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 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.
Readership
Graduate students and research mathematicians interested in
differential topology, functional analysis, and geometry; theoretical
physicists.
