On spaces of embeddings of circles in surfaces

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.