@article{SchickSeyedhosseini2021,
  author    = {Schick, Thomas and Seyedhosseini, Mehran},
  title     = {On an index theorem of Chang, Weinberger and Yu},
  series = {M{\"u}nster journal of mathematics},
  volume    = {14},
  journal   = {M{\"u}nster journal of mathematics},
  number    = {1},
  publisher = {WWU, Fachbereich Mathematik und Informatik},
  address   = {M{\"u}nster},
  issn      = {1867-5778},
  doi       = {10.17879/59019522628},
  pages     = {123 -- 154},
  year      = {2021},
  abstract  = {In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that (a slight variation of) the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger and Yu (rather: a slight variation). To take the fundamental groups of the manifold and its boundary into account requires working with maximal C*-completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions. On the other hand, we introduce and propose a more conceptual and more geometric completion, which still has all the required functoriality.},
  language  = {en}
}
@article{AzzaliGoetteSchick2015,
  author    = {Azzali, Sara and Goette, Sebastian and Schick, Thomas},
  title     = {Large time limit and local L-2-index theorems for families},
  series = {Journal of noncommutative geometry},
  volume    = {9},
  journal   = {Journal of noncommutative geometry},
  number    = {2},
  publisher = {EMS Publ.},
  address   = {Z{\"u}rich},
  issn      = {1661-6952},
  doi       = {10.4171/JNCG/203},
  pages     = {621 -- 664},
  year      = {2015},
  abstract  = {We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L-2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L-2-index formulas. As applications, we prove a local L-2-index theorem for families of signature operators and an L-2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L-2-eta forms and L-2-torsion forms as transgression forms.},
  language  = {en}
}