Embedding spaces of split links

2022/07/01 by Rachael Boyd, Boyd, Rachael, Corey Bregman +1 · 2 citations
Computer Science · Mathematics · #20F34 (primary) #20F36 (secondary) #55P15 #55U10 #57M07 #58D10 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2207.00619

openalex publication_date 2022/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the homotopy type of the space E(L) of unparametrised embeddings of a split link L=L₁\sqcup … \sqcup L_n in ℝ³. Our main result is a simple description of the fundamental group, or motion group, of E(L), and we extend this to a description of the motion group of embeddings in S³. The main tool we build is a semi-simplicial space of separating systems, which we show 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(L_i).

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