**31st NRW Topology Meeting**

## +++ Bochum (GERMANY) +++ Friday, May 10 and Saturday, May 11, 2019 +++

Johannes Ebert (Universität Münster)

#### The positive scalar curvature cobordism category

I report on recent joint with O. Randal--Williams concerning the cobordism category of manifolds equipped with positive scalar curvature. We carry over some of the surgery techniques develooped by Galatius and Randal--Williams to this setting. Together with an elaboration of the Gromov-Lawson surgery method, this allows us to identify the homotopy fibre of the forgetful map to the ordinary cobordism category in terms of spaces of metrics of positive scalar curvature. This allows us to conclude that a certain union of path components of the space of positive scalar curvature metrics on a sphere of dimension at least six has the homotopy type of an infinite loop space.
Furthermore, we can prove that the secondary index invariant map to real K-Theory is an infinite loop map, using a new extension of the Atiyah--Singer index theorem to cobordism categories.

*Organizers: Gerd Laures, Viktoriya Ozornova and Björn Schuster*