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On the exponent of the cokernel of the forget-control map on K₀-groups

100%

Connolly F., Prassidis S.

Fundamenta Mathematicae

2002

tom 172

nr 3

201-216

For groups that satisfy the Isomorphism Conjecture in lower K-theory, we show that the cokernel of the forget-control K₀-groups is composed by the NK₀-groups of the finite subgroups. Using this information, we can calculate the exponent of each element in the cokernel in terms of the torsion of the group.

Waldhausen’s Nil groups and continuously controlled K-theory

100%

Munkholm H. J., Prassidis S.

Fundamenta Mathematicae

1999

tom 161

nr 1-2

217-224

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.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Prassidis S.

1 Connolly F.

1 Munkholm H. J.

1 2002

1 1999

Wyniki wyszukiwania

On the exponent of the cokernel of the forget-control map on K₀-groups

Waldhausen’s Nil groups and continuously controlled K-theory