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E-grāmata: Homotopy Theory with Bornological Coarse Spaces

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  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 2269
  • Izdošanas datums: 03-Sep-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • Valoda: eng
  • ISBN-13: 9783030513351
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Homotopy Theory with Bornological Coarse SpacesPalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 2269
  • Izdošanas datums: 03-Sep-2020
  • Izdevniecība: Springer Nature Switzerland AG
  • Valoda: eng
  • ISBN-13: 9783030513351
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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.
1 Introduction
1(12)
Part I Motivic Coarse Spaces and Spectra
2 Bornological Coarse Spaces
13(8)
2.1 Basic Definitions
13(2)
2.2 Examples
15(2)
2.3 Categorical Properties of BornCoarse
17(4)
3 Motivic Coarse Spaces
21(14)
3.1 Descent
22(4)
3.2 Coarse Equivalences
26(2)
3.3 Flasque Spaces
28(3)
3.4 u-Continuity and Motivic Coarse Spaces
31(2)
3.5 Coarse Excision and Further Properties
33(2)
4 Motivic Coarse Spectra
35(18)
4.1 Stabilization
35(5)
4.2 Further Properties of Yos
40(4)
4.3 Homotopy Invariance
44(4)
4.4 Axioms for a Coarse Homology Theory
48(5)
5 Merging Coarse and Uniform Structures
53(42)
5.1 The Hybrid Structure
53(5)
5.2 Decomposition Theorem
58(7)
5.2.1 Uniform Decompositions and Statement of the Theorem
58(2)
5.2.2 Proof of the Decomposition Theorem
60(5)
5.2.3 Excisiveness of the Cone-at-Infinity
65(1)
5.3 Homotopy Theorem
65(7)
5.3.1 Statement of the Theorem
66(1)
5.3.2 Proof of the Homotopy Theorem
66(4)
5.3.3 Uniform Homotopies and the Cone Functors
70(2)
5.4 Flasque Hybrid Spaces
72(5)
5.5 Decomposition of Simplicial Complexes
77(5)
5.5.1 Metrics on Simplicial Complexes
77(2)
5.5.2 Decomposing Simplicial Complexes
79(3)
5.6 Flasqueness of the Coarsening Space
82(9)
5.6.1 Construction of the Coarsening Space
82(3)
5.6.2 Flasqueness for the Co-Structure
85(2)
5.6.3 Flasqueness for the Hybrid Structure
87(4)
5.7 The Motivic Coarse Spectra of Simplicial Complexes and Coarsening Spaces
91(4)
Part II Coarse and Locally Finite Homology Theories
6 First Examples and Comparison of Coarse Homology Theories
95(24)
6.1 Forcing u-Continuity
95(3)
6.2 Additivity and Coproducts
98(3)
6.2.1 Additivity
98(1)
6.2.2 Coproducts
99(2)
6.3 Coarse Ordinary Homology
101(5)
6.4 Coarsification of Stable Homotopy
106(11)
6.4.1 Rips Complexes and a Coarsification of Stable Homotopy
108(4)
6.4.2 Proof of Theorem 6.32
112(3)
6.4.3 Further Properties of the Functor Q and Generalizations
115(2)
6.5 Comparison of Coarse Homology Theories
117(2)
7 Locally Finite Homology Theories and Coarsification
119(38)
7.1 Locally Finite Homology Theories
119(19)
7.1.1 Topological Bomological Spaces
120(2)
7.1.2 Definition of Locally Finite Homology Theories
122(7)
7.1.3 Additivity
129(3)
7.1.4 Construction of Locally Finite Homology Theories
132(3)
7.1.5 Classification of Locally Finite Homology Theories
135(3)
7.2 Coarsification of Locally Finite Theories
138(3)
7.3 Analytic Locally Finite K-Homology
141(9)
7.3.1 Extending Functors from Locally Compact Spaces to TopBorn
141(4)
7.3.2 Cohomology for C*-Algebras
145(2)
7.3.3 Locally Finite Homology Theories from Cohomology Theories for C*-Algebras
147(3)
7.4 Coarsification Spaces
150(7)
8 Coarse K-Homology
157(78)
8.1 X-Controlled Hilbert Spaces
158(3)
8.2 Ample X-Controlled Hilbert Spaces
161(5)
8.3 Roe Algebras
166(8)
8.4 K-Theory of C*-Algebras
174(3)
8.5 C*-Categories and Their K-Theory
177(16)
8.5.1 Definition of C*-Categories
179(1)
8.5.2 From C*-Categories to C*-Algebras and K-Theory
180(5)
8.5.3 K-Theory Preserves Filtered Colimits
185(1)
8.5.4 K-Theory Preserves Unitary Equivalences
185(2)
8.5.5 Exactness of K-Theory
187(3)
8.5.6 Additivity of K-Theory
190(3)
8.6 Coarse K-Homology
193(10)
8.7 Comparison with the Classical Definition
203(9)
8.8 Additivity and Coproducts
212(12)
8.8.1 Additivity
212(9)
8.8.2 Coproducts
221(3)
8.9 Dirac Operators
224(6)
8.10 K-Theoretic Coarse Assembly Map
230(5)
References 235(4)
Index 239
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