@article{BeckerBeniniSchenkeletal.2019,
  author    = {Becker, Christian and Benini, Marco and Schenkel, Alexander and Szabo, Richard J.},
  title     = {Cheeger-Simons differential characters with compact support and Pontryagin duality},
  series = {Communications in analysis and geometry},
  volume    = {27},
  journal   = {Communications in analysis and geometry},
  number    = {7},
  publisher = {International Press of Boston},
  address   = {Somerville},
  issn      = {1019-8385},
  doi       = {10.4310/CAG.2019.v27.n7.a2},
  pages     = {1473 -- 1522},
  year      = {2019},
  abstract  = {By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology and differential forms with compact support, in full analogy to ordinary differential cohomology. We prove an excision theorem for differential cohomology using a suitable relative version. Furthermore, we use our model to give an independent proof of Pontryagin duality for differential cohomology recovering a result of [Harvey, Lawson, Zweck - Amer. J. Math. 125 (2003), 791]: On any oriented manifold, ordinary differential cohomology is isomorphic to the smooth Pontryagin dual of compactly supported differential cohomology. For manifolds of finite-type, a similar result is obtained interchanging ordinary with compactly supported differential cohomology.},
  language  = {en}
}
@article{Becker2016,
  author    = {Becker, Christian},
  title     = {Cheeger-Chern-Simons Theory and Differential String Classes},
  series = {Annales de l'Institut Henri Poincar{\~A}©},
  volume    = {17},
  journal   = {Annales de l'Institut Henri Poincar{\~A}©},
  publisher = {Springer},
  address   = {Basel},
  issn      = {1424-0637},
  doi       = {10.1007/s00023-016-0485-6},
  pages     = {1529 -- 1594},
  year      = {2016},
  abstract  = {We construct new concrete examples of relative differential characters, which we call Cheeger-Chern-Simons characters. They combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, Cheeger-Chern-Simons characters generalize Chern-Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger-Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf-Witten correspondence between 3-dimensional Chern-Simons theories and Wess-Zumino-Witten terms to fully extended higher-order Chern-Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger-Chern-Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class , we recover isomorphism classes of geometric string structures on Spin (n) -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger-Chern-Simons character associated with the class together with its transgressions to loop space and higher mapping spaces defines a Chern-Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern-Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger-Chern-Simons character and extended Chern-Simons theory. Differential trivialization classes yield trivializations of this extended Chern-Simons theory.},
  language  = {en}
}
@article{Becker2014,
  author    = {Becker, Christian},
  title     = {Relative differential cohomology},
  series = {Lecture notes in mathematics : a collection of informal reports and seminars},
  volume    = {2112},
  journal   = {Lecture notes in mathematics : a collection of informal reports and seminars},
  publisher = {Springer},
  address   = {Berlin},
  isbn      = {978-3-319-07034-6; 978-3-319-07033-9},
  issn      = {0075-8434},
  doi       = {10.1007/978-3-319-07034-6_2},
  pages     = {91 -- 180},
  year      = {2014},
  abstract  = {We study two notions of relative differential cohomology, using the model of differential characters. The two notions arise from the two options to construct relative homology, either by cycles of a quotient complex or of a mapping cone complex. We discuss the relation of the two notions of relative differential cohomology to each other. We discuss long exact sequences for both notions, thereby clarifying their relation to absolute differential cohomology. We construct the external and internal product of relative and absolute characters and show that relative differential cohomology is a right module over the absolute differential cohomology ring. Finally we construct fiber integration and transgression for relative differential characters.},
  language  = {en}
}
@article{BaerBecker2014,
  author    = {B{\"a}r, Christian and Becker, Christian},
  title     = {Differential characters and geometric chains},
  series = {Lecture notes in mathematics : a collection of informal reports and seminars},
  volume    = {2112},
  journal   = {Lecture notes in mathematics : a collection of informal reports and seminars},
  publisher = {Springer},
  address   = {Berlin},
  isbn      = {978-3-319-07034-6; 978-3-319-07033-9},
  issn      = {0075-8434},
  doi       = {10.1007/978-3-319-07034-6_1},
  pages     = {1 -- 90},
  year      = {2014},
  abstract  = {We study Cheeger-Simons differential characters and provide geometric descriptions of the ring structure and of the fiber integration map. The uniqueness of differential cohomology (up to unique natural transformation) is proved by deriving an explicit formula for any natural transformation between a differential cohomology theory and the model given by differential characters. Fiber integration for fibers with boundary is treated in the context of relative differential characters. As applications we treat higher-dimensional holonomy, parallel transport, and transgression.},
  language  = {en}
}