process

On K 1 of a Waldhausen category 992 Pairs of weak cofiber sequences and K 1Let C be a Waldhausen category. A pair of weak cofiber sequences is a diagram givenby two weak cofiber sequences with the first, second, and fourth objects in common:A C 1 B C C 2 . We associate to any such pair the following element89r C ˆ= B1 w Cj ˆ: 2 2D ŒA j 1 B r 1 Cr 2 1 CC ŒA j 2 B r 2 C 2 Cw 1 w 2 >;C 2D ŒA j 2 B r 2 C 2 C ŒA j 1 B r 1 C 1 C;w 2 w 1(2.1)which lies in K 1 C D Ker @, see diagram (A) in the introduction. For the secondequality in (2.1) we use that Ker @ is central, see Remark 1.2, so we can permute theterms cyclically.The following theorem is one of the main results of this paper.Theorem 2.1. Any element in K 1 C is represented by a pair of weak cofiber sequences.This theorem will be proved later in this paper. We first give a set of useful relationsbetween pairs of weak cofiber sequences and develop a sum-normalized version of themodel D C.3 Relations between pairs of weak cofiber sequencesSuppose we have six pairs of weak cofiber sequences in a Waldhausen category CA 0A 0A 001 w1Ajr A 1A 1 A A00 ; j A 2r A 2A 002w A 2C1 0 w10 j r 0 10 1 B 0 Cj20 0 ; r 0 2C 0 2w 0 2B 0j B 1j B 2B1 00 w1Br1B B B00 ; r B 2B 002w B 2r 1C 1 w 1A j 1 B C;j 2r 2 wC22C 0j C 1j C 2LC1 00 w1C r1C C C 00 ; r C 2LC200OC 00w C 21 w 100r jA 00 00 1 001 B 00 Cj200 00 ; r 002OC 002w 002