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

Abstract

top
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 B Γ 1 ← B G ← B Γ 2 . Denote by ξ the diagram I p ← H 1 → X = H , where p is the natural map onto the unit interval. We show that the N i l ∼ groups which occur in Waldhausen’s description of K * ( ℤ Γ ) coincide with the continuously controlled groups * c c ( ξ ) , defined by Anderson and Munkholm. This also allows us to identify the continuously controlled groups * c c ( 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.

How to cite

top

Munkholm, 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. [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. [2] D. R. Anderson and H. J. Munkholm, Geometric modules and algebraic K-homology theory, K-Theory 3 (1990), 561-602. Zbl0772.57036
  3. [3] D. R. Anderson and H. J. Munkholm, Geometric modules and Quinn homology theory, ibid. 7 (1993), 443-475. Zbl0907.55005
  4. [4] D. R. Anderson and H. J. Munkholm, Continuously controlled K-theory with variable coefficients, J. Pure Appl. Algebra, to appear. Zbl0942.19001
  5. [5] M. M. Cohen, A Course in Simple-Homotopy Theory, Grad. Texts in Math. 10, Springer, New York, 1973. 
  6. [6] F. Quinn, Geometric algebra, in: Lecture Notes in Math. 1126, Springer, Berlin, 1985, 182-198. 
  7. [7] A. A. Ranicki and M. Yamasaki, Controlled K-theory, Topology Appl. 61 (1995), 1-59. 
  8. [8] F. Waldhausen, Algebraic K-theory of generalized free products. Parts 1 and 2, Ann. of Math. (2) 108 (1978), 135-256. Zbl0407.18009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.