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Johannes Ebert (WWU Münster): The positive scalar curvature cobordism category
May 2nd, 2017


The category in the title has as objects the closed manifolds equipped with psc metrics (i.e. Riemannian metrics of positive scalar curvature) and as morphisms cobordisms, equipped with psc metrics as well. In addition, we require appropriate tangential data and connectivity conditions on objects and morphisms. Forgetting the psc metrics yields a forgetful functor to the ordinary cobordism category, whose homotopy type is of course well-understood. Using the Gromov-Lawson surgery method and abstract homotopy theory, we are able to identify the homotopy fibre of the forgetful functor. It turns out that the loop space of this homotopy fibre is homotopy equivalent to the space of psc metrics on a single cobordism. Moreover, the surgery method in cobordism categories by Galatius and Randal-Williams carries over to the psc setting. These results lead to interesting consequences, among them that the space of psc metrics on suitable manifolds has the homotopy type of an infinite loop space, and that the action of the diffeomorphism group of the manifold on the space of psc metrics factors through cobordism theory. There is also an index-theoretic side of the story, leading to a vast generalization of our results with Botvinnik.
(j.w. with O. Randal-Williams)

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