Homotopy theory with bornological coarse spaces

Bunke, Ulrich; Engel, Alexander

doi:10.48550/arxiv.1607.03657

Cited by 14 publications

(62 citation statements)

References 34 publications

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“…We will use a mildly adapted version of the definition of coarse homotopy introduced by MNS in [10] (see also [2] for a similar definition). Denote the space [0, ∞) ⊆ R by R + .…”

Section: Coarse Homotopiesmentioning

confidence: 99%

“…Among the very first proofs of the coarse Baum-Connes Conjecture for certain spaces were a proof for manifolds with "Lipschitz good covers" by Guoliang Yu [17] and a proof by Higson and Roe for Gromov hyperbolic spaces [9], both of which used some notion of coarse homotopy equivalence. More recently, coarse homotopy theory is being developed by Bunke, Engel and others using the context of ∞-categories [2].…”

Section: Introductionmentioning

confidence: 99%

“…In [3], Dranishnikov proposed that it would be interesting to define the coarse fundamental group using the upper half-plane, but the coarse fundamental group was only recently formally defined in a paper by Mitchener-Norouzizadeh-Schick (hereafter MNS) [10] using ideas from Mitchener's coarse homology theory. In that paper, MNS also introduce a notion of coarse homotopy very similar to one used by Bunke and Engel in [2], a definition which we will use with only a small modification (Definition 2.3). Another large-scale version of the fundamental group is the fundamental group at infinity studied in geometric group theory (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Lifting coarse homotopies

Weighill¹

2019

Preprint

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Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective maps. This class is wide enough to include quotients by coarsely discontinuous group actions, which allows us to obtain results concerning the coarse fundamental group of quotients which are analogous to classical topological results for the fundamental group. As an application, we compute the fundamental group of metric cones over negatively curved compact Riemannian manifolds.

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“…We will use a mildly adapted version of the definition of coarse homotopy introduced by MNS in [10] (see also [2] for a similar definition). Denote the space [0, ∞) ⊆ R by R + .…”

Section: Coarse Homotopiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lifting coarse homotopies

Weighill¹

2019

Preprint

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“…In what follows we assume that the reader is familiar with the notions of bornological coarse spaces and coarse homology theories [BE16], [BEKW17].…”

Section: Conventionsmentioning

confidence: 99%

“…Example 3.56. We consider the equivariant coarse topological K-homology functor KX G in Fun colim (GSpcX , Sp) (see [BE16,Definition 7.52] for the non equivariant construction and [BEa,Def. 5 The Baum-Connes conjecture for the group H asserts, that the assembly map Asbl KU r,H ,Fin is an equivalence, where Fin denotes the family of finite subgroups.…”

Section: The Coarse Abstract Localization Theoremsmentioning

confidence: 99%

Localization for coarse homology theories

Bunke¹,

Caputi²

2019

Preprint

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We introduce the notion of a Bredon-style equivariant coarse homology theory. We show that such a Bredon-style equivariant coarse homology theory satisfies localization theorems and that a general equivariant coarse homology theory can be approximated by a Bredon-style version. We discuss the special case of algebraic and topological equivariant coarse K-homology and obtain the coarse analog of Segal's localization theorem.

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Uniform K-theory, and Poincaré duality for uniform K-homology

Engel

2019

Journal of Functional Analysis

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We revisit Špakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology.We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K-homology and prove Poincaré duality between uniform K-theory and uniform K-homology on spin c manifolds of bounded geometry. Uniform K-homologyIn this section we will investigate uniform K-homology-a version of K-homology that incorporates into its definition uniformity estimates that one usually has for, e.g., Dirac operators over manifolds of bounded geometry (see Example 2.7). Uniform K-homology was introduced by Špakula [Špa08, Špa09]. 2 We will revisit it and we will prove additional 1 Though note that this approach does not give a concrete formula for how to find this vector bundle. It is an important observation of Atiyah and Singer (and later elaborated upon by Baum and Douglas in their geometric picture for K-homology) that if the K-homology class is given by an elliptic pseudodifferential operator, then one can use the symbol of the operator to get a representative as the class of a twisted Dirac operator. 2 But we have changed the definition slightly, see Section 2.2 for how and why.

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Homotopy theory with bornological coarse spaces

Abstract: We propose an axiomatic characterization of coarse homology theories defined on the category of bornological coarse spaces BornCoarse. We construct a stable ∞-category SpX of motivic coarse spectra which is the target of the universal coarse homology theory BornCoarse → SpX .

Cited by 14 publications

References 34 publications

Lifting coarse homotopies

Lifting coarse homotopies

Localization for coarse homology theories

Uniform K-theory, and Poincaré duality for uniform K-homology

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