to give a tentative definition of higher algebraic $K$-theory. Later Anderson, Karoubi and Wagoner proved that this definition is equivalent to Quillen’s definition of $K$-theory in [@AKW:HAK:73; @AKW:HAK:77; @Wag:EAK:77].

The complexes Wagoner defined are closely related to the building associated to the general linear group. Unlike the building, the Wagoner complex is not homotopy equivalent to a bouquet of spheres. Instead it has interesting homotopy and homology groups. Wagoner complexes have already been generalised to reductive $F$-groups by Petzold in [@Pet:HDW:04]. There, rational homology groups of Wagoner complexes are used to construct rational representations of reductive groups.

We define Wagoner complexes in the context of groups of Kac-Moody type (or groups with root data), which we believe to be the appropriate level of generality. All definitions are very natural and the connection to the theory of buildings is clearly visible. Important examples of such groups are semi-simple Lie groups and algebraic groups as well as Kac-Moody groups.

We exhibit interesting substructures in Wagoner complexes analogous to apartments in buildings and present some general properties of Wagoner complexes. The main result is the following characterisation of low-dimensional homotopy groups of Wagoner complexes in the case where the associated root datum is 