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 Events
Stanford University
Tue 12 Dec 2017, 3:00pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Parametrized morse theory, cobordism categories, and positive scalar curvature
ESB 4133 (PIMS Lounge)
Tue 12 Dec 2017, 3:00pm-4:00pm

Abstract

In this talk I will show how to use parametrized Morse theory to construct a map from the infinite loopspace of certain Thom spectrum, MTSpin(d), into the space of positive scalar curvature metrics on a closed spin manifold of dimension d > 4. My main novel construction is a cobordism category consisting of cobordisms equipped with a choice of Morse function, whose critical points occupy a prescribed range of degrees. My first result identifies the homotopy type of the classifying space of this topological category with the infinite loopspace of another Thom spectrum related that is related to MTSpin(d), and built out of the space of Morse jets on Euclidean space. The result can viewed as an analogue of the well known theorem of Galatius, Madsen, Tillmann, and Weiss, for manifolds equipped with the extra geometric structure of a choice of admissible Morse function.

In the second part of the talk I will show how to use this cobordism category to probe the homotopy type of the space of positive scalar curvature metrics, R^{+}(M), on a closed, spin manifold M when dim(M) > 4. This uses a parametrized version of the Gromov-Lawson construction developed by Walsh and Chernysh. Our main result detects many non-trivial homotopy groups in the space of positive scalar curvature metrics R^{+}(M). It in particular gives an alternative proof and extension of a recent breakthrough theorem of Botvinnik, Ebert, and Randal-Williams.
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Université de Sherbrooke
Thu 14 Dec 2017, 3:00pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Heegaard Floer homology as immersed curves.
ESB 4133 (PIMS Lounge)
Thu 14 Dec 2017, 3:00pm-4:00pm

Abstract

The Heegaard Floer homology of a manifold with torus boundary can be expressed as a collection of immersed curves (possibly decorated with local systems). This provides a geometric structure theorem, interpreting the algebraic invariants that arise in bordered Floer homology. From this point of view, the Heegaard Floer homology of a closed manifold obtained by gluing manifolds (with boundary) along a torus may be recovered as the Lagrangian intersection Floer homology of the associated curves. In practice, this reduces gluing problems to simple minimal intersection counts. I'll set up this machinery and describe some of the applications that follow. This is joint work with Jonathan Hanselman and Jake Rasmussen.
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