Project B9: Cobordism categories and applications to geometric topology



Principal investigators

Participating scientists

Summary

Specific goals we wish to pursue in this project are:
  • Use cobordism categories to improve on the currently poor understanding of rational homotopy properties of spaces of diffeomorphisms of disks fixing the boundary pointwise. Use smoothing theory in reverse to deduce results on the rational characteristic classes of fiber bundles with fiber R^n.
  • Improve our understanding of tautological classes of manifold bundles; we aim at computations that refine known detection results.
  • Study the homotopical properties of the space of positive scalar curvature metrics on a high-dimensional spin manifold. Prove a better detection theorem on its homotopy and homology groups.
  • Develop index theory of elliptic operators in cobordism categories and relate it to spaces of positive scalar curvature metrics.
  • Establish and use further techniques to investigate the ko-homology of classifying spaces in order to investigate questions concerning existence and classification of positive scalar curvature metrics on smooth manifolds.
We believe that these goals have a common feature, in that they have become much more accessible thanks to recent advances in the theory of cobordism categories. We hope to combine the force of these advances with well-established techniques in each problem area.

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