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

Hans Munkholm; Stratos Prassidis

Fundamenta Mathematicae (1999)

- Volume: 161, Issue: 1-2, page 217-224
- ISSN: 0016-2736

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topMunkholm, Hans, and Prassidis, Stratos. "Waldhausen’s Nil groups and continuously controlled K-theory." Fundamenta Mathematicae 161.1-2 (1999): 217-224. <http://eudml.org/doc/212401>.

@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 -

## References

top- [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. Zbl0853.19002
- [2] D. R. Anderson and H. J. Munkholm, Geometric modules and algebraic K-homology theory, K-Theory 3 (1990), 561-602. Zbl0772.57036
- [3] D. R. Anderson and H. J. Munkholm, Geometric modules and Quinn homology theory, ibid. 7 (1993), 443-475. Zbl0907.55005
- [4] D. R. Anderson and H. J. Munkholm, Continuously controlled K-theory with variable coefficients, J. Pure Appl. Algebra, to appear. Zbl0942.19001
- [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. Zbl0407.18009

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