process

108 F. Muro and A. TonksC Œ S1 ;:::;S nW L 2 C E 2! L1 C E 1 C ŒA L 2 C E 2 L 2 C D C ŒL C D ! L 2 C D(R7) D ŒL C D ! L 1 C D ŒA L 1 C E 1 L 1 C DC ŒA L 1 C E 1 L 2 C D C ŒL C D ! L 2 C Drenaming D ŒC ! C 1 ŒA B C 1 C ŒA B C 2 C ŒC ! C 2 D ŒC ! C 1 ŒA ŒA B C 1 C ŒA B C 2 C ŒC ! C 2 ŒAD ŒA B C 1CC ŒA B C2C mod h; i;i.e. there is y 2 D C 1C in the image of h; i such that8C ˆ=C y:>;Now assume that @.x/ D 0. Since the pair of weak cofiber sequences is also inthe kernel of @ we have @.y/ D 0. In order to give the next step we need a technicallemma.Lemma 5.1. Let C be a stable quadratic module such that C 0 is a free group ofnilpotency class 2. Then any element y 2 Ker @ \ Imageh; iis of the form y Dha; aifor some a 2 C 0 .Proof. For any abelian group A let y˝2Abe the quotient of the tensor square A ˝ A bythe relations a ˝ b C b ˝ a D 0, a; b 2 A, and let ^2A be the quotient of A ˝ A bythe relations a ˝ a D 0, a 2 A, which is also a quotient of y˝2A.The projection ofa ˝ b 2 A ˝ A to y˝2Aand ^2A is denoted by a y˝b and a ^ b, respectively.There is a commutative diagram of group homomorphismsC0 ab ˝ C 0ab 0 ˝ Z=2 N y˝2C ab q^2C 00abh;i c 1c 0 @C 1 C 0C abwhere N.a ˝ 1/ D a y˝ a, the factorization c 1 of h; i is given by c 1 .a y˝ b/ Dha; bi,which is well defined by Definition 1.1 (3), and c 0 .a ^ b/ D Œb; a is well known to beinjective in the case C 0 is free of nilpotency class 2.