Large scale index, positive scalar curvature and manifold topology

• Applicant: Professor Dr. Thomas Schick
• Subject Area: Mathematics
• Term: from 2016 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 321324296

• Final Report Year: 2023

Final Report Abstract

In the project we studied which types of geometry a given compact manifold M without boundary can carry. Specifically - motivated by cosmology - are we interested in scalar curvature and we studied in particular the space of metrics of positive scalar curvature. When is this space empty and when does a metric of positive scalar curvature exist? More generally: what can we say about its topology? For example: how do its homotopy groups look like? A fundamental differential equation of quantum physics, the Dirac equation for spinors, is closely related to (positive) scalar curvature. Modern refinements, in part obtained in the research project, use operator algebras and K-theory to detect subtle global information about the geometry which is encoded in this relation. In the project we further developed the paradigm of large-scale index theory. We have identified new geometric situation where it can be applied. Furthermore, we constructed new geometric-topological spaces which allow to apply the method to answer questions for the classical case of compact manifolds. To achieve this, we developed refined analytic methods. These have been combined with tool of homotopy theory and homology to better describe the space of metrics of positive scalar curvature.

Publications

• A note on invertibility of the Dirac operator twisted with Hilbert- A-module coefficients
  Thomas Schick
  
• Adiabatic groupoid and secondary invariants in K-theory. Advances in Mathematics, 347, 940-1001.
  Zenobi, Vito Felice
  
• Singular spaces, groupoids and metrics of positive scalar curvature. Journal of Geometry and Physics, 137, 87-123.
  Piazza, Paolo & Zenobi, Vito Felice
  
• Positive Scalar Curvature due to the Cokernel of the Classifying Map. Symmetry, Integrability and Geometry: Methods and Applications.
  Schick, Thomas & Zenobi, Vito Felice
  
• Non-negative versus positive scalar curvature. Journal de Mathématiques Pures et Appliquées, 146, 218-232.
  Schick, Thomas & Wraith, David J.
  
• On an index theorem of Chang, Weinberger and Yu, Münster J. Math. 14, no. 1, 123–154.
  Thomas Schick & Mehran Seyedhosseini
  
• The adiabatic groupoid and the Higson–Roe exact sequence. Journal of Noncommutative Geometry, 15(3), 797-827.
  Zenobi, Vito Felice
  
• The Gromov–Lawson codimension 2 obstruction to positive scalar curvature and the C∗–index. Geometry & Topology, 25(2), 949-960.
  Kubota, Yosuke & Schick, Thomas
  
• Transfer maps in generalized group homology via submanifolds. Documenta Mathematica, 26(2021), 947-979.
  Nitsche, Martin; Schick, Thomas & Zeidler, Rudolf
  
• A variant of Roe algebras for spaces with cylindrical ends with applications in relative higher index theory. Journal of Noncommutative Geometry, 16(2), 595-624.
  Seyedhosseini, Mehran
  
• On positive scalar curvature bordism. Communications in Analysis and Geometry, 30(9), 2049-2058.
  Piazza, Paolo; Schick, Thomas & Zenobi, Vito Felice
  
• Mapping Analytic Surgery to Homology, Higher Rho Numbers and Metrics of Positive Scalar Curvature. Memoirs of the American Mathematical Society, 309(1562).
  Piazza, Paolo; Schick, Thomas & Zenobi, Vito Felice