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

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

2009-07-29

arxiv.org/abs/0907.5188v3

Geometry & Topology 14 (2010) 2047-2076

Mathematics

Geometric Topology

19 pages, pdf-figures; version 2: Bernhard Hanke joined as coauthor, the paper contains a substantial new result: non-trivial

Scientific paper

10.2140/gt.2010.14.2047

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.

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