Abstract:
In the previous talks by means of bundle gerbes a functor $bg$ from
the category of compact
topological spaces to the category of 2-groupoids has been constructed.
Furthermore, using modules over bundle
gerbes a functor $bg(X)-->Ab$ ("twisted $K$-theory")
which is functorial on the base $X$ and coincides with $K^0(X)$
in case of trivial bundle gerbe has been defined.
The group of isomorphism classes of objects of the category $bg(X)$
is isomorphic to $H^3(X,\mathbb{Z})$, the third cohomology group
of $X$ with integer coefficients. However twists from $H^3(X,\mathbb{Z})$
are not most general
possible for K-theory. Therefore it seems natural to extend the
above construction of twisted $K$-theory to more general twists.
In order to do this, the notion of a bundle gerbe itself has to be
generalized: first, in place of line bundles we have to consider
arbitrary finite dimensional vector bundles, and second, replace
isomorphisms by homotopies.
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