Download E-books $C^*$-Algebras and Elliptic Operators in Differential Topology (Translations of Mathematical Monographs) PDF

The purpose of this ebook is to give a few purposes of useful research and the idea of differential operators to the research of topological invariants of manifolds. the most topological program mentioned within the publication issues the matter of the outline of homotopy-invariant rational Pontryagin numbers of non-simply hooked up manifolds and the Novikov conjecture of homotopy invariance of upper signatures. The definition of upper signatures and the formula of the Novikov conjecture are given in bankruptcy three. during this bankruptcy, the authors additionally provide an outline of alternative ways to the facts of the Novikov conjecture. First, there's the Mishchenko symmetric signature and the generalized Hirzebruch formulae and the Mishchenko theorem of homotopy invariance of upper signatures for manifolds whose primary teams have a classifying area, being a whole Riemannian non-positive curvature manifold. Then the authors current Solovyov's evidence of the Novikov conjecture for manifolds with primary team isomorphic to a discrete subgroup of a linear algebraic workforce over a neighborhood box, in accordance with the suggestion of the Bruhat-Tits development. ultimately, the authors speak about the process as a result of Kasparov in accordance with the operator $KK$-theory and one other facts of the Mishchenko theorem. In bankruptcy four, they define the method of the Novikov conjecture because of Connes and Moscovici related to cyclic homology. that permits one to end up the conjecture within the case while the basic team is a (Gromov) hyperbolic workforce. The textual content offers a concise exposition of a few subject matters from sensible research (for example, $C^*$-Hilbert modules, $K$-theory or $C^*$-bundles, Hermitian $K$-theory, Fredholm representations, $KK$-theory, and practical integration) from the speculation of differential operators (pseudodifferential calculus and Sobolev chains over $C^*$-algebras), and from differential topology (characteristic classes). The publication explains simple principles of the topic and will function a path textual content for an advent to the learn of unique works and targeted monographs.

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