Equivariant coarse homotopy theory and coarse algebraic $\mathbf{K}$-homology

Abstract

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 equivariant homology theories in equivariant topology, and assembly and forget-control maps. This is a preparation for applications in subsequent papers aiming at split-injectivity results for the Farrell-Jones assembly map.

arXiv comment110 pages, minor improvements

References in corpus (6)

• The K-theoretic Farrell-Jones Conjecture for hyperbolic groups
• Universality of multiplicative infinite loop space machines
• Cohomological Finiteness Conditions in Bredon Cohomology
• The General Notion of Descent in Coarse Geometry
• On the finiteness of the classifying space for the family of virtually cyclic subgroups
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Cited by in corpus (12)

• Split injectivity of A-theoretic assembly maps
• The coarse index class with support
• Controlled objects in left-exact $\infty$-categories and the Novikov conjecture
• Topological equivariant coarse K-homology
• Generalized bornological coarse spaces and coarse motivic spectra
• Localization for coarse homology theories
• Controlled objects as a symmetric monoidal functor
• Coarse cohomology theories
• An equivariant PPV theorem and Paschke-Higson duality
• Paschke duality and assembly maps
• On the Farrell-Jones conjecture for localising invariants
• Coarse cohomology of configuration space and coarse embedding