A+ CATEGORY SCIENTIFIC UNIT

Waldhausen’s Nil groups and continuously controlled K-theory

Volume 161 / 1999

Hans J. Munkholm, Stratos Prassidis Fundamenta Mathematicae 161 (1999), 217-224 DOI: 10.4064/fm-161-1-2-217-224

Abstract

Let $Γ = Γ_1 *_G Γ_2$ 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.

Authors

  • Hans J. Munkholm
  • Stratos Prassidis

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