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Homotopy groups of the observer moduli space of Ricci positive metrics

Boris Botvinnik, Mark G Walsh and David J Wraith

Geometry & Topology 23 (2019) 3003–3040

The observer moduli space of Riemannian metrics is the quotient of the space (M) of all Riemannian metrics on a manifold M by the group of diffeomorphisms Diffx0(M) which fix both a basepoint x0 and the tangent space at x0. The group Diffx0(M) acts freely on (M) provided that M is connected. This offers certain advantages over the classic moduli space, which is the quotient by the full diffeomorphism group. Results due to Botvinnik, Hanke, Schick and Walsh, and Hanke, Schick and Steimle have demonstrated that the higher homotopy groups of the observer moduli space x0s>0(M) of positive scalar curvature metrics are, in many cases, nontrivial. The aim in the current paper is to establish similar results for the moduli space x0Ric>0(M) of metrics with positive Ricci curvature. In particular we show that for a given k, there are infinite-order elements in the homotopy group π4kx0Ric>0(Sn) provided the dimension n is odd and sufficiently large. In establishing this we make use of a gluing result of Perelman. We provide full details of the proof of this gluing theorem, which we believe have not appeared before in the literature. We also extend this to a family gluing theorem for Ricci positive manifolds.

positive Ricci curvature, moduli space, Riemannian metrics, Perelman gluing construction, Hatcher bundles
Mathematical Subject Classification 2010
Primary: 53C21, 53C27, 57R65, 58J05, 58J50
Secondary: 55Q52
Received: 9 January 2018
Revised: 21 December 2018
Accepted: 21 January 2019
Published: 1 December 2019
Proposed: Tobias H Colding
Seconded: Bruce Kleiner, Ralph Cohen
Boris Botvinnik
Department of Mathematics
University of Oregon
Eugene, OR
United States
Mark G Walsh
Department of Mathematics and Statistics
Maynooth University
David J Wraith
Department of Mathematics and Statistics
Maynooth University