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114 F. Muro and A. Tonksis given by the product of the following two homomorphisms,c 0 W F.E 0 ;E 1 / ab0 ˝ F.E 0;E 1 / ab0 ! hE 0 i ab ˝ Z=2;defined on the generators x;y 2 E 0 [ @E 1 by c 0 .x; y/ D x ˝ 1 if x D y 2 E 0 andc 0 .x; y/ D 0 otherwise, andc W F.E 0 ;E 1 / ab0 ˝ F.E 0;E 1 / ab0 ! Ker ıinduced by the commutator bracket, c.a;b/ D Œb; a.It is now straightforward to define explicitly the stable quadratic module C presentedby generators E i and relations R i F.E 0 ;E 1 / i in degrees i D 0; 1, byC 0 D F.E 0 ;E 1 / 0 =N 0 ;C 1 D F.E 0 ;E 1 / 1 =N 1 :Here N 0 F.E 0 ;E 1 / 0 is the normal subgroup generated by the elements of R 0 and@R 1 , and N 1 F.E 0 ;E 1 / 1 is the normal subgroup generated by the elements ofR 1 and hF.E 0 ;E 1 / 0 ;N 0 i. The boundary and bracket on F.E 0 ;E 1 / induce a stablequadratic module structure on C which satisfies the following universal property:given a stable quadratic module C 0 , any pair of functions E i ! Ci 0 .i D 0; 1/ suchthat the induced morphism F.E 0 ;E 1 / ! C 0 annihilates R 0 and R 1 induces a uniquemorphism C ! C 0 .In [1] Baues considers the totally free stable quadratic module C with basis givenby a function g W E 1 !hE 0 i nil . In the language of this paper C is the stable quadraticmodule with generators E i in degree i D 0; 1 and degree 0 relations @.e 1 / D g.e 1 / forall e 1 2 E 1 .References[1] H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, de Gruyter Exp.Math. 2, Walter de Gruyter, Berlin 1991.[2] H. Gillet, D. R. Grayson, The loop space of the Q-construction, Illinois J. Math. 31 (1987),574–597.[3] T. Gunnarsson, R. Schwänzl, R. M. Vogt, F. Waldhausen, An un-delooped version of algebraicK-theory, J. Pure Appl. Algebra 79 (1992), 255–270.[4] F. Muro, A. Tonks, The 1-type of a Waldhausen K-theory spectrum, Adv. Math. 216 (2007),178–211, .[5] A. Nenashev, Double short exact sequences produce all elements of Quillen’s K 1 ,inAlgebraicK-theory (Poznań, 1995), Contemp. Math. 199, Amer. Math. Soc., Providence, R.I.,1996, 151–160.[6] A. Nenashev, Double short exact sequences and K 1 of an exact category, K-Theory 14(1998), 23–41.[7] A. Nenashev, K 1 by generators and relations, J. Pure Appl. Algebra 131 (1998), 195–212.