A metric for the space of submanifolds of Galatius and Randal-Williams

TL;DR

This paper introduces explicit metrics for the topology on the space of submanifolds defined by Galatius and Randal-Williams, comparing it with the Fell topology and utilizing the Hausdorff distance.

Contribution

It provides explicit metrics for the submanifold space topology and compares it with existing topologies, enhancing understanding of its structure.

Findings

Explicit metric derived for the Galatius-Randal-Williams topology.

Comparison between the new metric and the Fell topology.

Use of Hausdorff distance to analyze the space of submanifolds.

Abstract

Galatius and Randal-Williams defined a topology on the set of closed submanifolds of ${\mathbb R}^n$. B\"okstedt and Madsen proved that a $C^1$-version of this topology is metrizable by showing that it is regular and second countable. Using that the scanning map of a topological sheaf on manifolds is an embedding, we give an explicit metric to the space considered by B\"okstedt and Madsen. Then, we compare this topology with the Fell topology and we use the Hausdorff distance to give another metric to the space of Galatius and Randal-Williams.