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## Fundamenta Mathematicae

1999 | 161 | 1-2 | 217-224
Tytuł artykułu

### Waldhausen’s Nil groups and continuously controlled K-theory

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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.
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Tom
Numer
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217-224
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Daty
wydano
1999
otrzymano
1997-12-17
poprawiono
1999-03-13
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autor
autor
Bibliografia
• [1] D. R. Anderson, F. X. Connolly and H. J. Munkholm, A comparison of continuously controlled and controlled K-theory, Topology Appl. 71 (1996), 9-46.
• [2] D. R. Anderson and H. J. Munkholm, Geometric modules and algebraic K-homology theory, K-Theory 3 (1990), 561-602.
• [3] D. R. Anderson and H. J. Munkholm, Geometric modules and Quinn homology theory, ibid. 7 (1993), 443-475.
• [4] D. R. Anderson and H. J. Munkholm, Continuously controlled K-theory with variable coefficients, J. Pure Appl. Algebra, to appear.
• [5] M. M. Cohen, A Course in Simple-Homotopy Theory, Grad. Texts in Math. 10, Springer, New York, 1973.
• [6] F. Quinn, Geometric algebra, in: Lecture Notes in Math. 1126, Springer, Berlin, 1985, 182-198.
• [7] A. A. Ranicki and M. Yamasaki, Controlled K-theory, Topology Appl. 61 (1995), 1-59.
• [8] F. Waldhausen, Algebraic K-theory of generalized free products. Parts 1 and 2, Ann. of Math. (2) 108 (1978), 135-256.
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