process

On K 1 of a Waldhausen category 113The equation 2ŒA D 0 follows from (S2) and Lemma 7.2 by using the secondformula for ŒA above and the fact that 2 A;A D 1.We have already shown that is a well-defined homomorphism. The composite of and (6.1) coincides with the action of the stable Hopf map by the main result of [4].Appendix. Free stable quadratic modules and presentationsFree stable quadratic modules, and also stable quadratic modules defined by generatorsand relations, can be characterized up to isomorphism by obvious universal properties.In this appendix we give more explicit constructions of these notions.Let squad be the category of stable quadratic modules and letU W squad! Set Setbe the forgetful functor, U.C / D .C 0 ;C 1 /. The functor U has a left adjoint F , and astable quadratic module F.E 0 ;E 1 / is called a free stable quadratic module on the setsE 0 and E 1 . In order to give an explicit description of F.E 0 ;E 1 / we fix some notation.Given a set E we denote by hEi the free group with basis E, and by hEi ab the freeabelian group with basis E. The free group of nilpotency class 2 with basis E, denotedby hEi nil , is the quotient of hEi by triple commutators. Given a pair of sets E 0 andE 1 , we write E 0 [ @E 1 for the set whose elements are the symbols e 0 and @e 1 for eache 0 2 E 0 , e 1 2 E 1 .To define F.E 0 ;E 1 /, consider the groupsF.E 0 ;E 1 / 0 DhE 0 [ @E 1 i nil ;F.E 0 ;E 1 / 1 D .hE 0 i ab ˝ Z=2/ Ker ı:Here ı W F.E 0 ;E 1 / 0 !hE 0 i ab is the homomorphism given by ıe 0 D e 0 and ı@e 1 D 0.In the notation of the proof of lemma 5.1 there are isomorphismsKer ı Š^2hE0 i ab hE 0 E 1 i ab hE 1 i nil ;.hE 0 i ab ˝ Z=2/ Ker ı Š Ő 2 hE 0 i ab hE 0 E 1 i ab hE 1 i nil ;and intuitively we think of F.E 0 ;E 1 / 1 as a group generated by symbols he 0 ;e0 0 i,he 0 ;@e 1 i and e 1 . The symbol h@e 1 ;@e1 0 i is unnecessary since it will be given by thecommutator Œe1 0 ;e 1.We define structure homomorphisms on F.E 0 ;E 1 / as follows. The boundary@W F.E 0 ;E 1 / 1 ! F.E 0 ;E 1 / 0is the projection onto Ker ı followed by the inclusion of the kernel. The bracketh; iW F.E 0 ;E 1 / ab0 ˝ F.E 0;E 1 / ab0 ! F.E 0 ;E 1 / 1