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Product Code:  MMONO/192 
213 pp 
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Electronic ISBN:  9781470446062 
Product Code:  MMONO/192.E 
213 pp 
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Book DetailsTranslations of Mathematical MonographsVolume: 192; 2001MSC: Primary 46; Secondary 19; 35; 47; 57;
The aim of this book is to present some applications of functional analysis and the theory of differential operators to the investigation of topological invariants of manifolds.
The main topological application discussed in the book concerns the problem of the description of homotopyinvariant rational Pontryagin numbers of nonsimply connected manifolds and the Novikov conjecture of homotopy invariance of higher signatures. The definition of higher signatures and the formulation of the Novikov conjecture are given in Chapter 3. In this chapter, the authors also give an overview of different approaches to the proof of the Novikov conjecture. First, there is the Mishchenko symmetric signature and the generalized Hirzebruch formulae and the Mishchenko theorem of homotopy invariance of higher signatures for manifolds whose fundamental groups have a classifying space, being a complete Riemannian nonpositive curvature manifold. Then the authors present Solovyov's proof of the Novikov conjecture for manifolds with fundamental group isomorphic to a discrete subgroup of a linear algebraic group over a local field, based on the notion of the BruhatTits building. Finally, the authors discuss the approach due to Kasparov based on the operator \(KK\)theory and another proof of the Mishchenko theorem. In Chapter 4, they outline the approach to the Novikov conjecture due to Connes and Moscovici involving cyclic homology. That allows one to prove the conjecture in the case when the fundamental group is a (Gromov) hyperbolic group.
The text provides a concise exposition of some topics from functional analysis (for instance, \(C^*\)Hilbert modules, \(K\)theory or \(C^*\)bundles, Hermitian \(K\)theory, Fredholm representations, \(KK\)theory, and functional integration) from the theory of differential operators (pseudodifferential calculus and Sobolev chains over \(C^*\)algebras), and from differential topology (characteristic classes).
The book explains basic ideas of the subject and can serve as a course text for an introduction to the study of original works and special monographs.ReadershipGraduate students and research mathematicians interested in differential topology, functional analysis, and geometry; theoretical physicists.

Table of Contents

Chapters

$C^*$algebras and $K$theory

Index theorems

The higher signatures

Noncommutative differential geometry


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 Table of Contents
 Additional Material

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The aim of this book is to present some applications of functional analysis and the theory of differential operators to the investigation of topological invariants of manifolds.
The main topological application discussed in the book concerns the problem of the description of homotopyinvariant rational Pontryagin numbers of nonsimply connected manifolds and the Novikov conjecture of homotopy invariance of higher signatures. The definition of higher signatures and the formulation of the Novikov conjecture are given in Chapter 3. In this chapter, the authors also give an overview of different approaches to the proof of the Novikov conjecture. First, there is the Mishchenko symmetric signature and the generalized Hirzebruch formulae and the Mishchenko theorem of homotopy invariance of higher signatures for manifolds whose fundamental groups have a classifying space, being a complete Riemannian nonpositive curvature manifold. Then the authors present Solovyov's proof of the Novikov conjecture for manifolds with fundamental group isomorphic to a discrete subgroup of a linear algebraic group over a local field, based on the notion of the BruhatTits building. Finally, the authors discuss the approach due to Kasparov based on the operator \(KK\)theory and another proof of the Mishchenko theorem. In Chapter 4, they outline the approach to the Novikov conjecture due to Connes and Moscovici involving cyclic homology. That allows one to prove the conjecture in the case when the fundamental group is a (Gromov) hyperbolic group.
The text provides a concise exposition of some topics from functional analysis (for instance, \(C^*\)Hilbert modules, \(K\)theory or \(C^*\)bundles, Hermitian \(K\)theory, Fredholm representations, \(KK\)theory, and functional integration) from the theory of differential operators (pseudodifferential calculus and Sobolev chains over \(C^*\)algebras), and from differential topology (characteristic classes).
The book explains basic ideas of the subject and can serve as a course text for an introduction to the study of original works and special monographs.
Graduate students and research mathematicians interested in differential topology, functional analysis, and geometry; theoretical physicists.

Chapters

$C^*$algebras and $K$theory

Index theorems

The higher signatures

Noncommutative differential geometry