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Title page for ETD etd-12022013-143841


Type of Document Dissertation
Author Zhou, Dapeng
URN etd-12022013-143841
Title The coarse Baum-Connes conjecture and controlled operator K-theory
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Gennadi Kasparov Committee Chair
Bruce Hughes Committee Member
Dietmar Bisch Committee Member
Guoliang Yu Committee Member
Thomas Kephart Committee Member
Keywords
  • Baum-Connes
  • K-theory
Date of Defense 2013-11-22
Availability unrestricted
Abstract
The Atiyah-Singer index theorem has been vastly generaized to higher index theory for elliptic operators in the context of noncommutative geometry. Higher index theory has important applications in differential topology and differential geometry such as the Novikov conjecture on homotopy invariance of higher signature and the existence preoblem of Riemannian metrics with positive scalar curvature.

The Baum-Connes conjecture and the coarse Baum-Connes conjecture are algorithms to compute the higher indices of elliptic differential operators. The controlled operator K-theory, a refined version of classical operator K-theory, is a powerful tool to study these conjectures.

In this paper, we study the controlled operator K-theory, give a characterization of the K-theory elements in the image of the Baum-Connes map, and verify the coarse Baum-Connes conjecture for a large class of spaces, especially for spaces with finite decomposition complexities.

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