The action of the mapping class group on metrics of positive scalar curvature

Abstract

We present a rigidity theorem for the action of the mapping class group ${\pi }_0(\mathrm{Diff}(M))$ on the space ${\mathcal{R}}^{+}(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a great number of cases, for example to simply connected $6$-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the $2$-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected $7$-dimensional $\mathrm{Spin}$-manifolds.

34 pages, 10 figures

References in corpus (2)

• The Gromov-Lawson-Chernysh surgery theorem
• Infinite loop spaces and positive scalar curvature in the presence of a fundamental group

Cited by in corpus (6)

• The positive scalar curvature cobordism category
• Bundles with non-multiplicative $\hat{A}$-genus and spaces of metrics with lower curvature bounds
• H-Space structures on spaces of metrics of positive scalar curvature
• Diffeomorphisms and positive curvature
• Spaces of Positive Scalar Curvature metrics on totally nonspin Manifolds with spin boundary
• Positive $(p,n)$-intermediate scalar curvature and cobordism