Cohomogeneity one RCD-spaces

2024/05/15 by Diego Corro, Corro, Diego, Jesús Núñez‐Zimbrón +3
Mathematics · #Advanced Banach Space Theory #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2405.09448

Abstract

We study RCD-spaces (X,d,\mathfrakm) with group actions by isometries preserving the reference measure \mathfrakm and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that when X is non-collapsed the slices are homeomorphic to metric cones over homogeneous spaces with Ric ≥ 0. As a consequence we obtain complete topological structural results (also in the collapsed case) and a regular orbit representation theorem. Conversely, we show how to construct new RCD-spaces from a cohomogeneity one group diagram, giving a complete description of RCD-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed RCD-spaces of essential dimension at most 4.

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