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Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2012-10-03

2012

Ζαβιτσάνος Γιώργος

Ανδρουλιδάκης Ιάκωβος Επικ. Καθηγ.

Gerbes Δεσμών και Συνεστραμμένη Κ-Θεωρία

Greek

We introduce the notion of bundle grebes and give the basic features of their

theory. As we explicitly show, these are geometric objects that are associated

with degree 3 integral Cech cohomology class on a manifold M, known as

Dixmier-Douady class, in analogy to the 2 integral Cech cohomology class that

is associated to every line bundle through the Chern class isomorphism. We give

the constructions of lifting bundle gerbe as well as the tautological bundle

gerbe and we show that the Dixmier-Douady class induces a bijection between the

classes of stable isomorphic bundle gerbes and the group H^3(M,Z).

We define the K-theory of bundle grebes and we investigate the relation to

twisted K-theory which is defined according to Atiyah’s view, making use of the

classifying space of Fredholm operators. When the Dixmier-Douady class is

torsion, we show that the K-theory of bundle gebres and twisted K-theory

coincide. In the case of non-torsion class, K-theory of bundle gerbes is

defined in analogous way considering the classifying space of zero-index

Fredholm operators.

theory. As we explicitly show, these are geometric objects that are associated

with degree 3 integral Cech cohomology class on a manifold M, known as

Dixmier-Douady class, in analogy to the 2 integral Cech cohomology class that

is associated to every line bundle through the Chern class isomorphism. We give

the constructions of lifting bundle gerbe as well as the tautological bundle

gerbe and we show that the Dixmier-Douady class induces a bijection between the

classes of stable isomorphic bundle gerbes and the group H^3(M,Z).

We define the K-theory of bundle grebes and we investigate the relation to

twisted K-theory which is defined according to Atiyah’s view, making use of the

classifying space of Fredholm operators. When the Dixmier-Douady class is

torsion, we show that the K-theory of bundle gebres and twisted K-theory

coincide. In the case of non-torsion class, K-theory of bundle gerbes is

defined in analogous way considering the classifying space of zero-index

Fredholm operators.

Bundle gerbes, Τwisted K-theory, Deligne cohomology, Fredholm operators, Dixmier-Douady class

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