Operator algebras and topology

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dc.creatorSchick, Thomas
dc.date2002-09-13
dc.date.accessioned2026-07-07T04:50:50Z
dc.date.available2026-07-07T04:50:50Z
dc.descriptionThese notes cover the contents of three survey lectures held at the ICTP Trieste Summer school on High dimensional manifold theory 2001. They introduce techniques coming from the theory of operator algebras. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. An central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. It implies the Novikov conjecture. In the first talk, the Baum-Connes conjecture will be explained and put into our context. One application of the Baum-Connes conjecture is to the positive scalar curvature question. It implies the so called ``stable Gromov-Lawson-Rosenberg conjecture''. The unstable version of this conjecture said that, given a closed spin manifolds M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterexamples will be presented in the second talk. The third talk introduces $L^2$-cohomology, $L^2$-Betti numbers and other $L^2$-invariants. These invariants, their basic properties, and the central questions about them are introduced.
dc.descriptionto appear in the Proceedings of the ICTP Trieste Summer School on High dimensional manifold theory 2001
dc.identifierhttps://arxiv.org/abs/math/0209164
dc.identifierhttp://arxiv.org/abs/math/0209164
dc.identifierTopology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), 571--660, ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/64936
dc.subjectGeometric Topology
dc.subjectDifferential Geometry
dc.subjectK-Theory and Homology
dc.titleOperator algebras and topology
dc.typetext

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