Embedding Spaces of Split Links

Abstract

We study the homotopy type of the space E(L) of unparametrised embeddings of a split link L=L1⊔…⊔Ln in R³. Inspired by work of Brendle and Hatcher, we introduce a semi-simplicial space of separating systems and show that this is homotopy equivalent to E(L). This combinatorial object provides a gateway to studying the homotopy type of E(L) via the homotopy type of the spaces E(Li). We apply this tool to find a simple description of the fundamental group, or motion group, of E(L), and extend this to a description of the motion group of embeddings in S³.