University of Cambridge > > Differential Geometry and Topology Seminar > Infinite loop spaces and positive scalar curvature

Infinite loop spaces and positive scalar curvature

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  • UserOscar Randal-Williams, Cambridge
  • ClockWednesday 16 October 2013, 16:00-17:00
  • HouseMR13.

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It is well known that there are topological obstructions to a manifold $M$ admitting a Riemannian metric of everywhere positive scalar curvature (psc): if $M$ is Spin and admits a psc metric, the Lichnerowicz–Weitzenböck formula implies that the Dirac operator of $M$ is invertible, so the vanishing of the $\hat{A}$ genus is a necessary topological condition for such a manifold to admit a psc metric. If $M$ is simply-connected as well as Spin, then deep work of Gromov—Lawson, Schoen—Yau, and Stolz implies that the vanishing of (a small refinement of) the $\hat{A}$ genus is a sufficient condition for admitting a psc metric. For non-simply-connected manifolds, sufficient conditions for a manifold to admit a psc metric are not yet understood, and are a topic of much current research.

I will discuss a related but somewhat different problem: if $M$ does admit a psc metric, what is the topology of the space $\mathcal{R}$ of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows that this problem is unchanged when modifying $M$ by certain surgeries, and I will explain how this can be used along with work of Galatius and the speaker to show that the algebraic topology of $\mathcal{R}(M)$ for $M$ of dimension at least 6 is “as complicated as can possibly be detected by index-theory”. This is joint work with Boris Botvinnik and Johannes Ebert.

This talk is part of the Differential Geometry and Topology Seminar series.

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