Ebook: Topological and Bivariant K-Theory

Topological K-theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory.

We describe a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, we discuss other approaches to bivariant K-theories for operator algebras. As applications, we study K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem.

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Topological K-theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory.

We describe a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, we discuss other approaches to bivariant K-theories for operator algebras. As applications, we study K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem.

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Topological K-theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory.

We describe a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, we discuss other approaches to bivariant K-theories for operator algebras. As applications, we study K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem.

Content:
Front Matter....Pages i-xi
The elementary algebra of K-theory....Pages 1-18
Functional calculus and topological K-theory....Pages 19-44
Homotopy invariance of stabilised algebraic K-theory....Pages 45-62
Bott periodicity....Pages 63-74
The K-theory of crossed products....Pages 75-90
Towards bivariant K-theory: how to classify extensions....Pages 91-122
Bivariant K-theory for bornological algebras....Pages 123-140
A survey of bivariant K-theories....Pages 141-172
Algebras of continuous trace, twisted K-theory....Pages 173-183
Crossed products by ? and Connes’ Thom Isomorphism....Pages 185-193
Applications to physics....Pages 195-202
Some connections with index theory....Pages 203-223
Localisation of triangulated categories....Pages 225-239
Back Matter....Pages 241-262

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Topological K-theory is one of the most important invariants for noncommutative algebras equipped with a suitable topology or bornology. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory.

We describe a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, we discuss other approaches to bivariant K-theories for operator algebras. As applications, we study K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem.

Content:
Front Matter....Pages i-xi
The elementary algebra of K-theory....Pages 1-18
Functional calculus and topological K-theory....Pages 19-44
Homotopy invariance of stabilised algebraic K-theory....Pages 45-62
Bott periodicity....Pages 63-74
The K-theory of crossed products....Pages 75-90
Towards bivariant K-theory: how to classify extensions....Pages 91-122
Bivariant K-theory for bornological algebras....Pages 123-140
A survey of bivariant K-theories....Pages 141-172
Algebras of continuous trace, twisted K-theory....Pages 173-183
Crossed products by ? and Connes’ Thom Isomorphism....Pages 185-193
Applications to physics....Pages 195-202
Some connections with index theory....Pages 203-223
Localisation of triangulated categories....Pages 225-239
Back Matter....Pages 241-262
....