String Topology for Embedding Spaces

TL;DR

This paper extends string topology to embedding spaces using operad actions, providing new algebraic structures on their homology and analyzing their relation to mapping spaces.

Contribution

It introduces an operad-based framework for string topology on embedding spaces, generalizing previous results from mapping spaces.

Findings

Established an action of the cleavage operad on embedding spaces

Derived algebraic structures on the homology of embedding spaces

Analyzed the relationship between embedding and mapping space string topology

Abstract

We extend the structure of string topology from mapping spaces to embedding spaces $Emb(S^n,M)$. This extension comes from an action of the cleavage operad, a coloured $E_{n+1}$-operad. For all values of $n \in \mathbb{N}$, this gives an action of certain Thom spectra that provides an action on the level homology. We provide an analysis of the string topology structure derived from the embedding $Emb(S^n,M) \hookrightarrow Map(S^n,M)$. In the $1$-dimensional case it specializes a morphism of BV-algebras $\mathbb{H}_*(Emb(S^1,M)) \to \mathbb{H}_*(Map(S^1,M)$.