# Equivariant coarse homotopy theory and coarse algebraic K-homology

@inproceedings{Bunke2020EquivariantCH, title={Equivariant coarse homotopy theory and coarse algebraic K-homology}, author={Ulrich Bunke and Alexander Engel and Daniel Kasprowski and Christoph Winges}, year={2020} }

We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the category of equivariant coarse motivic spectra. As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic K-homology. Moreover, we discuss the cone functor, its relation with… Expand

#### 12 Citations

Homotopy Theory with Bornological Coarse Spaces

- Mathematics
- 2020

We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the… Expand

Topological equivariant coarse K-homology

- Mathematics
- 2020

For a $C^{*}$-category with a strict $G$-action we construct examples of equivariant coarse homology theories. To this end we first introduce versions of Roe categories of objects in… Expand

Coarse K-Homology

- Mathematics
- 2020

This final section of the book is devoted to the construction and investigation of coarse K-homology. This coarse homology theory and its applications to index theory, geometric group theory and… Expand

Split Injectivity of A-Theoretic Assembly Maps

- Mathematics
- International Mathematics Research Notices
- 2019

We construct an equivariant coarse homology theory arising from the algebraic $K$-theory of spherical group rings and use this theory to derive split injectivity results for associated assembly… Expand

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… Expand

Additive C*-categories and K-theory.

- Mathematics
- 2020

We introduce and study the notion of an orthogonal sum of a (possibly infinite) family of objects in a $C^{*}$-category. Furthermore, we construct reduced crossed products of $C^{*}$-categories with… Expand

Cyclic homology for bornological coarse spaces

- Mathematics
- 2019

We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field $k$ and group $G$, these are lax symmetric monoidal functors $\mathcal{X}HH_{k}^G$ and… Expand

Controlled objects as a symmetric monoidal functor

- Mathematics
- 2019

The goal of this paper is to associate functorially to every symmetric monoidal additive category $\mathbf{A}$ with a strict $G$-action a lax symmetric monoidal functor… Expand

Equivariant coarse (co-)homology theories

- Mathematics
- 2020

We present an Eilenberg--Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological… Expand

#### References

SHOWING 1-10 OF 51 REFERENCES

Homotopy Theory with Bornological Coarse Spaces

- Mathematics
- 2020

We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces. We construct a category of motivic coarse spectra. Our focus is the… Expand

The general notion of descent in coarse geometry

- Mathematics
- 2010

In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive ‐ a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking… Expand

Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory.

- Mathematics
- 1998

We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K- and L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory of… Expand

SPECTRAL MACKEY FUNCTORS AND EQUIVARIANT ALGEBRAIC K-THEORY (I)

- 2016

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable∞-category, and we use… Expand

Coarse homology theories

- Mathematics
- 2001

In this paper we develop an axiomatic approach to coarse ho- mology theories. We prove a uniqueness result concerning coarse homology theories on the category of \coarse CW-complexes". This… Expand

Spectral Mackey functors and equivariant
algebraic K-theory, II

- Mathematics
- 2020

We study the "higher algebra" of spectral Mackey functors, which the first named author introduced in Part I of this paper. In particular, armed with our new theory of symmetric promonoidal… Expand

On the Isomorphism Conjecture in algebraic K -theory

- Mathematics
- 2001

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RΓ, where Γ is an infinite group. In this paper we prove the conjecture in… Expand

Injectivity results for coarse homology theories.

- Mathematics
- 2018

We show injectivity results for assembly maps using equivariant coarse homology theories with transfers. Our method is based on the descent principle and applies to a large class of linear groups or,… Expand

∞-Categories for the Working Mathematician

- 2018

homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,… Expand

Negative K-theory of derived categories

- Mathematics
- 2006

We define negative K-groups for exact categories and for ``derived categories'' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen-Weibel and Thomason.… Expand