Homotopy types of spaces of submanifolds of ${\mathbb R}^n$

TL;DR

This paper determines the homotopy types of spaces of proper submanifolds in Euclidean space, extending previous results to labeled submanifolds and providing a new proof of a key theorem in the field.

Contribution

It computes the homotopy type of submanifold spaces with a smooth topology and extends these results to labeled submanifolds, offering a new proof of an existing theorem.

Findings

Homotopy type of submanifold spaces computed

Extension to labeled submanifolds achieved

New proof of Galatius--Randal-Williams theorem provided

Abstract

We compute the homotopy type of the space of proper d-dimensional submanifolds of ${\mathbb R}^n$ with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the space of submanifolds with summable labels too, and to give a new proof of the Galatius--Randal-Williams theorem on the homotopy type of their space of submanifolds