Homotopy theory with Bornological coarse spaces /

Homotopy theory with Bornological coarse spaces /

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Bibliographic Details
Author / Creator:Bunke, Ulrich, 1963-
Imprint:Cham : Springer, 2020.
Description:1 online resource
Language:English
Series:Lecture notes in mathematics ; v. 2269
Lecture notes in mathematics (Springer-Verlag) ; 2269.
Subject:
Homotopy theory.
Bornological spaces.
Bornological spaces.
Homotopy theory.
Electronic books.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/12607425
Hidden Bibliographic Details
Other authors / contributors:Engel, Alexander.
ISBN:9783030513351
3030513351
3030513343
9783030513344
Digital file characteristics:text file PDF
Notes:Includes bibliographical references and index.
Summary:Providing a new approach to assembly maps, this book develops the foundations of coarse homotopy using the language of infinity categories. It introduces the category of bornological coarse spaces and the notion of a coarse homology theory, and further constructs the universal coarse homology theory. Hybrid structures are introduced as a tool to connect large-scale with small-scale geometry, and are then employed to describe the coarse motives of bornological coarse spaces of finite asymptotic dimension. The remainder of the book is devoted to the construction of examples of coarse homology theories, including an account of the coarsification of locally finite homology theories and of coarse K-theory. Thereby it develops background material about locally finite homology theories and C*-categories. The book is intended for advanced graduate students and researchers who want to learn about the homotopy-theoretical aspects of large scale geometry via the theory of infinity categories.
Other form:Original 3030513343 9783030513344
Standard no.:10.1007/978-3-030-51335-1
10.1007/978-3-030-51
Table of Contents:
  • Intro
  • Contents
  • 1 Introduction
  • Part I Motivic Coarse Spaces and Spectra
  • 2 Bornological Coarse Spaces
  • 2.1 Basic Definitions
  • 2.2 Examples
  • 2.3 Categorical Properties of BornCoarse
  • 3 Motivic Coarse Spaces
  • 3.1 Descent
  • 3.2 Coarse Equivalences
  • 3.3 Flasque Spaces
  • 3.4 u-Continuity and Motivic Coarse Spaces
  • 3.5 Coarse Excision and Further Properties
  • 4 Motivic Coarse Spectra
  • 4.1 Stabilization
  • 4.2 Further Properties of Yo-s
  • 4.3 Homotopy Invariance
  • 4.4 Axioms for a Coarse Homology Theory
  • 5 Merging Coarse and Uniform Structures
  • 5.1 The Hybrid Structure
  • 5.2 Decomposition Theorem
  • 5.2.1 Uniform Decompositions and Statement of the Theorem
  • 5.2.2 Proof of the Decomposition Theorem
  • 5.2.3 Excisiveness of the Cone-at-Infinity
  • 5.3 Homotopy Theorem
  • 5.3.1 Statement of the Theorem
  • 5.3.2 Proof of the Homotopy Theorem
  • 5.3.3 Uniform Homotopies and the Cone Functors
  • 5.4 Flasque Hybrid Spaces
  • 5.5 Decomposition of Simplicial Complexes
  • 5.5.1 Metrics on Simplicial Complexes
  • 5.5.2 Decomposing Simplicial Complexes
  • 5.6 Flasqueness of the Coarsening Space
  • 5.6.1 Construction of the Coarsening Space
  • 5.6.2 Flasqueness for the C0-Structure
  • 5.6.3 Flasqueness for the Hybrid Structure
  • 5.7 The Motivic Coarse Spectra of Simplicial Complexes and Coarsening Spaces
  • Part II Coarse and Locally Finite Homology Theories
  • 6 First Examples and Comparison of Coarse Homology Theories
  • 6.1 Forcing u-Continuity
  • 6.2 Additivity and Coproducts
  • 6.2.1 Additivity
  • 6.2.2 Coproducts
  • 6.3 Coarse Ordinary Homology
  • 6.4 Coarsification of Stable Homotopy
  • 6.4.1 Rips Complexes and a Coarsification of Stable Homotopy
  • 6.4.2 Proof of Theorem 6.32
  • 6.4.3 Further Properties of the Functor Q and Generalizations
  • 6.5 Comparison of Coarse Homology Theories
  • 7 Locally Finite Homology Theories and Coarsification
  • 7.1 Locally Finite Homology Theories
  • 7.1.1 Topological Bornological Spaces
  • 7.1.2 Definition of Locally Finite Homology Theories
  • 7.1.3 Additivity
  • 7.1.4 Construction of Locally Finite Homology Theories
  • 7.1.5 Classification of Locally Finite Homology Theories
  • 7.2 Coarsification of Locally Finite Theories
  • 7.3 Analytic Locally Finite K-Homology
  • 7.3.1 Extending Functors from Locally Compact Spaces to TopBorn
  • 7.3.2 Cohomology for Cstar-Algebras
  • 7.3.3 Locally Finite Homology Theories from Cohomology Theories for Cstar-Algebras
  • 7.4 Coarsification Spaces
  • 8 Coarse K-Homology
  • 8.1 X-Controlled Hilbert Spaces
  • 8.2 Ample X-Controlled Hilbert Spaces
  • 8.3 Roe Algebras
  • 8.4 K-Theory of C*-Algebras
  • 8.5 C*-Categories and Their K-Theory
  • 8.5.1 Definition of Cstar-Categories
  • 8.5.2 From Cstar-Categories to Cstar-Algebras and K-Theory
  • 8.5.3 K-Theory Preserves Filtered Colimits
  • 8.5.4 K-Theory Preserves Unitary Equivalences
  • 8.5.5 Exactness of K-Theory
  • 8.5.6 Additivity of K-Theory

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