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92 F. Muro and A. TonksIn this paper we produce representatives for all elements in K 1 of an arbitraryWaldhausen category W which are as close as possible to Nenashev’s representativesfor exact categories. They are given by pairs of cofiber sequences where the cofibrationshave the same source and target. The cofibers, however, may be non-isomorphic, butthey are weakly equivalent via a length 2 zig-zag.A C 1 B C: C 2This diagram is called a pair of weak cofiber sequences. It corresponds to a 2-sphere injwS.Wj obtained by pasting the 2-simplices associated to the cofiber sequences alongthe common part of the boundary. The two edges which remain free after this operationare filled with a disk made of two pieces corresponding to the weak equivalences. BA C 1 C C 2Figure 2. Our representative of an element in K 1 W.In addition we show in Section 3 that pairs of weak cofiber sequences satisfy acertain generalization of Nenashev’s relations.Let K1 wcs W be the group generated by the pairs of weak cofiber sequences in Wmodulo the relations in Section 3. The results of this paper show in particular that thereis a natural surjectionK1 wcs W K 1 Wwhich, by [7], is an isomorphism in case W D E is an exact category. See Section 6for a comparison of K1 wcs E with Nenashev’s presentation. We do not know whetherthis natural surjection is an isomorphism for an arbitrary Waldhausen category W,butwe prove here further evidence which supports the conjecture: the homomorphism W K 0 W ˝ Z=2 ! K 1 Winduced by the action of the Hopf map 2 s in the stable homotopy groups of spheresfactors asK 0 W ˝ Z=2 ! K1 wcs W K 1 W:We finish this introduction with some comments about the method of proof. We donot rely on any previous similar result because we do not know of any generalization of