Topology of non-collapsed three-dimensional RCD spaces

TL;DR

This paper characterizes the topology of non-collapsed three-dimensional RCD spaces without boundary, showing they are orbifolds with specific singularities, and studies stability properties of their topological and orientability features under Gromov-Hausdorff convergence.

Contribution

It establishes that such RCD spaces are orbifolds with cone singularities over $ ext{RP}^2$, and introduces a stable notion of non-orientability and double covers in this context.

Findings

Non-collapsed RCD( K,3 ) spaces are orbifolds with cone singularities.

Non-orientability is stable under Gromov-Hausdorff convergence.

Non-orientable spaces admit stable ramified double covers that are orientable.

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.