On \(K_1\) of an exact category (Q1386677)

Let \({\mathcal M}\) be an exact category, \(A\overset \alpha \rightarrowtail X\) and \(B\overset \beta \rightarrowtail Y\) two admissible monomorphisms, and \(\Theta:A\oplus X/A\oplus Y\to B\oplus Y/B\oplus X\) an isomorphism in \({\mathcal M}\). In 1994 the author constructed an element \(G(\alpha, \beta, \Theta)\) of \(K_1({\mathcal M})\), where \(G(\alpha, \beta, \Theta)\) is the homotopy class in \(\pi_1(| G{\mathcal M}|)\simeq K_1({\mathcal M})\) represented by the loop [see \textit{C. Sherman}, J. Algebra 163, No. 2, 568-582 (1994; Zbl 0798.19001)]. In this paper the author proves that every element of \(K_1({\mathcal M})\) is of this form. In addition, several geometric representations of \(G(\alpha, \beta, \Theta)\) are constructed, and then he uses these results to determine the description of elements arising from Bass's group \(K_1^{\text{det}}({\mathcal M})\).