## Splitting with Continuous Control in Algebraic K-theory (2002)

Citations: | 10 - 5 self |

### BibTeX

@TECHREPORT{Rosenthal02splittingwith,

author = {David Rosenthal},

title = {Splitting with Continuous Control in Algebraic K-theory},

institution = {},

year = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. In this work, the continuously controlled assembly map in algebraic K-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups Γ that satisfy certain geometric conditions. The group Γ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K0(kΓ) is proved to be isomorphic to the colimit of K0(kH) over the finite subgroups H of Γ, when Γ is a virtually polycyclic group and k is a field of characteristic zero. 1.

### Citations

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(Show Context)
Citation Context ...le compact metrizable space by a finite group is trivial. The first step toward proving this is Theorem 5.1 below, which is a previously known special case of Oliver’s result. A proof can be found in =-=[3]-=-. Theorem 5.1. [3, Theorem III.7.12] Let G be a finite group acting on a compact Hausdorff space X. If the reduced Čech cohomology of X is trivial, then the reduced Čech cohomology of X/G is also triv... |

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Citation Context ...actible if H is finite, and that it is empty otherwise. This is true since p(H) is finite if and only if H is finite. □ All virtually polycyclic groups contain a torsion-free subgroup of finite index =-=[19]-=-. Therefore if H is a finite normal subgroup of a virtually polycyclic group Γ, then it must be contained in a maximal finite normal subgroup, N. This implies that Γ ′ = Γ/N is a virtually polycyclic ... |

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(Show Context)
Citation Context ...duction map ⊕ G0(RH) → G0(RΓ) H∈f is a surjection. If R is a regular ring then G0(R) ∼ = K0(R). This is the case for kΓ, when k is a field of characteristic zero and Γ is a virtually polycyclic group =-=[2]-=-. We have the following proposition as a special case of Moody’s result. Proposition 9.1. Let Γ be a virtually polycyclic group and k be a field of characteristic zero. Then the induction map is a sur... |

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(Show Context)
Citation Context ... y such that g ∈ Γ and gK ∩ V ̸= ∅ implies gK ⊂ U. Then H Γ ∗ (E; K−∞ (RΓx)) → K∗(RΓ) is split injective for every ring R. If Γ is torsion-free, then EΓ(f) = EΓ, and Theorem 6.1 is [8, Theorem A]. In =-=[10]-=-, Davis-Lück provide a different formulation of the assembly map. However, in recent work of Hambleton-Pedersen [14], it is shown that the Davis-Lück version is homotopy equivalent to the continuously... |

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(Show Context)
Citation Context ...fies certain geometric conditions. There are many ways to build assembly maps. The following theorem is proved in this paper using the continuously controlled model developed by Carlsson and Pedersen =-=[8]-=-. Theorem (Theorem 6.1). Let Γ be a discrete group, and let E = EΓ(f) be the universal space for Γ-actions with finite isotropy, where f denotes the family of finite subgroups of Γ. Assume that E is a... |

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Citation Context ...r whose value on S 0 is ΣK −∞ (R). Furthermore, if X is a finite CWcomplex then ΩK −∞ (B(CX, X; R)) is weakly homotopy equivalent to X ∧K −∞ (R). Clever use of Theorem 3.2 yields the following result =-=[1, 14]-=-. Theorem 3.9. The functor K −∞ ( B(− × (0, 1], − × 1; R) −×1) Γ , from the category of Γ-spaces to the category of spectra, is Γ-homotopy invariant and Γ-excisive. Theorem 3.9 implies that π∗(K −∞ (B... |

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(Show Context)
Citation Context ...ing R. If Γ is torsion-free, then EΓ(f) = EΓ, and Theorem 6.1 is [8, Theorem A]. In [10], Davis-Lück provide a different formulation of the assembly map. However, in recent work of Hambleton-Pedersen =-=[14]-=-, it is shown that the Davis-Lück version is homotopy equivalent to the continuously controlled version. The continuously controlled model of the assembly map that is used in [14] is actually differen... |

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Citation Context ...Schochet axioms. Given any generalized homology theory, there is a unique Steenrod homology extension. Existence of such extensions was proved by Kahn, Kaminker, and Schochet and Edwards and Hastings =-=[16, 11]-=-. Uniqueness was proved by Milnor [20]. Definition 3.6. A functor k from the category of compact metrizable spaces to the category of spectra is called a reduced Steenrod functor if it satisfies the f... |

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(Show Context)
Citation Context ...ame as the one on the product in Lemma 4.2. The projection maps induce a map K −∞( ∏ B((0, 1], 1; R) 1) → ∏ K −∞ ( B((0, 1], 1; R) 1) Γ/H Γ/H that is Γ-equivariant, and a weak homotopy equivalence by =-=[7]-=-. Consider the following commutative diagram: K−∞ ( ∏ ) Γ Γ/H B((0, 1], 1; R)1 �� K−∞ ( ∏ Γ/H B((0, 1], 1; R)1 a b ) hfΓ d � �� ( ∏ Γ/H K−∞ ( B((0, 1], 1; R) 1)) Γ c �� ( � ∏ Γ/H K−∞ ( B((0, 1], 1; R)... |

7 |
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(Show Context)
Citation Context ...ology theory, there is a unique Steenrod homology extension. Existence of such extensions was proved by Kahn, Kaminker, and Schochet and Edwards and Hastings [16, 11]. Uniqueness was proved by Milnor =-=[20]-=-. Definition 3.6. A functor k from the category of compact metrizable spaces to the category of spectra is called a reduced Steenrod functor if it satisfies the following conditions. (i) The spectrum ... |

7 |
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(Show Context)
Citation Context ... of finite cohomological dimension with finitely many orbit types or compact Hausdorff, then X/G has trivial reduced Čech cohomology if X does [9]. This latter conjecture was proved by Oliver in 1976 =-=[22]-=-. In Section 7, we will want to know that the reduced Steenrod homology (from Theorem 3.8) of the quotient of a contractible compact metrizable space by a finite group is trivial. The first step towar... |

6 |
On the Karoubi filtration of a category, K-theory 12
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(Show Context)
Citation Context ... by restricting to isomorphisms. Then K −∞ is a functor from the category of small additive categories to the category of spectra. Cárdenas and Pedersen give a detailed description of this functor in =-=[5]-=-. Karoubi filtrations become the main tool in the theory of continuously controlled algebra via Theorem 3.2 below. A proof of it can also be found in [5]. Theorem 3.2. [8, Theorem 1.28] The sequence i... |

6 |
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(Show Context)
Citation Context ...f φ. This completes the proof of the first part of this lemma. The second part is proved precisely the same way, replacing cones with suspensions. □ 3. Homology and Continuously Controlled Algebra In =-=[17]-=-, Karoubi introduced the notion of an A-filtered additive category U. Definition 3.1. Let A be a full subcategory of an additive category U. Denote the objects of A by the letters A through F and the ... |

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4 |
Retraction properties of the orbit space of a compact topological transformation group
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(Show Context)
Citation Context ...Conjecture and the Vanishing of Steenrod Homology In 1960 Conner conjectured that the orbit space of any action of a compact Lie group on Euclidean n-space — or on the closed n-disk — is contractible =-=[9]-=-. This conjecture was motivated by a result of Floyd in 1951 that whenever a finite group operates as a group of topological transformations on a compact finite dimensional ANR, then the orbit space i... |

3 |
Some retraction properties of the orbit decomposition space of periodic maps
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(Show Context)
Citation Context ...ure was motivated by a result of Floyd in 1951 that whenever a finite group operates as a group of topological transformations on a compact finite dimensional ANR, then the orbit space is also an ANR =-=[13]-=-. To prove his conjecture, Conner further conjectured that if a compact Lie group G acts on a space X, where X is either paracompact of finite cohomological dimension with finitely many orbit types or... |