# Assembly maps

@article{Lueck2018AssemblyM, title={Assembly maps}, author={Wolfgang Lueck}, journal={Handbook of Homotopy Theory}, year={2018} }

We introduce and analyze the concept of an assembly map from the original homotopy theoretic point of view. We give also interpretations in terms of surgery theory, controlled topology and index theory. The motivation is that prominent conjectures of Farrell-Jones and Baum-Connes about K- and L-theory of group rings and group C^*-algebras predict that certain assembly maps are weak homotopy equivalences.

## 13 Citations

### A certain structure of Artin groups and the isomorphism conjecture

- MathematicsCanadian Journal of Mathematics
- 2020

Abstract We observe an inductive structure in a large class of Artin groups of finite real, complex and affine types and exploit this information to deduce the Farrell–Jones isomorphism conjecture…

### Universal spaces for proper actions play an important role in geometric group theory , equivariant homotopy theory , and the Baum – Connes and Farrell – Jones Conjectures

- Mathematics
- 2019

We formalize an equivariant version of Bestvina–Brady discrete Morse theory, and apply it to Vietoris–Rips complexes in order to exhibit finite universal spaces for proper actions for all…

### Generalizations of Loday's assembly maps for Lawvere's algebraic theories

- Mathematics
- 2021

Loday’s assembly maps approximate the K-theory of group rings by the K-theory of the coefﬁcient ring and the corresponding homology of the group. We present a generalization that places both…

### Assembly and Morita invariance in the algebraic K-theory of Lawvere theories.

- Mathematics
- 2020

The algebraic K-theory of Lawvere theories provides a context for the systematic study of the stable homology of the automorphism groups of algebraic structures, such as the symmetric groups, the…

### Homotopy ribbon discs with a fixed group

- Mathematics
- 2022

. In the topological category, the classiﬁcation of homotopy ribbon discs is known when the fundamental group G of the exterior is Z and the Baumslag-Solitar group BS (1 , 2). We prove that if a…

### Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

- Mathematics
- 2019

We associate to every $G$-bornological coarse space $X$ and every left-exact $\infty$-category with $G$-action a left-exact infinity-category of equivariant $X$-controlled objects. Postcomposing with…

### Quasifibrations in configuration Lie groupoids and orbifold braid groups

- Mathematics
- 2021

. In [19] we studied a Fadell-Neuwirth type ﬁbration theorem for orbifolds, and gave a short exact sequence of fundamental groups of conﬁguration Lie groupoids of Lie groupoids corresponding to the…

### Conﬁguration Lie groupoids and orbifold braid groups. Discriminantal varieties and conﬁguration spaces in algebraic topology

- Mathematics
- 2022

Summary: We propose two deﬁnitions of conﬁguration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type ﬁbration theorem for a class of Lie groupoids. We show that this is the best…

### TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

- MathematicsGlasgow Mathematical Journal
- 2021

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with…

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In this paper we study the index theoretic interpretation of the analytical assembly map that appears in the Baum-Connes conjecture. In its general form it may be constructed using Kasparov's…

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In this paper we prove the equivalence of various algebraically or geometrically defined assembly maps used in formulating the main conjectures in K- and L-theory, and C*-theory.

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We compare the domain of the assembly map in algebraic K { theory with respect to the family of nite subgroups with the domain of the assembly map with respect to the family of virtually cyclic…

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We show that the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic K-theory is split injective in the setting where the…

### Assembly maps for topological cyclic homology of group algebras

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- 2017

Abstract We use assembly maps to study 𝐓𝐂 ( 𝔸 [ G ] ; p ) \mathbf{TC}(\mathbb{A}[G];p) , the topological cyclic homology at a prime p of the group algebra of a discrete group G with…

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We first construct a classifying space for defining equivariant K-theory for proper actions of discrete groups. This is then applied to construct equivariant Chern characters with values in Bredon…