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On K 1 of a Waldhausen categoryFernando Muro and Andrew Tonks IntroductionA general notion of K-theory, for a category W with cofibrations and weak equivalences,was defined by Waldhausen [10] as the homotopy groups of the loop space of a certainsimplicial category wS.W,K n W D n jwS.Wj Š nC1 jwS.Wj ; n 0:Waldhausen K-theory generalizes the K-theory of an exact category E, defined as thehomotopy groups of jQEj, where QE is a category defined by Quillen.Gillet and Grayson defined in [2] a simplicial set G.E which is a model for jQEj.This allows one to compute K 1 E as a fundamental group, K 1 E D 1 jG.Ej. Usingthe standard techniques for computing 1 Gillet and Grayson produced algebraic representativesfor arbitrary elements in K 1 E. These representatives were simplified bySherman [8], [9] and further simplified by Nenashev [5]. Nenashev’s representativesare pairs of short exact sequences on the same objects,A B C:Such a pair gives a loop in jG.Ej which corresponds to a 2-sphere in jwS.Ej, obtainedby pasting the 2-simplices associated to each short exact sequence along their commonboundary. BA C Figure 1. Nenashev’s representative of an element in K 1 E.Nenashev showed in [6] certain relations which are satisfied by these pairs of shortexact sequences, and he later proved in [7] that pairs of short exact sequences togetherwith these relations yield a presentation of K 1 E. The first author was partially supported by the MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the MEC postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research cantract; and thesecond author by the MEC-FEDER grant MTM2004-03629.