$C^{*}$-Algebras and Elliptic Operators in Differential Topology
( Translations of Mathematical Monographs )
Publication series ： Translations of Mathematical Monographs
Author： Yu. P. Solovyov;E. V. Troitsky
Publisher： American Mathematical Society
Publication year： 2018
E-ISBN: 9781470446062
P-ISBN(Paperback): 9780821813997
Subject： O177.5 Banach algebras; Normed algebras (), algebraic topology, abstract harmonic analysis
Keyword： 数学
Language： ENG
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$C^{*}$-Algebras and Elliptic Operators in Differential Topology
Description
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 homotopy-invariant rational Pontryagin numbers of non-simply 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 non-positive 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 Bruhat-Tits 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 funda
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