F i s given by (g,w)(g',w') = ( g o H W ' ( g ! ) , ww') f o r (g,w), (g',w') e G x a F. In p a r t i c u l a r , i f F i s an i n f i n i t e c y c l i c group T = <t> generated by t , we have the s e m i - d i r e c t product G x T of G and T w i t h respect to a. Twisted group r i n g s Let R be a r i n g and a an automorphism of R. Let F be a f r e e group (or f r e e semigroup) generated by { x j ^ ' The a - t w i s t e d R group  r i n g of F, denoted by R a [ F ] , i s d e f i n e d as f o l l o w s : a d d i t i v e l y R a [ F ] = R[F] so that i t s elements are f i n i t e l i n e a r combinations of elements i n F w i t h c o e f f i c i e n t s i n R. M u l t i p l i c a t i o n i n R a [ F ] i s given by - 6 -(rwXr'w*) = r a | w | ( r ' ) w w ' f o r any rw, r'w' e R a [ F ] , In p a r t i c u l a r , i f F i s a f r e e group (resp. f r e e semigroup) generated by t , we have R a[T] (resp. R a [ t ] ) and we c a l l i t the a - t w i s t e d f i n i t e Laurent s e r i e s r i n g (resp. a - t w i s t e d polynomial  r i n g ) . Let R = Z(G) and a an automorphism of G. Then a i s a l s o used to denote the induced automorphism on Z(G) defined by I geG & I v( g ) geG 8 where g e G and X' e Z. Note that there i s a standard isomorphism • & between Z ( G ) a [ F ] (resp. Z ( G ) Q [ T ] ) and Z ( G x a F ) (resp. Z(G x a T)) which i s the i d e n t i t y map on Z(G) and maps e Z ( G ) a [ F ] (resp. t e Z ( G ) a [ T ] ) onto x^ e Z(G x a F) (resp. t £ Z(G x a T ) ) . In [ 8 ] , F a r r e l l and Hsiang obtained a formula f o r K 1 R a [ T ] , and, as an a p p l i c a t i o n , they deduced a decomposition formula f o r Wh(G x Q T), which we w i l l review i n the next s e c t i o n . §1.2. The F a r r e l l - H s i a n g Decomposition Formula f o r Wh(G x a T) In t h i s s e c t i o n , we r e c a l l some of the r e s u l t s i n F a r r e l l and Hsiang ([8]) which w i l l be subsequently used. For more d e t a i l , we r e f e r to [ 8 ] . F i r s t , we r e c a l l the f o l l o w i n g terminology. I f M i s a r i g h t R-module, we denote by r g : M — > M a r i g h t m u l t i p l i c a t i o n by s e R, i . r s(m) = ms f o r a l l m e M. I f M i s a r i g h t module over R a [ T ] , then r t : M — > M i s an a - l i n e a r endomorphism. The f o l l o w i n g i n c l u s i o n maps are r i n g homomorphisms : -i • TJ — > v f T l J "a1 J» k : R — > R„[t] and k - : R — > R„[t _ 1], i : R a [ t ] — > R a[T] and i " : R a [ t - 1 ] — > R a [ T ] ; and the f o l l o w i n g two r i n g homomorphisms are augmentations : e : R a [ t ] — > R de f i n e d by e ( t ) = 0, e~ : R a t t _ 1 ] — > R d e f i n e d by e ~ ( t _ 1 ) = 0. For any a b e l i a n group G w i t h an automorphism a, we i n t r o d u c e the f o l l o w i n g two subgroups : G a = : { g | a(g) = g, g e G } 1(a) = { g - q(g) | g e G }. - 8 -Consider now the homomorphism i : R a [ t ] — > R a [ T ] . We i d e n t i f y the element P e l ( R a [ t ] ) w i t h by sending x to x 8 1 f o r x e P. We w i l l then give a d e s c r i p t i o n of a homomorphism from K^R^T] i n t o C'(R.,a). Let a e GL(n, R a[T]) and l e t v : R a [ T ] n > R a [ T ] n be the l i n e a r isomorphism a s s o c i a t e d w i t h a. We have the n a t u r a l i n c l u s i o n R a [ t ] n c R a [ T ] n Thus, there i s an i n t e g e r N >_ 0 such that r N v ( R a [ t ] n ) C R a [ t ] n . Let M = R a [ t ] n / r N v ( R a [ t ] n ) . Then M e 1(R) and r induces an a - l i n e a r n i l p o t e n t endomorphism on M, i . e . (M,r t) e C(R,a)' ( c f . [ 8 ] , Theorem 8 ( b ) ) . Let p : K ^ t T ] — > C (R,a) be d efined by (1) p[a] = [M,r t] - t R a [ t ] n / r N ( R a [ t ] n ) , r j . Then one can v e r i f y that p i s a homomorphism ( c f . [ 8 ] , Theorem 8 ( c ) ) . In - 9 -p a r t i c u l a r , i f a = ( t ) , the l x l m a t r i x determined by the generator t of T, then n = 1 and N = 0 so th a t i n ( 1 ) , M = R and the second term on the r i g h t hand s i d e i s zero. Thus p [ ( t ) ] = [R, r t ] . By combining p w i t h the map F* : C(R,a) — > K QR, we get the homomorphism F*p : K'Ra'["T] — > K QR and that F*p[(t)] = [R]. Next, l e t p' = p i * : K R ^ t " 1 ] — > C'(R,a). Then Theorem 1.2.1. ( [ 8 ] , Theorem 13) Image p 1 = C(R,a) and the f o l l o w i n g sequence i s s p l i t exact : k * _ i 0 >. KjR ~ — > K 1 R a [ t ] > C(R,a) > 0 Li k e w i s e , the sequence 0 > K R ~ > K 1 R a t t ] > C(R,a ) > 0 i s s p l i t exact. (Here, we i d e n t i f y C(R,a) w i t h the image I(C(R,a)) i n C'(R,a) where I i s given i n Theorem 1.1.1.). Note that the k e r n e l of i s I (a*) where j f t : -> K 1R ( X[T] i s the homomorphism induced by the i n c l u s i o n j : R — > . R a [ T ] , ( c f . [ 8 ] , Theorem 14). Moreover, we have : - 10 -Theorem 1.2.2. ( [ 8 ] , Theorem 19 and [4]) ( F a r r e l l - H s i a n g decomposition f o r -mula f o r K j R ^ f T ] ; Bass-Heller-Swan decomposition formula when a = i d e n t i t y ) K 1 R a [ T ] = X 9 C(R,a) €> C(R,cf *) and X i s given by the f o l l o w i n g exact sequence <J> ty <** 0 > K 1 R / I ( a A ) — > X > (K DR) > 0, where $ i s induced by j A and 'ty i s induced by F*p. Remark 1.2.3. Note t h a t , i n Theorem 1.2.2, * [ ( t ) ] = [R]. F i n a l l y , we r e c a l l : Theorem 1.2.4. ( [ 8 ] , Theorem 21 and [4]) ( F a r r e l l - H s i a n g decomposition f o r -mula f o r Wh(G * a T); Bass-Heller-Swan decomposition.formula when a = i d e n t i t y ) Wh(G x a T) = X $ C(Z(G),a) O C ( Z ( G ) , a _ 1 ) where X i s given by the f o l l o w i n g exact sequence 0 > Wh G/I(a*) — > X — > ( K Q Z ( G ) ) a * >'0 i n which 4> and are induced by the corresponding maps i n Theorem 1.2.2. - 1 1 -Note that C(Z(G),ct) = CCZCG)^" 1) i n the case of a group r i n g Z(G). As an a p p l i c a t i o n , F a r r e l l and Hsiang have a l s o shown that : Theorem 1 . 2 . 5 . ( [ 8 ] , Theroem 3 1 ) Let F be a f r e e group. Then Wh (F xa T) = 0. § 1 . 3 . S t a l l i n g s ' Decomposition Formula f o r Free Products In t h i s s e c t i o n , we r e c a l l those d e f i n i t i o n s and r e s u l t s i n [ 1 8 ] which we need i n our l a t e r work. For more d e t a i l , we r e f e r to [ 1 8 ] . Let R be a r i n g . A r i n g A i s c a l l e d an R - r i n g , i f A contains R, the i n c l u s i o n i : R —> A i s a r i n g homomorphism, and there e x i s t s a r i n g homomorphism : A — > R such that E ^ ( r ) = r r o r a H r e R. i s c a l l e d an augmentation of A. Any group r i n g R(G) of a group G i s an R- r i n g w i t h augmentation : R(G) — > R defined by e Q ( S ) = 1 f ° r a l l g e G. I f A i s an R - r i n g , we denote by K^A the cok e r n e l of the homomorphism i * : K 1R — > K^A, induced by the i n c l u s i o n i : R — > A . Let A and r be R-rings w i t h augmentations and E p r e s p e c t i v e l y . I f a r i n g homomorphism f : A — > r i s such that f ( r ) = r f o r a l l r e R, we c a l l f a homomorphism of R-rings or simply j u s t R-homomorphism. We say that the R r i n g tt i s the f r e e product of A and r i f there are given R-homomorphisms f : A — > tt and g : r —> tt - 12 -such t h a t f o r any R - r i n g £ and R-homomorphisms f' : A —> E and g' : r — > £, there e x i s t s a unique c o n s i s t e n t R-homomorphism h : tt —> E. We abbreviate i t as tt = A * T. I t i s c l e a r that R(G X * G 2) = R(G X) * R(G 2) f o r any two .groups G^ and G 2, where G^  * G 2 denotes the f r e e product of Gj and G 2. Note that the f r e e product of R-rings i s j u s t the coproduct i n the category whose ob j e c t s are R-rings and whose morphisms are R-homomorphisms. Now, l e t A = Ker and V = Ker . Then A and T are R-bimodules and as bimodules A = R e A , r = R # r~. Moreover, the m u l t i p l i c a t i o n s on A and Y d e f i n e a s s o c i a t i v e maps A ®_ A — > A and T 8 _ T —> r. We have the f o l l o w i n g s t r u c t u r e theorem f o r A * r ( [ 1 8 ] , §3.2) : A * r = R 0 A « T " < B (I s D r) « (r 8 . I) e (I «„ r e n I) « (r 0 D I 0 O r) e • • • The m u l t i p l i c a t i v e s t r u c t u r e i s determined by m u l t i p l y i n g components by the tensor product and then c o l l a p s i n g i f p o s s i b l e using the m u l t i p l i c a t i o n s A 0 A — > A and T 8 T — > V d e r i v e d from A and V. R R R e c a l l t h a t , i f M i s an R-bimodule, the tensor a l g e b r a T (M) K of M over R i s de f i n e d to be T_ (M) = R <B M <B (M 8 M) « (M 8 M 8 M) e • • • . R K R R I t i s c l e a r t h a t A * V contains as a s u b r i n g the tensor a l g e b r a T ( A 8 p T) - 13 -of the bimodule A 8 I". Moreover, we have : K Theorem 1.3.1. ( [ 1 8 ] , §5) The group K ^ ( A * V) i s generated by the images, under the obvious maps, of K ^ A , K^T and K j T ^ ( A *^ T). The f o l l o w i n g r e s u l t i s due to Gersten ( c f . [ 2 ] , p.646). I f KjRf-t] = 0 where R [ t ] i s the polynomial extension of R, ( i n other words, i f k* : KjR — > K j R [ t ] i s an isomorphism or C(R,id) = 0 by Theorem 1.2.1), then K.T (M) = 0 f o r any f r e e R-bimodule M. Therefore : Theorem 1.3.2. ( [ 1 8 ] , Theorem 6.2) i f C(R,id) = 0,. then Kj(A * T) s K A « KjT. F i n a l l y , l e t G be a group and R(G) the group r i n g w i t h augmentation z . I t i s known that Ker e i s a f r e e R-bimodule. We c l o s e G g our review by the f o l l o w i n g theorem, which i s a d i r e c t consequence of Theorem 1.3.2. Theorem 1.3.3. Let R be a r i n g such t h a t C(R,id) = 0 and l e t and G 2 be groups. Then K ^ G j * G 2) S K J ^ G J ) 6 K 2 R ( G 2 ) . In p a r t i c u l a r , i f F i s a f r e e group of rank m, we w r i t e F = T * ... * T I f A 8. T i s a f r e e R-bimodule and 14 as a f r e e product of m i n f i n i t e c y c l i c groups, so that m j = l K,R(F) £ « KjRCT.). As an a p p l i c a t i o n , S t a l l i n g s has shown that Wh (G x ft G 2) = Wh Gj ® Wh G 2 and so Wh F = 0 f o r a f r e e group F. CHAPTER 2 COHERENT RINGS §2.1. I n t r o d u c t i o n The present chapter i s devoted to the study of a s p e c i a l c l a s s of r i n g s , c a l l e d coherent r i n g s , which are of importance i n A l g e b r a i c K-theory. We f i r s t r e c a l l the d e f i n i t i o n of such r i n g s . L e t R be a r i n g . A r i g h t R-module M i s s a i d to be f i n i t e l y presented i f there i s an exact sequence 0 — > K — > F — > M — > 0 of r i g h t R-modules, where F i s f r e e and both F and K are f i n i t e l y generated. N o t i c e that i f M i s f i n i t e l y presented and i f there i s another exact sequence 0 — > K' — > F 1 ~ > M — > 0 of r i g h t R-modules, w i t h F' f r e e and f i n i t e l y generated, then K' i s n e c e s s a r i l y f i n i t e l y generated. For a proof, see ( [ 2 ] ) . D e f i n i t i o n 2.1.1. A r i n g R i s c a l l e d r i g h t coherent i f any f i n i t e l y generated submodule of a f r e e r i g h t R-module i s f i n i t e l y presented. An e q u i v a l e n t property i s : Any homomorphism f : R n — > R m of r i g h t R-modules R n and R m has f i n i t e l y generated k e r n e l . A general reference f o r coherent r i n g s i s Chase [ 7 ] , Bourbaki [6] and S o u b l i n [16]. Of course, any r i g h t Noetherian r i n g i s r i g h t coherent. - 16 -However, there are important examples of coherent r i n g s , which are not Noetherian. Theorem 2.1.2. Let F be a f r e e group. Then the i n t e g r a l group r i n g Z(F) i s r i g h t . c o h e r e n t . The proof of t h i s theorem i s i m p l i c i t l y contained i n the argument of ( [ 8 ] , Theorem 31). We do not reproduce the argument i n [ 8 ] , s i n c e the r e s u l t w i l l f o l l o w from our main theorem i n the next s e c t i o n . Note that Z(F) i s not r i g h t Noetherian unless F i s c y c l i c . We c a u t i o n that there i s no " H i l b e r t b a s i s theorem" f o r coherent n'naC Tt l f loprl R n i l K I "f T l -l-n T 1 7 1 rrnirp pry Cfyarory 1 r\€ r> P A ^ i o i - p n H - y> r* •i/Hri'^ f* — - J-J '— 1 — J " — " t ^ • j v.. V C... * »1.* .. 1 \^ J- V.l 1 1 1— j_ i — l_ L l_ L J. £j vv 11 U V—• polynomial extension i s not coherent. N e v e r t h e l e s s , we w i l l see i n s e c t i o n 2 that the polynomial extension of Z(F) i s r i g h t coherent. We have the f o l l o w i n g r e s u l t , the proof of which i s contained i n [9] ( a l s o c f . proof of Theorem 31 i n [8]) : Theorem 2.1.3. I f R i s r i g h t coherent and has f i n i t e r i g h t g l o b a l dimension, then k* : K j R — > K ^ R ^ t ] i s an isomorphism f o r any, automorphism a of R. In other words C(R,a) = 0. This theorem i m p l i e s , f o r example, that f o r a group G w i t h Z(G) r i g h t coherent and of f i n i t e r i g h t g l o b a l dimension, the e x o t i c summand C(Z(G),a) i n the F a r r e l l - H s i a n g decomposition formula f o r Wh (G * a T) - 17 -becomes zero. This g r e a t l y s i m p l i f i e s the determination of Wh (G x Q T). Before proving the coherence of Z(G) f o r c e r t a i n c l a s s e s of groups G, we c l o s e t h i s s e c t i o n w i t h the f o l l o w i n g r e s u l t on d i r e c t l i m i t o f coherent r i n g s . Lemma 2.1.4. ( [ 6 ] , p.63) Let (R^) be a d i r e c t e d system of r i n g s and l e t R be t h e i r d i r e c t l i m i t . Suppose that R i s f l a t as a l e f t R^-module f o r each A. I f each R^ i s r i g h t coherent, then R i s r i g h t coherent. §2.2. The Group Ring of a Free Group over a Noetherian Ring main purpose of t h i s s e c t i o n i s to show th a t the group r i n g R(F) i s r i g h t coherent. In f a c t , we w i l l prove the f o l l o w i n g more general r e s u l t . Theorem 2.2.1. Let R be a r i g h t Noetherian r i n g and l e t G^ and G 2 be groups such that R(Gj) and R(G 2) are r i g h t coherent. Let G = Gj * G 2 be the f r e e product of G1 and G 2. Then R(G) i s r i g h t coherent. Before proving the theorem, we i n t r o d u c e some r e l e v a n t terminology. Let R be a r i n g and N = ( r j . j ) a n m x n m a t ri x over R. I f f : R n — > R1 i s the homomorphism (of r i g h t R-modules) a s s o c i a t e d to N, then the k e r n e l of f i s p r e c i s e l y the s o l u t i o n space of N. We c a l l N a ( r i g h t ) coherent matr i x i f i t s s o l u t i o n space i s f i n i t e l y generated as a r i g h t R-module. I t - 18 -f o l l o w s that a r i n g R i s r i g h t coherent i f and only i f a l l m x n matrices over R are ( r i g h t ) coherent. Now, l e t N be an m x n m a t r i x over R. Let Nj (resp. N 2) be the m x n m a t r i x over R obtained from N by an elementary row ope r a t i o n (resp. elementary column operation) and l e t N 3 be the (m + 1) x (n 4- 1) matrix I 0 i 1 J Then the f o l l o w i n g lemma i s t r i v i a l : Lemma 2.2.2. For each i , N^ i s coherent i f and only i f N i s coherent. Let R be a r i n g . Let and G 2 be groups and G = Gj * G 2 .their f r e e product ..so that there...are.„na>tural i n c l u s i o n s R(G X) R R(G) R(G 2) R e c a l l that each element 1 ^ g e G can be uniquely expressed as a product (1) 8 82 §2 * * * where g^ 4 1, g^ i s i n Gj or G 2 and g^, g^ +^ are not i n the same f r e e f a c t o r G 1 or G 2. I f g e G i s expressed i n the form ( 1 ) , then we c a l l n the s y l l a b l e l e n g t h |g| of g ( c f . [13], p.182). - 19 -Our next lemma i s a key step towards the proof of Theorem 2.2.1. Lemma 2.2.3. Let be a submodule of R(G) (as r i g h t R(G)-module) generated by c e r t a i n elements i n R ( G 1 ) m and M 2 a submodule o f R ( G ) m generated by c e r t a i n elements i n R ( G 2 ) m . Let K = (M 1 + M 2) 0 R m. Then (2) (M 1 + K-R(G)) O (M + K-R(G)) = K-R(G) Here K>R(G) denotes the r i g h t R(G)-module generated by K. Proof : Let M? be the R(G^)-submodule of R ( G ^ ) m generated by the same s e t of elements which generate ( i = 1,2). Then M? C ( i = 1,2). Let 'K'R^') be the r i g h t R(G^)-module generated by "K. One d i r e c t i o n of i n c l u s i o n s i n (2) i s obvious. So, l e t x e (Mj + K-R(G)) f> (M 2 + K'R(G)). Then, c o n s i d e r i n g x as an element i n R(G ) m , we can express x uniquely as (3) x = 7 c.w. x w i t h c^ e R m and w^ e G such that [w-J >_ |w2| _^ • • • ' • A l s o , x can be expressed uniquely as (4) x = / a.u. i 3 3 and - 20 -( 5 ) x - l b k v k , k where a. (resp. b ) i s i n M° + K-R(G 1) (resp. M° + K-R(G 2)) and u (resp. v. ) i s an element i n G s t a r t i n g w i t h a n o n t r i v i a l element J k i n G 2 (resp. Gj) f o r each j (resp. k) except that one of them may be t r i v i a l . We a s s e r t t h a t c^ e K f o r each i . Since c^ e R m, i t s u f f i c e s to prove that c. e M. + M„ f o r each i . r x 1 2 Without l o s s of g e n e r a l i t y , we can assume that w 1 i s an element i n G s t a r t i n g w i t h a n o n t r i v i a l element i n G 2. Then, i n the expression ( 4 ) , there i s a j (say j = 1 ) , such t h a t = w j • W e c l a i m that a^ = c^. For t h i s purpose, w r i t e where cJ, d^ e R m and "u^ i s a n o n t r i v i a l element i n G j , f o r each I. I f d^ ^ 0 f o r some iy then d^TT^Wj must appear i n the e x p r e s s i o n ( 3 ) , which c o n t r a d i c t s the f a c t that wl i s of maximum s y l l a b l e l e n g t h . Hence, a l l d^ = 0 so that ax = cJ = c1. Therefore c1 c M° + K-R(G 1) c M j + M 2. (Note t h a t , . i f w. i s an element s t a r t i n g w i t h a n o n t r i v i a l element i n G., - 2 1 -then we w i l l c onsider ( 5 ) and i t w i l l l e a d to the c o n c l u s i o n t h a t Cj E M ° + K-R(G 2) C M I + M 2 . ) Consequently c^ e K. Having proved c 1 e K, we can apply the same procedure to x - CjW^ to conclude i n d u c t i v e l y that c^ e K f o r a l l i . Hence x e K-R(G). This completes the proof. For convenience, we s t a t e the f o l l o w i n g r e s u l t , the proof of which i s t r i v i a l . •Lemma 2 . 2 . - 4 . -Let N be an -m x n -matrix over •R(G1) (resp. R ( G 2 ) ) . Let f : R ( G 1 ) n — > R ( G 1 ) m (resp. f : R ( G 2 ) n — > R ( G 2 ) m ) be the homomorphism a s s o c i a t e d w i t h N and g : R ( G ) n — > R ( G ) m the homomorphism a s s o c i a t e d w i t h N (considered as a m a t r i x over R(G)). I f Ker f i s a f i n i t e l y generated r i g h t R(G 1)-module (resp. R(G 2)-module), then Ker g i s a f i n i t e l y generated r i g h t R(G)-module. As a consequence, we have : C o r o l l a r y 2 . 2 . 5 . I f R(Gj) (resp. R(G 2)) i s a r i g h t coherent r i n g and i f N i s a m a t r i x over R(G) w i t h e n t r i e s from R ( G ^ (resp. R ( G 2 ) ) , then the s o l u t i o n space of N i s f i n i t e l y generated as a r i g h t R(G)-module. - 22 -Now, we make the f o l l o w i n g important remark : Remark 2.2.6. (Modified Higman's t r i c k ) Let N' be an m' x n' ma t r i x over R(G). Each e n t r y of N' i s a f i n i t e l i n e a r combination of the form / r g w i t h r e R and g e G. g S -8 -Note t h a t g i s of the form ( 1 ) . We perform s u c c e s s i v e s i m p l i f i c a t i o n s of N' by changing i t to N' | 0 an (m' + 1) x (n' + 1) ma t r i x over R(G), and reduce t h i s m a t r i x by elementary row and column operations (where we m u l t i p l y row from the l e f t and column from the r i g h t ) so as to . ( i ) make l i n e a r combinations s h o r t e r and ( i i ) reduce the s y l l a b l e lengths of g. This r e d u c t i o n process can be i l l u s t r a t e d by : ft * * 0 ft ft ft 0 ft ft * o * * * -> * I + 8 1 8 2 * 0 ft ft 0 -> * I * -&1 *_ _0 0 0 1_ _0 g 2 0 1_ 0 1 F i n a l l y , we reduce N' to the m a t r i x of the form (6) N = - 2 3 -where N, a. • • • a. 11 IX a a , mi mX and N, b l l b ! 11 l y b , b ml my are m x X ma t r i x and m x y ma t r i x over R ( G 1 ) and R ( G 2 ) r e s p e c t i v e l y , and X + y = n, f o r some m and n. Proof of Theorem 2 . 2 . 1 : We need to show that any m x n m a t r i x over R ( G ) i s r i g h t coherent. By Lemma 2 . 2 . 2 and Remark 2 . 2 . 6 , we can assume without l o s s of ..generality t h a t . t h e .given „m x ,n .matrix N over R . (G) i s of the .form . . ( . 6 ) . Now, l e t a. = ( a , a .) ( i = 1,•••,X) and b. = (b .,•• • ,b .) ' l l x ' ' mi 5 ' j 1J m j ( j = l , - - ' , y ) . Then l e t be the submodule of R ( G ) M generated by the elements a ^ - ' - j a ^ i n R C G J)™ and M 2 the submodule of R ( G ) M generated by the elements b j , •••-,!> i n R ( G 2 ) M . I f f : R ( G ) N —> R ( G ) M i s the homomorphism a s s o c i a t e d w i t h N, we have the f o l l o w i n g p r e s e n t a t i o n f o r M 1 + M 2 : (7) 0 > ker f > R ( G ) n -^-> M + M 2 > 0. Next, l e t K = ( M ] [ + M 2 ) 0 R™. Then K i s a submodule of R™ (as r i g h t R-module) and so K i s f i n i t e l y generated s i n c e R i s r i g h t Noetherian. Suppose that K i s generated by c v = ( c ^ , • • • , c m t ) (k = a) "mk m^ i n R , and l e t - 24 -N R = '11 '10 C • • • C ml ma be the m x a ma t r i x over R determined by the generators of K. Consider the new m x (n + a) ma t r i x N = [ % \ N R i NG, 1 over R(G), and l e t N_ = [ N ! N_ ] and N_ = [ N ! N ]. Since G G^ 1 K K | ^ K C Mj + M 2, i t f o l l o w s that a 1 >-'-,a^, c^,''•,c^, b ^ - . - j b y s t i l l generate M 1 + M 2. I f g : R(G) n +° — > R ( G ) m i s the homomorphism a s s o c i a t e d w i t h N, we have another p r e s e n t a t i o n f o r Mj + M 2 : (8) 0 — > Ker g — > R ( G ) n + ° — > M' + M 2 — > 0 To prove that Ker f i n (7) i s f i n i t e l y generated, we only need to show that Ker g i n (8) i s f i n i t e l y generated (compare remark i n §2.1). For t h i s purpose, l e t ( x ^ ' - ' . x - , z ,-..,z , y ^ ' " ^ ' ) e Ker g. Then a j X j + ... + a x x A + c l Z l + ••• + c a z a + b ^ + ••• + b p y u = 0 so t h a t (9) a l X l + ... + a x x A + c ^ + ••• + c a z a = - ( b ^ + ... + b y y y ) . Let x be the element of the l e f t hand s i d e of (9). Then (9) t e l l s us that x e (M 1 + K-R(G)) f l (M + K-R(G)) and so x e K-R(G), by Lemma 2.2.3. Thus - 25 -(10) x = c l Z { + ... + cazxa f o r some zl, >..,z' e R(G). By (9) and (10), we have a j X l + ••• + a x x x + c l Z l + ... + caza = c l Z { + ••• + c a z ^ and C l z J + ••• + c ^ = - ( b l Y l + + b u y u ) That i s , a,x, + ... + a->x, + c . ( z , - zl) + ••• + c„(z_ - z') = 0 and c! z{ + ••• + c 0 z ^ + b j y j + • . . + b u y y = 0. These mean that (x,.•••.x,. z, - z , ' z - z') i s i n the s o l u t i o n snace v 1 - - A- i J- ' ' o a of N and (z',.«.,z', y , •••,y 1 1) i s i n that of N . Since R(G,) and R(G„) are r i g h t coherent, the s o l u t i o n spaces of N and N are Z G l G 2 f i n i t e l y generated r i g h t R(G)-modules ( C o r o l l a r y 2.2.5). F i n a l l y , note that i f ( x j , . - . , x x , z^t" • tz^) i s i n the s o l u t i o n y terms space of , then ( x ^ , . . . , x x , z " , • • •, zJJ, 0,-..,0) i s i n that of N and i f (z^1, • • • , z ^ , , * * • »Yy) i s i n the s o l u t i o n space of N^ , then X terms (6, 0, z",...,Zp, y^,''',y^) i s i n that of N. Since ( x 1 , . - . , x x , z 1 , " . , z a , y l t " . , y p ) y terms X terms = ( x l f . . . , x x , z ^ z ^ - . - . z ^ z ^ , 0,-..,0)•+ (0,-..,0, z j , - . . , z ^ , y 1 , - - - , y y ) : - 26 -and s i n c e the s o l u t i o n spaces of N_, and N are f i n i t e l y generated, G l G2 i t f o l l o w s from the above ob s e r v a t i o n that Ker g i s f i n i t e l y generated. This completes the proof. Remark 2.2.7. Waldhausen i n [19] considers the question of the coherence of a group r i n g Z(G) when G i s an amalgamated product. The proof of Theorem 2.2.1 i s p a r t l y i n s p i r e d by h i s arguments. When s u i t a b l y adapted, our present proof w i l l a l s o show the coherence of the f r e e product of two r i n g s , under appropriate hypothesis. C o r o l l a r y 2.2.8. Let R be a r i g h t Noetherian r i n g and F n a f r e e group of f i n i t e rank. Then R ( F n ) i s r i g h t coherent. In p a r t i c u l a r , i f A i s a f i n i t e l y generated a b e l i a n group, then Z(A x F n) i s r i g h t coherent. Proof : Since F n i s the f r e e product of a f r e e group ^ n_^ °f rank n - 1 and an i n f i n i t e c y c l i c group T, the f i r s t a s s e r t i o n f o l l o w s from Theorem 2.2.1 by i n d u c t i o n on n. The second a s s e r t i o n f o l l o w s from the f i r s t s i n c e Z(A x F) = Z(A)(F) and Z(A) i s Noetherian. As a consequence of Lemma 2.1.4 and C o r o l l a r y 2.2.8, we have : C o r o l l a r y 2.2.9. Let R be a r i g h t Noetherian r i n g and F a f r e e group Then R(F) i s r i g h t coherent. Moreover, i f A i s a a b e l i a n group, then Z(A x F) i s r i g h t coherent. - 27 -We w i l l use the f o l l o w i n g r e s u l t to prove the t r i v i a l i t y of Wh(G x a T) f o r a c e r t a i n c l a s s of groups G. C o r o l l a r y 2.2.10. Let F be a f r e e group and A a fr e e a b e l i a n group of f i n i t e rank. Then C(Z(A x F),a) = 0 f o r any automorphism a of A x F. Proof : Since r t . g l . dim Z(F) <_ 2 ( c f . [ 8 ] , Lemma 33), i t f o l l o w s that r t . g l . dim Z(A x F) = r t . g l . dim Z(F) + rank of A < 0 0 ( c f . [ 1 ] , Lemma 2 ) . The a s s e r t i o n now f o l l o w s from Theorem 2.1.3, s i n c e Z(A x F) i s r i g h t coherent. We clos e t h i s chapter by the f o l l o w i n g remark : Remark 2.2.11. Let R be a r i n g and G be a group. Let a : G — > Aut R be a homomorphism of G i n t o the automorphism group Aut R of R. The t w i s t e d group r i n g R a(G) i s defined as f o l l o w s : a d d i t i v e l y R a(G) = R(G) and m u l t i p l i c a t i o n i s given by (rg)(r»g') = r ( a ( g ) ) ( r ' ) g g ' f o r any r g , r'g' e R a(G). Then the r e s u l t s i n Theorem 2.2.1 and Lemma 2.2.3 remain true i f we replace group r i n g s by t w i s t e d group r i n g s . As a consequence, we see that the group r i n g Z(A x a F ) , of a s e m i - d i r e c t product A-* a F, i s r i g h t coherent. CHAPTER 3 WHITEHEAD GROUPS OF SOME SEMI-DIRECT PRODUCTS OF FREE GROUPS n §3.1. The Whitehead Group of a D i r e c t Product IT F. of Free Groups F. i = l 1 1 Bass, H e l l e r and Swan ([4]) proved t h a t Wh A = 0 f o r a f r e e a b e l i a n group A, and S t a l l i n g s ([18]) and Gersten ([10]) have shown that Wh F = 0 f o r a f r e e group F. The main purpose of t h i s s e c t i o n i s to g e n e r a l i z e these r e s u l t s to the f o l l o w i n g : n Theorem 3.1.1. Let D = n F. be a d i r e c t product of n f r e e groups F.. i = l 1 1 Then Wh D = 0. Proof : Let the number of n o n c y c l i c f a c t o r s i n D be k. We w i l l prove the theorem by i n d u c t i o n on k. For k = 0, D i s j u s t a f r e e a b e l i a n group so that Wh D = 0. This s t a r t s the i n d u c t i o n . Now, suppose i n d u c t i v e l y that the theorem holds f o r any such group w i t h k - 1 n o n c y c l i c f a c t o r s . To show Wh D = 0, w r i t e D = D' x F where F i s n o n c y c l i c , and the number of n o n c y c l i c f a c t o r s i n D1 i s k - 1. Then, f o r an i n f i n i t e c y c l i c group T, Wh (D' x T) = 0 by i n d u c t i o n hypothesis. I t f o l l o w s from the Bass-Heller-Swan decomposition formula f o r Wh ( D T x T) ( c f . Theorem 1.2.4) that (1) C ( Z ( D ' ) , i d ) = 0 and K Q Z ( D ' ) = 0. Next, suppose that F i s of f i n i t e rank m and w r i t e F = T"^  * ••• • *• T m as a f r e e product of m i n f i n i t e c y c l i c groups. Since C ( Z ( D ' ) , i d ) = 0, i t f o l l o w s from Theorem 1.3.3 that _ m _ (2) K Z ( D ' ) ( F ) £ 6 K , Z ( D ' ) ( T . ) . j = l 3 Again, u s i n g C ( Z ( D ' ) , i d ) = 0, we deduce from Theorem 1.2.2 that the sequence 0 —> K 1 Z ( D ' ) — > - K 1 Z ( D * ) ( T ) —> K D Z ( D ' ) — > 0 i s s h o r t exact f o r each j ; i . e . - K ^ Z C D ' H T ) = K D Z ( D ' ) f o r each j . Then i t f o l l o w s from (2) th a t K 1 Z ( D ' ) ( F ) = K 0 Z ( D » ) € • . . « K D Z ( D ' ) (m copies) ; i . e . , the sequence ty (3) 0 —> K 1 Z ( D ' ) — > K Z ( D ' ) ( F ) —•> K O Z ( D ' ) « . . . « K Q Z ( D ' ) — > 0 i s s h o r t exact. P a s s i n g to Whitehead groups, we have ( c f . Remark 1.2.3) ty -0 — > Wh D 1 — > Wh ( D ' x F ) —> K 0 Z ( D ' ) « ••• « K Q Z ( D ' ) —^> 0 Hence, by (1) and the i n d u c t i o n hypothesis f o r D ' , we have - 30 -Wh D = Wh (D' x F) = 0 . The case when F has i n f i n i t e rank does not need to worry us s i n c e a m a t r i x over Z(D)(F) i n v o l v e s e n t r i e s which are sums of words i n v o l v i n g only a f i n i t e number of f r e e generators of F. This completes the proof. In a d d i t i o n to the v a n i s h i n g of Wh D, we have the f o l l o w i n g v a n i s h i n g r e s u l t s , as i n (1) : C o r o l l a r y 3.1.2. C(Z(D),id) = 0 and K QZ(D) = 0. §3.2. The Whitehead Group of n n F . i = i 1 xa T Let F be a f r e e group, a an automorphism of F and T an i n f i n i t e c y c l i c group. Then F a r r e l l and Hsiang ([8]) have shown that Wh (F x a T) = 0 . This s e c t i o n i s devoted to the f o l l o w i n g g e n e r a l i z e d r e s u l t : n Theorem 3.2.1. Let D = II F. be a d i r e c t product of n f r e e groups F. . i = l 1 . 1 Let a be an automorphism of D which leaves a l l but one of the n o n c y c l i c f a c t o r s i n D p o i n t w i s e f i x e d . Then Wh (D x a T) = 0 . Proof :. Let k be the number of n o n c y c l i c f a c t o r s i n D. We prove by i n d u c t i o n on k. ° - 31 -For k = 0, D i s j u s t a f r e e a b e l i a n group A and so, by the F a r r e l l - H s i a n g decomposition formula f o r Wh (D x a T) ( c f . Theorem 1.2.4), Wh (D x a T) = 0 s i n c e C(Z(A),a) = 0 , Wh A = .0 and K QZ(A) = 0. For k = 1, D i s of the form A x F w i t h A f r e e a b e l i a n of f i n i t e rank and F n o n c y c l i c . Then, i n the F a r r e l l - H s i a n g decomposition formula f o r Wh ( ( A x F) x a T), the term Wh (A x F) = 0 by Theorem 3.1.1 and K QZ(A x F) = 0 by C o r o l l a r y 3.1.2. A l s o , thanks to the coherence property of Z(A x F) and the f a c t that Z(A x F) i s of f i n i t e r i g h t g l o b a l dimension, C(Z(A x F),a) = 0 ( C o r o l l a r y 2.2.10). Hence Wh((A x F) x a T) = 0. Now, suppose i n d u c t i v e l y that the theorem holds f o r any such group w i t h k - 1 n o n c y c l i c f a c t o r s . Let D = H x F w i t h F n o n c y c l i c and a f i x e d on F w h i l e H has k - 1 n o n c y c l i c f a c t o r s . To show th a t Wh (D x a T) = 0, w r i t e D x a T = D' x F w i t h D' = H x a T. The s i t u a t i o n i s now completely analogous to Theorem 3.1.1 and the same argument as there gives Wh (D x a T) = 0. This completes the proof. In a d d i t i o n to the t r i v i a l i t y of Wh (D x a T), we have, by the F a r r e l l - H s i a n g decomposition formula f o r Wh (D *-a T) t h a t C(Z(D),a) = 0. Moreover, by c o n s i d e r i n g Wh ((D x a T) x T j ) which i s j u s t Wh ((D x Tj) x a X ^ ( j • T) > w e have the f o l l o w i n g v a n i s h i n g r e s u l t s : T l C o r o l l a r y 3.2.2. ( i ) C(Z(D x a T ) , i d ) = 0 . ( i i ) K QZ(D x a T) = 0. - 32 -The r e s u l t ( i i ) of C o r o l l a r y 3.2.2 i m p l i e s , i n p a r t i c u l a r , that f o r the fundamental group i r 1 ( M ) of a cl o s e d surface M (other than the r e a l p r o j e c t i v e p l a n e ) , the p r o j e c t i v e c l a s s group of Z(T T 1(M)) i s t r i v i a l . We a l s o need the f o l l o w i n g s l i g h t g e n e r a l i z a t i o n of C o r o l l a r y 3.2.2 ( i ) i n § 3. 4. C o r o l l a r y 3.2.3. Let D and a be as given i n Theorem 3.2.1. Then C(Z(D x a T),a y) = 0 f o r any i n t e g e r u, where a y denotes the automorphism a y x i d of D x a T induced by a y of D. Proof : Let = <t^> be another i n f i n i t e c y c l i c group generated by t j . Consider the s e m i - d i r e c t product (D x a T) x T^. Then, by change of a)1 generators i n T x (D x^ T) x T can be seen to be isomorphic to a (D x S) x a X ^ ( j T, where S = <t Vt^> i s an i n f i n i t e c y c l i c group generated s by t ~ y t , . By Theorem 3.2.1, Wh((D x S) x . , T) = 0 and so J 1 J ' a X l d S Wh((D x Q T) x Tj) = 0. Hence C(Z(D x a T),a y) = 0. This completes the proof. There i s a t o p o l o g i c a l a p p l i c a t i o n of Theorem 3.2.1. I f M i s a cl o s e d s u r f a c e (other than the r e a l p r o j e c t i v e plane) and (S 1)^" i s the l k k-dimensional t o r u s , then the fundamental group of M * (S ) i s of the - 33 -form D xa T. Hence Theorem 3.2.1 implies the following (cf. [14], p.393) : Corollary 3.2.4. If N is a differentiable or PL manifold of dim >_ 5 1 k which is h-cobordant to M x (S ) , then N i s actually diffeomorphic or 1 k PL-homeomorphic to M x (S ) respectively. §3.3. K1 of Twisted Free Associative Algebras Let R be a ring and X a set of non-commuting variables {x,} .. A A£ A Let R{X} be the free associative algebra on X over R. Gersten has shown that i f KjR —> K^Rtt] i s an isomorphism, where R[t] is the polynomial extension of R, then K^R —> K^R{X} is an isomorphism (cf. [10] and [2], p.646). This section presents a generalization of Gersten's result to twisted free associative algebras which we w i l l apply in §3.4. Let X be a set of non-commuting variables {x.. } and l e t A A E A a = {a,}, . be a set of automorphisms a, of R. The a-twisted free A AeA A associative algebra on X over R, denoted by R^X}, i s defined as follows : Additively, Ra(X} = R{X} so that i t s elements are f i n i t e linear combinations of words w(x^) in x^ with coefficients in R. If w(x^) = x^ ••• x^ is a word in x^, we denote the l k automorphism a, ••• a, by w(a.). - 34 -M u l t i p l i c a t i o n i n i s given by (rw( x ^ ) ) ( r ' w ' ( x ^ ) ) = rw ( o t ^ ) 1 ( r ' ) w ( x x ) w ' ( x ^ ) , f o r any rw(x^), r'w'(x^) e R a x X } . We s h a l l consider R {X} as an R-r i n g w i t h augmentation e X : ^ a ^ ^ — > R d e f i n e d by e x ( x ^ ) = 0 f o r each x^ e X. Denote by KjR {X} the cokernel of the homomorphism i ^ : KjR — > K^R^X} induced by the i n c l u s i o n i : R —-> R a(X}. Note that the augmentation e„ induces a homomo rphism e^* : K Rfl{X} — > KjR which s p l i t s i A . Now, l e t N" be an i n v e r t i b l e m a t r i x over R^{X}. By Higman's t r i c k , we can make N" eq u i v a l e n t i n K^R {X} to N' = N i + N j x x + . •. + N ^ o where X j , x n are d i s t i n c t elements of X and N| ( i = 0,1,-'^n) e W m(R) f o r some i n t e g e r m. (Here w? m( R) denotes the r i n g of a l l m x m matrices over R). By a p p l y i n g the homomorphism to N', i t f o l l o w s that N^ i s i n v e r t i b l e . Hence N" can be made e q u i v a l e n t i n KjR {X} to (1) N = I + N x x j + • + N n X n , where N = N ^ V and N ± = N£ _ 1Nj ( i = l , - - . , n ) . The i n v e r s e of t h i s m a t r i x N e x i s t s and can be w r i t t e n e x p l i c i t l y i n the r i n g of formal power s e r i e s . Since t h i s i n v e r s e e x i s t s i n R a(X}, - 35 -a l l but a f i n i t e number of i t s c o e f f i c i e n t s are zero. That i s , i f n M = M„ + M.x. + • • • + M x + J M. .x.'x. + • • • o 1 1 n n L l j 3 x 3 1 > J X i s a m a t r i x over R„{X} where a l l M., M. ., ••• are matrices over R, such that MN = NM = I , then there i s an i n t e g e r K > G such t h a t M. = 0 f o r a l l k > K, where i , , i„, i , run over 1, ...,n r e s p e c t i v e l y . From NM = I , we get, by equating c o e f f i c i e n t s of monomials i n the x's, the f o l l o w i n g r e l a t i o n s : M D = I ; M. = -N. ( i = 1. •••.n) : l l - ' • ' M , = N a T V ) ( i , j = 1, n) ; M. . = (-1) £N. aT^N. ) ••• ( a A . 1 ••• aT1 )(N. ) V V " 1 * . xi xi x2 \ \ V i \ ( i 1 5 i 2 , *••> i £ = i» ••',n) Hence, f o r a l l k > K, (2) N. a.^N. ) ••• ( a . ^ x . 1 ••• a . 1 ) (N, ) = 0. 1 1 1 1 H 1 1 x2 x k - l k Let us c a l l an element P e "1 (R) B-twisted n i l p b t e n t (g i s any automorphism of R) i f there e x i s t s an i n t e g e r k > 0 such that P B _ 1 ( P ) ... 3 ~ ( k ^ ( P ) = 0. - 36 -Hence, i t f o l l o w s from (2) that each ISL ( i = l,--',n) i n (1) i s a^ - t w i s t e d n i l p o t e n t . Our next lemma i s the key to the main r e s u l t : Lemma 3.3.1. The matrix N i n (1) i s a product of matrices of the form I + PwCxj,•••,x n), where P i s an wCo^ , • • • , a n ) - t w i s t e d n i l p o t e n t m a t r i x over R. (w(x 1,•••,x n) denotes a word i n x 1 , . . . , x n ) . Proof : R e c a l l from (1) and (2) that each N ± ( i = l,..-,n) i n (1) i s a.- t w i s t e d n i l p o t e n t . Consider I + Q = ( I - N,x') ••• ( I - N x )N . a. -TJl 'U Then Q i s of the form \ ^ j s j ' w n e r e each s.. i s a monomial of degree at l e a s t two i n the x i > - , , » x n a n& Q. = ± N. aT 1(N. ) ••• ( a T 1 ••• aT 1 )(N. ) ( i j , i 2 , i = -1, • • •, n) where £ >_ 2. Hence, f o r k > K/2, Q j e " 1 ( Q J ) 3 _ ( k ~ 1 ) ( Q j ) = °> fo r each j , where g i s an automorphism obtained i n r e p l a c i n g the x^ i n s by a r e s p e c t i v e l y . That i s , Q. i s s. (o^, • • • , o t n ) - t w i s t e d n i l p o t e n t f o r each j . Now, consider - 37 -I + Q' = H (I - Q,s.)(I + Q). j J J Then Q' i s of the form J Q'yo > where each y^ i s a monomial of degree a at l e a s t four i n the X j , x n and f o r k > K/4, Q A Y _ 1 ( Q : ) • • • Y ( k _ 1 ) ( Q : ) = o, f o r each cr, where Y i s an automorphism obtained i n r e p l a c i n g the x^ i n y a by r e s p e c t i v e l y . That i s , i s y 0(o' 1 > • • • , a n ) - t w i s t e d n i l p o t e n t f o r each a. L e f t m u l t i p l y i n g I + Q' by 11(1 - Q*y ) , and r e p e a t i n g the o above argument, we w i l l f i n a l l y a r r i v e at the c o n c l u s i o n that n ( I + Pw ( X l , - . . , x n ) ) . N = I where P i s an w(cij, • • • , a n ) - t w i s t e d n i l p o t e n t m a t r i x over R and wCx^.'-jX^) i s a word i n x 1 , - - - , x n . This completes the proof. The above d i s c u s s i o n s are m o d i f i c a t i o n s of those given i n [10] and ( [ 2 ] , p.647) f o r (untwisted) f r e e a s s o c i a t i v e algebras ; and the f o l l o w i n g r e s u l t i s already contained i n the above proof ( a l s o , c f [ 4 ] ) . - 38 -Lemma 3.3.2. For any automorphism g of R, K^R^ft] i s generated by the elements of the form I + Pt where P i s an g-twisted n i l p o t e n t m a t r i x over R. The f o l l o w i n g main theorem then f o l l o w s immediately from Lemma 3.3.1 and Lemma 3.3.2 : Theorem 3.3.3. The group K^R^fX} i s generated by the homomorphic images of K j R g t t ] under the homomorphisms KjRpEt] > K l R a { X } induced by the homomorphism Rett] —> R^tX} which maps t i n t o a word w(x^) i n x-^  w i t h g = w(a^) and w(x^) runs over a l l the words i n x^. C o r o l l a r y 3.3.4. I f R i s a r i n g such that C(R,g) = 0 f o r any automorphism g of R, then K^R^X} = 0; i n other words, i f K^R — > K j R g f t ] i s an isomorphism, then KjR — > K^R^fX} i s an isomorphism. Now, l e t a be an automorphism of R and f o r each X e A, l e t mX °x = a f o r some i n t e g e r m^ . In t h i s case, we denote R#{X} by R a(X} . Thus, we have the f o l l o w i n g c o r o l l a r i e s which w i l l be needed i n §3.4. - 39 -C o r o l l a r y 3.3.5. The group K 1R a(X} i s generated by the homomorphic images of K-^ R [ t ] (y any i n t e g e r ) under the homomorphisms K R• ;[t] > K,R a{X} induced by the homomorphisms R [ t ] — > R a(X} which map t i n t o a word w(x^) such t h a t y i s the t o t a l exponent sum of a appearing i n . w(oc x), and w(x^) runs over a l l the words i n x^ . C o r o l l a r y 3.3.6. I f R i s a r i n g such that CCRja'"1) = 0 f o r any i n t e g e r y, then K 1R a(X} = 0. f n 1 §3.4. The whitehead Group of .\ F i x a T x ., F . a x i d m I 1 = 1 J Tn Let D = II F. be a d i r e c t product of n f r e e groups F. and i = l a an automorphism of D which leaves a l l but one of the n o n c y c l i c f a c t o r s i n D pointwise f i x e d . In Theorem 3.2.1, we proved that Wh(D x a T) = 0, where T i s an i n f i n i t e c y c l i c group. Let F be another f r e e group. The purpose of t h i s s e c t i o n i s to g e n e r a l i z e Theorem 3.2.1 to : Theorem 3.4.1. Wh((D x„ T) x F) = 0 . a a x i d T The f a c t o r T i n the theorem can presumably be dropped, although t h i s i s not e n t i r e l y obvious. Compare the r e s u l t s at the end of t h i s s e c t i o n . - 40 -From now on, we denote a l s o by a, the automorphism a x i d ^ , of D x Q T induced by the automorphism a on D. Suppose that F i s generated by ( t ^ } and l e t R a t F ] be the a-twisted group r i n g of F over a r i n g R w i t h automorphism a. Not i c e t h a t , i n g e n e r a l , there i s no augmentation from R a [ F ] i n t o R. Now, l e t R = Z(D x a T) . Then R ^ F ] i s c a n o n i c a l l y i s omorphic to Z((D x a T) x Q F) ( c f . §1.1). Define a mapping : R a [ F ] — > R by e F ( r t A > " r t f o r a l l r E R and t ^ e F, where t i s the generator of T. Then i t i s c l e a r that e i s a homomorphism of R„[F] onto R w i t h e„(r) = r f o r a l l r E R, i . e . , we can consider R a[F] as an R-ring w i t h augmentation E j , . Note that the homomorphism i& : K^R — > K j R ^ F ] induced by the i n c l u s i o n R — > R a [ F ] i s one-to-one, s i n c e the homomorphism E p , v : KjRpjF] — > KjR, induced by E p , s p l i t s i A . Moreover, the k e r n e l of E f , denoted by R a [ F ] , i s a f r e e R-bimodule generated by where w runs over a l l the words i n t ^ and |w| i s the t o t a l exponent sum of t ^ appearing i n w, and as bimodules, c R o[F] = R « R j F ] . A l s o , we have R a [ F * F 1 ] = R a [ F ] * R^tF'] where F' i s another f r e e group. - 41 -Let T R ( R a [ F ] 0 R R a [ F 1 ]) be the tensor algebra of R 0 [ F ] 8 R R e ([F l] over R. Then i t i s easy to see that t h i s i s nothing but the t w i s t e d f r e e a s s o c i a t i v e algebra R a(X} over a s e t X of non-commuting v a r i a b l e s x given by x = (w - t ' w ' ) e (w' - tl w ' l ) where w - (resp. |w'| - JW )^ runs over the generators of R a [ F ] (resp. R a [ F ' ] ) . Since C(R,a y) = C(Z(D x a T),a y) = 0 f o r any i n t e g e r u ( C o r o l l a r y 3.2.3), i t f o l l o w s from C o r o l l a r y 3.3.6 th a t K ^ R J F ] 8 R R a [ F ' ] ) = 0. Hence, by Theorem 1.3.1, we have K 1 R a [ F * F' ] = K ^ C F ] <B K 1 R a [ F ' ] . That i s , (1) K XZ((D x a T) x a (F ft F')) = K XZ((D x a T) x a F) € KjZCCD x a T) x a F'). F i n a l l y we give the proof of Theorem 3.4.1. I t i s s i m i l a r to t h a t of Theorem 3.1.1. Proof of Theorem 3.4.1 : F i r s t , suppose that F i s of f i n i t e rank m and w r i t e F = T.. ft • • • * T m as a f r e e product of m i n f i n i t e c y c l i c groups. Then, i t f o l l o w s from (1) that _ m (2) KjZ((D x a T) x a F) = <B KjZ((D x a T) x a T ) j = l 3 Using C ( Z(D x a T),a y) = 0 f o r any i n t e g e r u, we deduce, from Theorem 1.2 tha t the sequence 0 > KjZ(D x a T) KjZCCD x a T) x a T ) > (K Q Z(D x a T ) ) " * > 0 J i s s p l i t s h o r t exact, f o r each j . That i s K.7,((T) x.. T~) x,' T 1 = (K Z(n xr. I ) ) " * f o r each j , and so, i t f o l l o w s from (2) that KjZ((D x a T ) x a F ) = (K D Z(D x a T))™* «••••© (K Q Z(D x a T ) ) ° * (m copies) I n other words, the sequence i * 0 > K XZ(D x a T ) — > KjZCCD x a T ) x a F ) > (K Q Z(D x a I ) ) " * 0 ••• ti (K D Z(D x a T))°* > 0 i s s h o r t exact. Passing to Whitehead groups ( c f . Remark 1.2.3), we have the s h o r t exact sequence - 43 -0 > Wh(D x a T) — > Wh ( (D xa T) x a F) > ( K Q Z ( D x a T ) ) ™ * « . . . © ( K 0 Z ( D x a T ) ) " * > 0. But Wh(D x a T) = 0 (Theorem 3.2.1) and K Q Z ( D x a T) = 0 ( C o r o l l a r y 3.2.2). Hence Wh ( ( D x a T) x a F) = 0. This completes the proof. F i n a l l y , s i n c e ( D x a ? ) - * a > < i d F and ( D x a F) T are T F isomorphic, we have, i n a d d i t i o n to Theorem 3.4.1, that C o r o l l a r y ,3.4.2. ( i ) C ( Z , ( D x a .p.),, a x idp) = 0 (a x i d ) * ( i i ) ( K Q Z ( D x a F)) = 0 . ( i i i ) Wh(D x a F ) / I ( ( a y } - 0 . r §3.5. Concluding Remarks Let D be the d i r e c t product of n f r e e groups. We observe th a t the c o n d i t i o n we impose on a i n §3.2 and §3.4 can be dropped i f one can prove that the i n t e g r a l group r i n g Z ( D ) i s r i g h t coherent. In f a c t , the coherent property of Z ( D ) w i l l f o l l o w from the f o l l o w i n g c o n j e c t u r a l r e s u l t : Let R be a r i n g such t h a t R[T], the Laurent s e r i e s e x t e n s i o n of R, i s r i g h t coherent ( t h i s i m p l i e s that R i s coherent). L et and G 2 be groups such that R(Gj) and R(G 2) are r i g h t coherent. Then - 44 -R(G 1 ft G 2) i s r i g h t coherent. Next, l e t R be a r i n g and a an automorphism of R. Then i t i s an i n t e r e s t i n g and important question to ask whether the coherence of R[T] w i l l imply that of R a [ T ] . This r e s u l t , i f t r u e , w i l l g i v e the t r i v i a l i t y of Wh G f o r another c l a s s of groups G. F i n a l l y , l e t F j , F 2 , F n be f r e e groups. Let be an automorphism of F j . Form the s e m i - d i r e c t product F j x^ F 2. Then, l e t a 2 be an automorphism of F j x a F 2 and again form the s e m i - d i r e c t product (Fj^ F 2) x^ F . Repeating the same procedure, we a r r i v e at the group G - ( . . . x F 2) x F 3) x ...) x F n . 1 z n-1 I t would be u s e f u l to have a method of computing Wh G, K QZ(G) and C(Z(G),a). We have proved that Wh G = 0 and K QZ(G) = 0 when = a 2 = ••• = a n_2 = i d e n t i t y and f o r some other s p e c i a l cases ( c f . Theorem 3.1.1, Theorem 3.2.1 and Theorem 3.4.1), but the general case remains open. CHAPTER 4 GROUPS F x a T WITH ONE DEFINING RELATOR S4.1. I n t r o d u c t i o n In [ 8 ] , F a r r e l l and Hsiang have shown that the fundamental group ir^M) of a c l o s e d s u r f a c e M (not the sphere or the p r o j e c t i v e plane) i s of the form F x a T, where F i s a f r e e group and T an i n f i n i t e c y c l i c group, and so Wh ir^M) = 0. T h e i r proof i s t o p o l o g i c a l . A l s o , i t i s w e l l known that such a group TTJ (M) can be presented as a group w i t h one d e f i n i n g r e l a t o r . The purpose of t h i s chapter i s to o b t a i n c e r t a i n necessary and s u f f i c i e n t c o n d i t i o n s f o r a group w i t h one d e f i n i n g r e l a t o r to be of the form F x a T. This w i l l g i v e us an a l g e b r a i c proof of the r e s u l t f o r ir^M) p r e v i o u s l y mentioned. Now, l e t us r e c a l l those d e f i n i t i o n s , terminology and r e s u l t s from C o m b i n a t o r i a l Group Theory which w i l l be subsequently used. For more d e t a i l s and undefined terms, we r e f e r to K a r r a s s , Magnus and S o l i t a r ( [ 1 3 ] ) . Let G be a group w i t h generators a, b, c, ••• . A word R(a,b,c,'»«) which defines the i d e n t i t y element 1 i n G i s c a l l e d a r e l a t o r . Let P, Q, R, ••• be any r e l a t o r s of G. I f every r e l a t o r i n G i s d e r i v a b l e from P, Q, R, we c a l l P, Q, R, ••• a s e t of d e f i n i n g r e l a t o r s f o r G on a, b, c, ••• . I f P, Q, R, ••• i s a s e t - 46 -of d e f i n i n g r e l a t o r s f o r G, we c a l l < a,b,c,... ; P(a,b,c,•••)> Q(a,b,c,•••), R(a,b, c, • •• •) , ••• > a p r e s e n t a t i o n of G and w r i t e (1) G = < a,b,c, ••« ; P,Q,R, ••• > . The f r e e group F n on the n f r e e generators X j , x 2 , ^ i s the group w i t h generators x^, x 2 , and the empty s e t of d e f i n i n g r e l a t o r s . A c y c l i c a l l y reduced word i n , •••, x f l i s a word i n which the symbols x^, x^ (e = ±1, i = l,'-',n) do not occur c o n s e c u t i v e l y and i t £ — £ does not simultaneouslv beein w i t h x and end w i t h x. (e = ±1. i = l 4 ' - ' , n ) . I f w ( x j , •••,x n) i s a word i n X j , x n , denote by O w(x_^) the exponent sum of w on x. . Theorem 4.1.1. ( [ 1 3 ] , Theorem 1.3) Let F R be a f r e e group on the f r e e generators x^, x^ and l e t w^, w 2 be two c y c l i c a l l y reduced words. Then w^, w 2 d e f i n e conjugate elements of F R i f and only i f i s a c y c l i c permutation of w 2 . Theorem 4.1.2. ( [ 1 3 ] , Theorem 4.10) ( F r e i h e i t s s a t z Theorem) Let G be a group presented by G = < x x , x n ; R ( x l f x n ) > - 47 -where R ( X j , x n ) i s a c y c l i c a l l y reduced word i n x^ ( i = 1, •••, n ) , which i n v o l v e s xn. Then the subgroup of G generated by x 1 , x n _ ^ i s f r e e l y generated by them. Theorem 4.1.3. ( [ 1 3 ] , Theorem 4.11) (Conjugacy Theorem f o r Groups w i t h  One D e f i n i n g r e l a t o r ) Let G = < Xj , • • •, ^ ', R ( x 1 , • • •, x n ) > and H = < X j , •••, x n ; S ( x 1 , x n ) > . Then G i s isomorphic to H under the mapping x^ — > x^ i f and only i f R ( X j , x n ) and S e ( x 1 , x f l) are conjugate i n the f r e e group on x J , x^, f o r e = 1 or -1. We c l o s e t h i s s e c t i o n by r e c a l l i n g the T i e t z e transformations ( [ 1 3 ] , §1.5). Let G be a group presented by (1). Then H. T i e t z e has shown tha t any other p r e s e n t a t i o n of G can be obtained by a repeated a p p l i c a t i o n of the f o l l o w i n g transformations to (1) : (TI) I f the words S, U, ••• are d e r i v a b l e from P, Q, R, then add S, U, ••• to the d e f i n i n g r e l a t o r s i n ( 1 ) . (T2) I f some of the r e l a t o r s , say, S, U, l i s t e d among the d e f i n i n g r e l a t o r s P, Q, R, *••, are d e r i v a b l e from the o t h e r s , d e l e t e S, U, ••• from the d e f i n i n g r e l a t o r s i n (1 ) . (T3) I f K, M, • • • are any words i n a, b, c, then a d j o i n the symbols x, y, ... to the generators i n (1) and a d j o i n the r e l a t i o n s x = K, y = M, ••• to the d e f i n i n g r e l a t o r s i n (1 ) . (T4) I f some of the d e f i n i n g r e l a t o r s i n (1) take the form p = V, q = W, •.. where p, q, ••• are generators i n (1) and V, W are words - 48 -i n the generators other than p, q, then d e l e t e p, q, ••• from the generators, d e l e t e p = V, q = W, ••• from the d e f i n i n g r e l a t i o n s , and re p l a c e p, q, «•• by V, W, ••• r e s p e c t i v e l y , i n the remaining d e f i n i n g r e l a t o r s i n ( 1 ) . The transformations ( T l ) , (T2), (T3), and (T4) are c a l l e d T i e t z e  t r a n s f o r m a t i o n s . Let G be a group presented by (2) G = < x x , x 2 , • • •, x n ; R ( x x , x 2 , • • •, x n ) > (n >_ 2) where we assume that R i s c y c l i c a l l y reduced and i n v o l v e s a l l the generators x 1' x2 ' * " ' ^ Si ' R e c a l l that i f there i s a s p l i t s h o r t exact sequence (3) 1 > N > G T > 1 , where T = <t> i s an i n f i n i t e c y c l i c group, then G i s the s e m i - d i r e c t product N * A T of N and T w i t h respect to the automorphism a : N ->- N defi n e d by a(g) = t g t f o r a l l g e N. I f G i s a group given by ( 2 ) , i t i s easy to f i n d a homomorphism <}> : G — > T from G onto T. Let N be the k e r n e l of <j>. Then G s a t i s f i e s (3) and so i s of the form N x Q T. Of course, i n general there e x i s t many such homomorphisms <j> and t h e r e f o r e many s p l i t t i n g s . - 49 -§4.2. Groups w i t h Two Generators and One D e f i n i n g R e l a t o r Let G be a group presented by (1) G = < a,b ; R(a,b) > where R i s c y c l i c a l l y reduced and i n v o l v e s both a and b. Then we know that G i s of the form N x a T. The purpose of t h i s s e c t i o n i s to o b t a i n c e r t a i n necessary and s u f f i c i e n t c o n d i t i o n s f o r the f a c t o r N i n N * a T to be f r e e corresponding to c e r t a i n n a t u r a l choices of the epimorphism <j> : G — > T. We d i s t i n g u i s h the f o l l o w i n g three cases. Case 1 : c (a) ^ 0 and c (b) = 0 or v i c e v e r s a ; K R Case 2 : ffn(a) ^ 0 and 0 o(b) 4 0 ; Case 3 : a., (a) = o_(b) = 0. We are able to s e t t l e cases 1 and 2, but not q u i t e case 3. F i r s t , l e t us consider case 1. In t h i s case, the epimorphism 4> : G — > T i s uniquely defined up to s i g n by (2) cb(a) = 1, (f>(b) = t e (e = 1 or - 1 ) . N o t i c e t h a t i n case 3, the above <j> (given by (2)) i s one of the choices of the homomorphisms from G onto T. Of course, there are many other homomorphisms from G onto T ; f o r example, one of them w i l l be - 50 -the homomorphism <f>1 : G — > T defined by (^(a) = t , ^ ( b ) - 1. We w i l l see l a t e r , by an example, t h a t i n case 3, G may be of the form N x a T and x Q T w i t h N f r e e and N 1 not f r e e . From now on, we w i l l c o nsider case 1 or case 3 w i t h the above homomorphism cf>. Thus, we may assume that the f a c t o r T, i n N x a T, i s j u s t the i n f i n i t e c y c l i c group generated by b i n G and then N i s nothing but the normal subgroup of G generated by a. To o b t a i n a p r e s e n t a t i o n - o f N, we make use of a Reidemeister-S c h r e i e r r e w r i t i n g process, and as S c h r e i e r r e p r e s e n t a t i v e s f o r G mod N we choose t 1 , where i runs over a l l i n t e g e r s ( [ 1 3 ] , §2.3 and §4.4). We f i n d that N i s generated by the elements a^ d e f i n e d by a^ = b 1ab 1 ( i , any i n t e g e r ) . Now, we r e w r i t e R(a,b) i n terms of a^ as f o l l o w s : Every symbol a e (e = ±1) i n R(a,b) i s rep l a c e d by a| where s i s the sum of the exponents of the b-symbols preceding the p a r t i c u l a r a e i n R(a,b). Thus R(a,b) can be expressed i n terms of a^ as : (3) R(a,b) = R D ( a A , a x + 1 , • • •, a y) w i t h X < • • • < y . Then N i s generated by and has as d e f i n i n g r e l a t o r s - 51 -(4) P = b 1R(a,b)b - l = R(ax+i> ax+i+i' V i } ( i , any i n t e g e r ) , and so N can be presented as (5) N =<•-., a _ 1 , a Q , a ^ ; • • •, P_ x, P Q, ? 1 , ••• > The f o l l o w i n g r e s u l t , mentioned i n ( [ 5 ] , §3) gives s u f f i c i e n t c o n d i t i o n s f o r N to be f r e e . Lemma 4.2.1. I f a^ and a y each appears j u s t once i n P Q (= R D) w i t h exponent 1 or -1, then N i s f r e e l y generated by a^, a^ +2. 5 ***> a u - l ' Proof : Since a^ and a u each appears j u s t once i n P 0 w i t h exponent 1 o r - 1 , i t f o l l o w s from (5) that a... and a"., each appears j u s t ' X+i u+i once i n P^ w i t h exponent 1 or -1, f o r each i . Thus, from ( 5 ) , we get ( 6 ) ax+i = w ( a x + i + r •••> V i ) f o r i < 0, where e = 1 or -1, and ( 7 ) i j ^ ' ^ j ' - ' V j J f o r j > 0, where n = 1 or -1. - 52 -Then, by app l y i n g T i e t z e t r a n s f o r m a t i o n (T4) r e p e a t e d l y , we can de l e t e P. and the corresponding generators a f o r i < 0 i n N, •L A+1 and we can del e t e P_. and the corresponding generators a u - j - j ^ o r J — ® i n N. Hence .N = < a,, a,.,, •••, a" , >, i . e . N i s f r e e l y generated A A+1 u-1 b y V AA+r V i * This completes the proof. We w i l l show that the c o n d i t i o n s given i n Lemma 4.2.1 are a l s o necessary f o r N to be f r e e . The next lemma i s the key to our main r e s u l t . Lemma 4.2.2. Let H be a group presented by (8) H = < ylt y 2 , y n ; S ( y x , y 2 , y n ) > where S i s c y c l i c a l l y reduced and suppose that y n appears a t l e a s t twice i n S. Then y n cannot be expressed as a word i n terms of y j , y 2 , •••» Y n_^ i n H. (Note that i f y^ appears i n S, then y n i s considered to appear twice i n S). Proof : Suppose that y n = V(yl>.'", y n _ 2 ) > a word i n y 1 5 yjx_1 • Then Q = y nV ^ i s a r e l a t o r i n H so t h a t , by T i e t z e t r a n s f o r m a t i o n ( T l ) , H = < y x , ••-> y n ; s , Q > . Therefore, by T i e t z e transformation (T4) H = < y l t y n _ 1 ; s ( Y l , • • , y ^ , v ( y i , y ^ ) ) > Thus, by F r e i h e i t s s a t z theorem (Theorem 4.1.2), S(yj,•••>y n - 1> V ( y 1 , # , , , y n must reduce to the empty word and so S i s d e r i v a b l e from Q. By T i e t z e t r a n s f o r m a t i o n (T2), we have H = < y v •", yn ; Q > . Hence, by Theorem 4.1.3, Q (e = 1 or -1) and S are conjugate i n the f r e e group on y , ' '"» v n • Therefore, by Theorem 4.1.1, Q (e = 1 or -must be a c y c l i c permutation of S, which i s impossible s i n c e y n appears at l e a s t twice i n S. This completes the proof. Now, without l o s s of g e n e r a l i t y , we can assume, i n ( 3 ) , that A = 1 and u = k f o r some i n t e g e r k > 1. Let H and H' be the groups presented r e s p e c t i v e l y by (9) H = < a l f a 2 , ••• ; P Q, P 2, • • • > and (10) H* = < a ^ , a x , a D , a_ j, ••• ; P_ i, P_ 2 > ••• > Then N i s the f r e e product of H and H' w i t h amalgamation over the - 54 -common subgroup f r e e l y generated by a^, ••«, a j c _ 1 ( c f . [13], §4.2). Thus, H and H' are subgroups of N. Let (ID A . = o p (a.) (j = 1, k ) . 3 R D 3 Let F Q = B D - < a l 5 a 2 , , ^  ; P Q (= R Q) > , Fm= < a l W a 2 W "••» ^ 5 Pm > • ( m > 0 ) = < a x , a 2 , ^ ; P D, P l , P m > , (m > 0 ) . Then H m i s the f r e e product of H m _ ^ a n a" F m w i t h amalgamation over the common subgroup f r e e l y generated by a ^ + m » ***» ^-fm-] " Hence, we have ascending chain of groups an H D C H L C . . . C H m C such t h a t H = U H. ( [ 1 3 ] , p.33). In t h i s way, we regard H m as subgroup j J of H f o r each i n t e g e r m > 0. We w i l l use the f o l l o w i n g lemma, due to E.S. Rapaport ( [ 1 5 ] ) . Lemma 4.2.3. Let E be a group presented by E '= < Xj, x 2, Xm ; Q > . Then m - l i s maximal f o r a l l p r e s e n t a t i o n s of E, t h a t i s , the d i f f e r e n c e - 55 -between the number of generators aud the number of r e l a t o r s i s always <_ m-l. Next, we are going to determine a necessary c o n d i t i o n f o r H to be f r e e . The f o l l o w i n g lemma may r e l a t e to case 3 w i t h the homomorphism <j>, but not to case 1. Lemma 4.2.4. I f a l l the exponent sums X.. ( j = 1,•••,k) given i n (11) are zero, then H given by (9) cannot be a f r e e group (and so N i s not f r e e ) . Proof : Consider the subgroup H Q = < a^, a 2 , •••, a^ ; P Q > of H. Suppose that H Q i s f r e e . Then, by Lemma 4.2.3, H Q i s f r e e of rank at most —k - -1. -Now,-since X 0 ( j =1, - • •, k ) , the a b e l i a n i z a t i o n of H Q can be presented by < ax, a 2 , a k ; a = a £ a ( j , A = 1, • • •, k ) , P Q > which i s < aj» a 2 » *""» a k » a j a £ ~ a £ a j (J » & = 1» ' *' > k) > by T i e t z e t r a n s f o r m a t i o n (T2). Hence t h i s a b e l i a n i z a t i o n i s f r e e a b e l i a n of rank k which c o n t r a d i c t s the f a c t that H Q i s f r e e of rank at most k - 1. Hence H i s not f r e e and so H i s not f r e e . This completes the proof. Thus, we can, i n a d d i t i o n , assume that some of the X^ ( j = l,«*«,k) i n (11) are not zero. Let H be the a b e l i a n i z a t i o n of H. Then H has generators - 56 -V 2 ! and has a s e t of d e f i n i n g r e l a t o r s : a.a. = a.a. ( i , i > 0) , l J J l J and k ( i 2 ) P : = y A.a... , 1 j = i J 1 + J where i >_ 0 and A ( j = 1, • • •, k) are given i n (11). (Here, we use a d d i t i v e n o t a t i o n s j u s t f o r convenience). Lemma 4.2.5. I f A_. = 0 f o r a l l j = 1, k except j = j Q , K i s not f r e e a b e l i a n (and so H i s not f r e e ) when j ^ 1 or -1. then Proof : Since , A^ = 0 except j = j Q , i t f o l l o w s from (12) that A. a... = 0 J o J 0 f o r l > 0. Sxnce A. ^ 1 or - 1 , the subgroup < a. : A. a. > of — 1 1 1 1 J o J o J o J o H , i s c l e a r l y not f r e e a b e l i a n . Hence H i s not fr e e a b e l i a n . Lemma 4.2.6. I f A = 0 f o r a l l j = 1, k except j - j Q and A. = 1 or -1, then H = <0> or H i s f r e e a b e l i a n on the generators -*o al» "**> a i _ i according as j c = 1 or j Q 4 1. J o ~ Proof : I t f o l l o w s from (12) that I f j = 1 , then a. J o ' i then a. = 0 f o r a l l l a x , • • •, a . J o This completes the proof. Next, suppose that at l e a s t two Xjs are not equal to zero. Without l o s s of g e n e r a l i t y , we can assume X^ -f 0. Lemma 4.2.7. I f H i s f r e e a b e l i a n , then i t s rank i s at most k - 1. Proof : Suppose that H i s f r e e a b e l i a n w i t h generators f^ , f 2 , * • • > f m 5 • * • I f any Z elements (£ <_ k - 1) i n t h i s s e t of generators are l i n e a r l y dependent, then there i s nothing to prove. Thus, suppose that there are k - 1 elements, say f 1 5 f 2 , f k _ 1 i n t h i s s e t , which are l i n e a r l y independent. Since X^ ^ 0, by a p p l y i n g the r e l a t o r s (12) repeatedly, we see that there e x i s t s some i n t e g e r j > 0 such t h a t e a . j . = 0 (e = 1 or -1, i > 0 ) . J o = 0 f o r a l l i > 1 and so H = <0> . I f j £ 1, i >_ j so that H i s f r e e a b e l i a n on the generators - 58 -(13) m, X f, = m. a. +m. , a. . + • • • + m. ,. ' a. ,, „ 1 1 J0>1 3Q J 0 + l , l J 0 + l J o ^ " 2 ' 1 J o + k ~ 2 X, f, = m. a. + m a + { 1 k - i . j 0 . , k - i j Q J 0 + l , k - l J o + l + m. . i , a. ,, j +k-2,k-l i^+k-2 and X l f k = m j , k a j + m j + l , k a j +1 + + m j +k-2,k aj +k-2 J o J o o J o J o J o where nu ( i = l,-««,k) are some i n t e g e r s > 0 and a l l m^  ^, are i n t e g e r s . Since f j , a r e assumed to be l i n e a r l y independent, the determinant D m. J 0 » l m j 0 + l , l m j Q + k - 2 , l m. . m. , , J o ' k - 1 J o + ^ k - 1 j n + k - 2 , k - i i s d i f f e r e n t from zero. Hence we can express each of the Da. , Da. ,,, •••, Da. ,, „ J o o J o m k i n terms of a l i n e a r combination of f^, • f ^ and so DX1 f ^ can be m k expressed as a l i n e a r combination of f ^ , f^. w i t h DX^ ^ 0 . Therefore f^, f ^ are l i n e a r l y dependent and hence H i s f r e e of rank at most k - 1 . - 59 -Lemma 4.2 .8 . I f appears at l e a s t twice i n R Q, then the group H given by (9) i s not f r e e (and so N i s not f r e e ) . Proof : Suppose that H i s f r e e . Then, by Lemma 4.2.6 and Lemma 4.2.7, i t f o l l o w s that H i s f r e e of rank at most k - 1. Let f ^ , • • •, f ^  (£ < k - 1) be a se t of f r e e generators f o r H ; i . e . (14) H = < f 1 5 f £ > . Note that each f^ ( j = 1, £) i s a word i n a^ ( i = 1, 2, «•*)• Let m be the maximal s u b s c r i p t of a l l the generators a^ that are i n v o l v e d i n f j , f £ . Then H i s f i n i t e l y generated by , •••, a m so that ( 1 5 ) H = Hm-k = < al» am J Po> P m - k > Thus each a^ = ^ ( f j , f £ ) ( i = 1, •••, m) i s a word i n f ^ , f ^ and the r e l a t o r s P Q, P ^ a l l reduce to the empty word upon r e p l a c i n g a^ by U\ ( i - 1,. •• • •, m) s i n c e (15) can be transformed i n t o (14) . Now consider Hm-k+l = K H l ' am+l ; Po> P m-k> Pm-k+l > ' Then H = H so that H can be presented a l s o by m-k+1 r J H = < f f „ , a , ; P . , > , 1' * £ ' m+l ' m-k+l ' - 60 -where P m_k. + 1 = Q(fi» ***> a m - t - i ^ w i t h a m + 1 appearing at l e a s t twice i n Q. Since H i s generated by f j , f ^ , am+l = V ( f l ' fl> ' a word i n f ^ , f ^ . But t h i s i s impossible by Lemma 4.2.2. Hence H i s not f r e e . This completes the proof. S i m i l a r l y , we can prove that : Lemma 4.2.9. I f a^ appears at l e a s t twice i n R Q, then the group II 1 given by (10) i s not f r e e (and so N i s not f r e e ) . Hence, we have proved : C o r o l l a r y 4.2.10. I f the normal subgroup N given by (5) i s f r e e , then each of a± and a^ must appear j u s t once i n R Q, w i t h exponent 1 or -1. Combining Lemma 4.2.1 and C o r o l l a r y 4.2.10, we o b t a i n the f o l l o w i n g : Theorem 4.2.11. Let G, R Q and N be as given by ( 1 ) , (3) and (5). Then N i s f r e e i f and only i f each of a, and a appears j u s t once i n A u R Q w i t h exponent 1 or -1. - 61 -The f o l l o w i n g example i n d i c a t e s that i n case 3, the s u i t a t i o n i s much l e s s d e f i n i t e . Example 4.2.12. Let G be a group presented by ? -1 - 2 G = < a, b ; R = ab^a b > Then a R ( a ) = 0, a R ( b ) = 0. Choose cb : G —> T to be the epimorphism de f i n e d by <fr(a) = 1, +(b) = t . Then G = N x T w i t h K the normal subgroup of G generated by a. Using the above n o t a t i o n s , we see that N = < • • • a a p • • • • • • • P P p . .•> » _j> o* 1' » _j> o ' l ' But P D = R Q = a 0 a 2 * so that the c o n d i t i o n s i n Theorem 4.2.11 are s a t i s f i e d . Hence N i s f r e e . Next, choose <j> : G — > T to be the epimorphism de f i n e d by cb^a) = t , 4» X (b> = 1 . Then G = Nj x a T x<rith N the normal subgroup of G generated by b. Let b^ = aXba 1 ( i , any i n t e g e r ) . Then N 2 has generators b^. Rewrite 2 -1 - 2 R = ab a b i n terms of b^, we have - 62 -R o = b l b o 2 ' Thus the c o n d i t i o n s i n Theorem 4.2.11 are not s a t i s f i e d . Hence ' Nj i s not f r e e . F i n a l l y , we r e t u r n to case 2. That i s , l e t (16) G = < a, b ; R(a,b) > w i t h a_(a) ^  0 and a n ( b ) 4 0. Let Xn = o_(a) and A 9 = cr (b). Then we can w r i t e X^ = ku^ and = k u 2 w i t h and u 2 r e l a t i v e l y prime. Then X 1 y 2 - X 2p^ = 0. Choose and v 2 w i t h v1 the minimum p o s i t i v e i n t e g e r such t h a t (17) vx\iz - \>2u1 = 0. I n f a c t , we can j u s t take v1 = {v^l • ( I f X 2 = k'X^ then = 1, P 2 = k 1 so tha t y 2 - k'pj = 0. I n t h i s case, v 1 = 1). Next, s i n c e and p 2 are r e l a t i v e l y prime, we can choose m^  and m2 w i t h the minimum p o s i t i v e i n t e g e r such t h a t mjVj + m 2p 2 = 1. In t h i s case, the homomorphism <j> : G — > T i s again uniquely defined up to exponent s i g n by Vz - H i <{>(a) = t and «J>(b) = t - 63 -I t i s easy to v e r i f y t h a t the elements V l _ 1 V l ~ 1 £ (18) a, a , b^, ab*", a b (£, any i n t e g e r ) form a S c h r e i e r r e p r e s e n t a t i v e s f o r G mod N. To o b t a i n a p r e s e n t a t i o n of N, we make use of a R e i d e m e i s t e r - S c h r e i e r r e w r i t i n g process and as S c h r e i e r r e p r e s e n t a t i v e s f o r G mod N, we choose (18) ([13],§2.3). We f i n d that N i s generated by the elements a^ defined by ' • £-v (19) a± = ( a ; , b £ ) a ( a j + 1 b £ ) " 1 or ( a j b £ ) a ( b 2 ) ~ 1 where i i s any i n t e g e r and 0 <_ j < v - 1 (or j = Vj — 1 ) , £ any i n t e g e r such t h a t (20) i = j u 2 - £p 2 Note that a. = ( a J ) a ( a ^ + 1 ) 1 = 1 f o r 0 < i < v, - 1. 3V2 ~ 1 Now, we r e w r i t e R(a,b) i n terms of as (21) R(a,b) = R Q ( a A , a x + 1 , a y) w i t h X < ••• < y ( c f . Example 4.2.14). Then N i s generated by a^ and has as d e f i n i n g r e l a t o r s a_.y (0 <_ j < - 1) and P = a : )b £R(a,b)(a jb 5')~ 1 = R o ( a X + i ' a X + l + i ' V i > ( i ' a n y Integer) - 64 -where i , j , I s a t i s f y (20), and so (22) N = <• • • , a _ i , a Q , a 1 , • • •; a ^ a ^ , • • • , a ^ -1)Vl^>?± C i , any i n t e g e r ) > Hence, by the same arguments as f o r case 1, we have : Theorem 4.2.13. Let G, R Q and N be given by (16), (21) and (22). Then N i s f r e e i f and only i f each of a^ and a y appears j u s t once i n R Q w i t h exponent 1 or -1. We c l o s e t h i s s e c t i o n w i t h the f o l l o w i n g example. Example 4.2.14. Let G be the group presented by \ X2 G = < a , b ; R = a b > w i t h X1 ± 0, A 2 f 0. Then G i s of the form N * a T w i t h N f r e e . Withour l o s s of g e n e r a l i t y , we can assume A^ > 0. Let A^ = ku^ and A 2 = k u 2 (k > 0 ) . Then v 1 = and v 2 = u 2 i n (17). We see that N can be presented by (22). Now, we r e w r i t e R = a b i n terms of a^ as f o l l o w s appearances of a v . - l , v - l -v. R Q = (b°)a(ab°)" 1(ab 0)a(a 2b 0) 1 ... (a 1 b°) 1 (a 1 b°)a(b V 1 * /-Vj appearances of a /> ~V2 ~V2 - n " V? o ~V9 1 V1 - 1 " V ? i v -1 -V„ x (b )a(ab ) X(ab ) a ( a 2 b ) ~ X ••• (a b ) (a 1 b )a - 65 -x • • • x V j . appearances of a / A  - ( k - l ) v ? - ( k - i ) v - ( k - l ) v 2 - ( k - l ) v 2 x (b )a(ab ) *(ab ) a ( a 2 b ) ••• v f l - ( k - l . ) v 2 -kv - k v 2 X2 (a b )a(b ) xb x b = a a . . . a . % a a • • • a / • \" • " ° y 2 ( v 1 - l ) y 2 +v 2y 1 p 2+v 2p 1 ( v 1 - i ) y 2 + v 2 y 1 a + 2 v 2 V ! a ( v 1 - i ) p 2 + ( k - i ) v 2 p 1 - k v 2 X 2 (Note that b b = 1 s i n c e X 2 = k v 2 ) . I t i s c l e a r that a l l the s u b s c r i p t s are d i f f e r e n t and hence we can apply Theorem 4..2.13 to .conclude that N i s f r e e . §4.3. Groups w i t h n (n > 2) Generators and One D e f i n i n g R e l a t o r s Let G be a group presented by (1) G = < X j , ^ ; R ( x x , x n ) > where R i s c y c l i c a l l y reduced and i n v o l v e s a l l the generators. Then G = N x a T. In t h i s s e c t i o n , we w i l l o b t a i n c e r t a i n necessary and s u f f i c i e n t c o n d i t i o n s f o r the f a c t o r N to be f r e e corresponding to c e r t a i n n a t u r a l choices of the epimorphism cb : G — > T. - 66 -I f a l l a (x.) ^ 0, then the s u i t a t i o n i s b a s i c a l l y s i m i l a r to R X case 2 of §4.2, but the number of cases to be t r e a t e d are so numerous th a t we w i l l not d e t a i l them here. Rather, we s h a l l take up the case where, say, o ^ (x^) = 0 . In t h i s case, we can choose (j> : G — > T by d e f i n i n g • (X j ) = 0 ( j = 1,2,...,n-1), <J)(xn) = t . (Note that there are many other choices of <f>). Thus we may assume that the f a c t o r T, i n N x a T, i s j u s t the i n f i n i t e c y c l i c group generated by 3 ^ i n G and N i s nothing but the normal subgroup of G generated by X j , x 2 , •••, x . To o b t a i n a p r e s e n t a t i o n of N, we make use of a Reidemexster-S c h r e i e r r e w r i t i n g process and as S c h r e i e r r e p r e s e n t a t i v e s f o r G mod N, we choose x^, where i runs over a l l i n t e g e r s ( [ 1 3 ] , §2.3). We f i n d t h a t N i s generated by the elements x^ defined by i - i x. . = x x.x i , J n j n where j = 1, n-1 and i any i n t e g e r . Now, we r e w r i t e ' R ( x j , • • • , x ) i n terms of x. . as f o l l o w s : Every symbol x? ( j = l,..«,n-l; e = 1 or -1) i n R ( x j , •••> x R ) i s repl a c e d by x E ^ where s i s the sum of the exponents of the x^-symbols preceding the p a r t i c u l a r x E i n R ( X j , x n ) • Thus R(x^, x n ) can be expressed i n terms of x^ as : - 67 -(2) R(K1,"-,xn) = R„(x.. ,••• ,x ,x, .•••,x ,•••,x. ,•••,x" ) w i t h X^ <_•••_< ( j = l,«>»,n-l). Then N i s generated by x^ ^ (1 runs over a l l i n t e g e r s and j runs over 1, n-1) and has a s e t of d e f i n i n g r e l a t o r s P i " *>o>C ( i any i n t e g e r ) and so N = < x. ( i runs over a l l i n t e g e r s and j runs over l,«-«,n-l) ; . . . P P p . . . > r _ 1 ' r O > r l > By adapting the arguments f o r the case of two generators, we can prove : Theorem 4.3.1. The normal subgroup N i s f r e e i f and only i f one of the x. , x, , • • •, x, and one of the x , x , • • •, x . ^ . i . * 2 ' 2 n - l ' n _ 1 y l ' 1 y 2 ' 2 y n - l ' n _ 1 each appears j u s t once i n R Q given by ( 2 ) , w i t h exponent 1 or -1. C o r o l l a r y 4.3.2. ([8]) Let G be the fundamental group of a c l o s e d o r i e n t a b l e s u r f a c e . Then G i s of the form N * a T w i t h N f r e e . - 68 -Proof : R e c a l l that G can be presented by G = < X j , - . . , ^ , y i , . . . , y n ; R = x ^ x ^ y ^ • • • x ^ x ^ y ^ >. Then a (y ) = 0, so that as discussed above, we can take T = <y > and R n n N the normal subgroup of G generated by x 1 , ' ' ' , x n , yi>'**>y n_2 • ^ e generators f o r N are given by x ± = y ^ y " 1 (i» any i n t e g e r ; j = 1, n) yi,£ = 7nySLynX ^±> 3 1 1 7 i n t e S e r > 1 = 1> n _ 1 ) By r e w r i t i n g R i n terms of x. . and y , we get l» 3 i» *> R o = ^ . l y o . ^ o j i y o l r - ^ o . n - i y o . n - ^ o l n - i y o l n - ^ o . n ^ n Since X q n and x^ ^ each appears j u s t once i n R Q w i t h exponent 1 or -1 , the c o n d i t i o n s i n Theorem 4.3.1 are s a t i s f i e d . Hence N i s f r e e . C o r o l l a r y 4.3.3. Let G be the group presented by G = < y,> y o » • • • > y « ; R = y s,y s 9 - - « s y s . (m 4 l ) > j l> iz* ,J,n» •'n-l H n - l 2 m-i^n-l ms v T ' m appearances of y where S , S , S are words i n y,, y n (some of them may be J. m—l l n-2 - 69 -t r i v i a l ) , and S m is a word in y 1 } *'*> y n 2> Y n with the condition that a (y ) = km for some integer k ^  0. Then G i s of the form N x a T, m with N free. Proof Let x. = y. (1 = 1, n-2, n) and. x , = y y Then, j j n-1 •'n-rn by applying Tietze transformations repeatedly, we see that G can be presented by ~ kS! ... S' x x" kS' > G = < x,, x 9, x n ; R' = x x .Si •••  1» 2» » n n-1 n 1 m-l n-1 n 1 m whe re S i, S1 , are words i n x, , • • •, x „ and S' is a word i n 1' ' m-l 1> ' n-2 m x, , •••• x . x.^  with a_.(x ) = km. Hence 0„.(x.) = 0. Let N be x> - n-z- a am n- R- I J / •m the normal subgroup generated by X j , •••, x^ . Then the generators for N are i - i X. . = X x.x i>J n j n ( i , any integer ; j = 1, n-1). By rewriting R' in terms of x. , , we get R1 = R_ = xrt _ ,S'x , Six . ... S1 x , >, S' o o,n-l l _k }n-i 2 -2k,n-l m-i -(m-i)k,n-i m where Sj, S 2, S m_ 1 > Sffl are words in x ^ ^ (i» anY integer ; 1=1, n-2). Since x„ _ , and x , , N 1 , each appears in R. ' o,n-i -(m-l)k,n-l r ° just once with exponent 1, the conditions in Theorem 4.3.1 are satisfied. Hence N i s free. - 70 -C o r o l l a r y 4.3.4. ([8]) Let G be the fundamental group of a c l o s e d n o n o r i e n t a b l e s u r f a c e . Then G i s of the form N * a T w i t h N f r e e . Proof : R e c a l l that G can be presented by G = < y 2 , yn ; y 2 y 2 •.. y 2 > Then apply C o r o l l a r y 4.3.3. BIBLIOGRAPHY 1. S. B a l c e r z y k , The G l o b a l Dimension of the Group Rings of A b e l i a n Groups, Fund. Math. LV, 293-301 (1964). 2. H. Bass, A l g e b r a i c K-Theory, W.A. Benjamin, Inc., New York (1968). 3. H. Bass, P r o j e c t i v e Modules over Free Groups are Free, J . of Algebra 1, 367-373 (1964). 4. H. Bass, A. H e l l e r and R.G. Swan, The Whitehead Group of a Polynomial Exte n s i o n , P u b l . I.H.E.S. No. 22, 61-79 (1964). 5. G. Baumslag and T. T a y l o r , The Centre of Groups w i t h One D e f i n i n g R e l a t o r , Math. Annalen 175, 315-319 (1968). 6. N. Bourbaki, Algebre Commutative, Chapters 1 and 2 (Fasc. 27), P a r i : Hermann and Cie (1961). 7. S.U. Chase. D i r e c t Products of Modules, Trans. Amer, Math. Soc. 97. 457-473 (I960). 8. F.T. F a r r e l l and W.C. Hsiang, A Formula f o r K j R ^ T ] , Proc. of Symposia i n Pure Math. 17, 192-219 (1970). 9. F.T. F a r r e l l , The O b s t r u c t i o n to F i b e r i n g a M a n i f o l d over a C i r c l e , Indiana U n i v e r s i t y Math. J . 21, No. 4, 315-346 (1971). 10. S. Gersten, Whitehead Groups of Free A s s o c i a t i v e Algebras, B u l l . Amer. Math. Soc. 71, 157-159 (1965). 11. S. Gersten, On Class Groups of Free Products, Ann. of Math. 85, 392-398 (1968). 12. G. Higman, The Units of Group Rings, Proc. London Math. Soc. 46, 231-248 (1940). 13. A. K a r r a s s , W. Magnus and D. S o l i t a r , C o m b i n a t o r i a l Group Theory, I n t e r s c i e n c e , New York (1966). 14. J . M i l n o r , Whitehead T o r s i o n , B u l l . Amer. Math. Soc. 72, 358-426 (1966). - i i -is. E.S. Rapaport, On the Defining Relations of a Free Product, Pacific J. Math. 14, 1389-1393 (1964). 16. J. Soublin, Anneaux Coherents, CR. Acad. Sc., Paris, t. 267 Ser. A, 183-186 (1968). 17. J. Soublin, Un Anneau Coherent dont l'anneau des Polynomes n'est pas Coherent, CR. Acad. Sc., Paris, t. 267 Ser. A, 241-243 (1968). 18. J. Stallings, .Whitehead Torsion of Free Products, Ann. of Math. 82, 354-363 (1965). 19. F. Waldhausen, Whitehead Groups of Generalized Free Products, Preliminary Report. 

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