Abstract
The observer moduli space of Riemannian metrics is the quotient of the space of all Riemannian metrics on a manifold by the group of diffeomorphisms which fix both a basepoint and the tangent space at . The group acts freely on provided that 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 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 of metrics with positive Ricci curvature. In particular we show that for a given , there are infinite-order elements in the homotopy group provided the dimension 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.
Citation
Boris Botvinnik. Mark G Walsh. David J Wraith. "Homotopy groups of the observer moduli space of Ricci positive metrics." Geom. Topol. 23 (6) 3003 - 3040, 2019. https://doi.org/10.2140/gt.2019.23.3003
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