@phdthesis{Mertsch2020,
  author    = {Darvin Mertsch},
  title     = {Geometric Models of Twisted K-Theory based Bundle Gerbes and Algebra Bundles},
  journal    = {Geometrische Modelle von Getwisteter K-Theorie basierend auf B{\"u}ndelgerben und Algebrab{\"u}ndel},
  url       = {https://nbn-resolving.org/urn:nbn:de:gbv:9-opus-40972},
  pages     = {196},
  year      = {2020},
  abstract  = {Twisted topological K-theory is a twisted version of topological K-theory in the sense of twisted generalized cohomology theories. It was pioneered by Donavan and Karoubi in 1970 where they used bundles of central simple graded algebras to model twists of K-theory. By the end of the last century physicists realised that D-brane charges in the field of string theory may be studied in terms of twisted K-theory. This rekindled interest in the topic lead to a wave of new models for the twists and new ways to realize the respective twisted K-theory groups. The state-of-the-art models today use bundles of projective unitary operators on separable Hilbert spaces as twists and K-groups are modeled by homotopy classes of sections of certain bundles of Fredholm operators. From a physics perspective these treatments are not optimal yet: they are intrinsically infinite-dimensional and these models do not immediately allow the inclusion of differential data like forms and connections. In this thesis we introduce the 2-stack of k-algebra gerbes. Objects, 1-morphisms and 2-morphisms consist of finite-dimensional geometric data simultaneously generalizing bundle gerbes and bundles of central simple graded k-algebras for k either the field of real numbers or the field of complex numbers. We construct an explicit isomorphism from equivalence classes of k-algebra gerbes over a space X to the full set of twists of real K-theory and complex K-theory respectively. Further, we model relative twisted K-groups for compact spaces X and closed subspaces Y twisted by algebra gerbes. These groups are modeled directly in terms of 1-morphisms and 2-morphisms of algebra gerbes over X. We exhibit a relation to the K-groups introduced by Donavan and Karoubi and we translate their fundamental isomorphism -- an isomorphism relating K-groups over Thom spaces with K-groups twisted by Clifford algebra bundles -- to the new setting. With the help of this fundamental isomorphism we construct an explicit Thom isomorphism and explicit pushforward homomorphisms for smooth maps between compact manifolds, without requiring these maps to be K-oriented. Further -- in order to treat K-groups for non-torsion twists -- we implement a geometric cocycle model, inspired by a related geometric cycle model developed by Baum and Douglas for K-homology in 1982, and construct an assembly map for this model.},
  language  = {en}
}