Embedding spaces of split links

Abstract

We study the homotopy type of the space $E\left(L\right)$ of unparametrised embeddings of a split link $L=L_1\bigsqcup \dots \bigsqcup L_n$ in $R^3$. Our main result is a simple description of the fundamental group, or motion group, of $E\left(L\right)$, and we extend this to a description of the motion group of embeddings in $S^3$. The main tool we build is a semi-simplicial space of separating systems, which we show is homotopy equivalent to $E\left(L\right)$. This combinatorial object provides a gateway to studying the homotopy type of $E\left(L\right)$ via the homotopy type of the spaces $E\left(L_i\right)$.