Waldhausen’s Nil groups and continuously controlled K-theory

@article{Munkholm1999,
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 $^\{cc\}_*(ξ)$, defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups $^\{cc\}_*(ξ^+)$ 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.},
author = {Munkholm, Hans, Prassidis, Stratos},
journal = {Fundamenta Mathematicae},
keywords = {Nil-groups; continuously controlled -groups; spectral sequence; Atiyah-Hirzebruch sequence},
language = {eng},
number = {1-2},
pages = {217-224},
title = {Waldhausen’s Nil groups and continuously controlled K-theory},
url = {http://eudml.org/doc/212401},
volume = {161},
year = {1999},
}

TY - JOUR
AU - Munkholm, Hans
AU - Prassidis, Stratos
TI - Waldhausen’s Nil groups and continuously controlled K-theory
JO - Fundamenta Mathematicae
PY - 1999
VL - 161
IS - 1-2
SP - 217
EP - 224
AB - 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 $^{cc}_*(ξ)$, defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups $^{cc}_*(ξ^+)$ 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.
LA - eng
KW - Nil-groups; continuously controlled -groups; spectral sequence; Atiyah-Hirzebruch sequence
UR - http://eudml.org/doc/212401
ER -