Homotopy groups of the moduli space of metrics of positive scalar curvature

Abstract

We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S^n and on other manifolds are non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to an exotic smooth families of spheres due to Hatcher. As described, this works for all manifolds of suitable dimension and for the quotient of the space of metrics of positive scalar curvature by the (free) action of the subgroup of diffeomorphisms which fix a point and its tangent space. We also construct special manifolds where the quotient of the space of metrcis of positive scalar curvature by the full diffeomorphism group has non-trivial higher homotopy groups.

19 pages, pdf-figures; version 2: Bernhard Hanke joined as coauthor, the paper contains a substantial new result: non-trivial higher homotopy groups for the full moduli space of metrics of positive scalar curvature on suitable manifolds

References in corpus (4)

• A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture
• Axioms for higher torsion invariants of smooth bundles
• Bordism, rho-invariants and the Baum-Connes conjecture
• Finite Groups and Hyperbolic Manifolds

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• On the topology of moduli spaces of non-negatively curved Riemannian metrics
• Metrics with non-negative Ricci curvature on convex three-manifolds
• Non-negative versus positive scalar curvature
• On moduli spaces of positive scalar curvature metrics on highly connected manifolds
• Spaces of positive intermediate curvature metrics
• Moduli spaces of invariant metrics of positive scalar curvature on quasitoric manifolds
• Invertible Dirac operators and handle attachments on manifolds with boundary
• Positive $(p,n)$-intermediate scalar curvature and cobordism