Publisher: American Mathematical Society (October 3, 2000)
Format: PDF / Kindle / ePub
Size: 8.6 MB
Downloadable formats: PDF
the purpose of this ebook is to offer a few purposes of sensible research and the speculation of differential operators to the research of topological invariants of manifolds. the most topological software mentioned within the publication matters 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 supply an outline of alternative ways to the evidence 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 house, being a whole Riemannian non-positive curvature manifold. Then the authors current Solovyov's facts of the Novikov conjecture for manifolds with primary crew isomorphic to a discrete subgroup of a linear algebraic staff over a neighborhood box, in line with the suggestion of the Bruhat-Tits development. eventually, the authors speak about the process as a result of Kasparov in line with the operator $KK$-theory and one other evidence of the Mishchenko theorem. In bankruptcy four, they define the method of the Novikov conjecture as a result of Connes and Moscovici related to cyclic homology. that permits one to turn out the conjecture within the case whilst the basic workforce is a (Gromov) hyperbolic workforce. The textual content offers a concise exposition of a few subject matters from practical research (for example, $C^*$-Hilbert modules, $K$-theory or $C^*$-bundles, Hermitian $K$-theory, Fredholm representations, $KK$-theory, and sensible integration) from the idea of differential operators (pseudodifferential calculus and Sobolev chains over $C^*$-algebras), and from differential topology (characteristic classes). The e-book explains easy principles of the topic and will function a path textual content for an advent to the research of unique works and distinctive monographs.