Waldhausen’s Nil groups and continuously controlled K-theory
Volume 161 / 1999
Abstract
Let be the pushout of two groups Γ_i, i = 1,2, over a common subgroup G, and H be the double mapping cylinder of the corresponding diagram of classifying spaces BΓ_1 ← BG ← BΓ_2. Denote by ξ the diagram I {p \over ←} H {1 \over →} X = H, where p is the natural map onto the unit interval. We show that the Nil^∼ groups which occur in Waldhausen's description of K_*(ℤΓ) coincide with the continuously controlled groups \widetildeK^{cc}_*(ξ), defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups \widetildeK^{cc}_*(ξ^+) which are known to form a homology theory in the variable ξ, with the "homology part" in Waldhausen's description of K_{*-1}(ℤ Γ). A similar result is also obtained for HNN extensions.