Topology of non-collapsed three-dimensional RCD spaces

Abstract

We show that non-collapsed $\text{RCD}(K,3)$ spaces without boundary are orbifolds whose topological singularities are locally finite and locally homeomorphic to cones over $\mathbb{RP}^2$, and that the topology of such spaces is stable under non-collapsed Gromov-Hausdorff convergence. We study the notion of non-orientability on these spaces as a key part of our analysis and show that the property of non-orientability (on uniformly sized balls) is stable under non-collapsed Gromov-Hausdorff convergence. Finally, we show that any non-orientable non-collapsed $\text{RCD}(K,3)$ space without boundary admits a ramified double cover which is itself an orientable non-collapsed $\text{RCD}(K,3)$ space without boundary, and that such ramified double cover is stable under non-collapsed Gromov-Hausdorff convergence.