On spaces of embeddings of circles in surfaces

TL;DR

This paper studies the topology of embedding spaces of circles in non-positively curved surfaces, classifying their components via rooted trees and automorphism groups, with a focus on geometric deformation retractions.

Contribution

It introduces a novel classification of embedding space components using rooted trees and automorphism groups, and constructs equivariant deformation retractions.

Findings

Connected components correspond to classifying spaces of automorphism groups.

A strong deformation retraction onto geometric circles is constructed.

The deformation retraction is equivariant under surface transformations.

Abstract

We consider the space of embeddings of finitely many circles that bound disks in non-positively curved surfaces. We index the connected components of this space with finite rooted trees and show that the connected components are classifying spaces of the ``braided" automorphism groups of the associated trees. An intermediate step to proving these results is to construct a strong deformation retract onto the subspace of geometric circles; moreover, this strong deformation retraction is equivariant with respect to transformations of the surface.