Certain racks associated with the braid groups (Q2725542)

A rack is a non-empty set with a binary operation such that right multiplication is automorphism. This paper introduces certain racks closely related to the braid groups and the mapping class groups through chords on surfaces. The main emphasis lies on a special rack \( X_{n+1}(\Sigma),\) which is the set of isotopy classes of chords on \((\Sigma, Q_{n+1}),\) where \(\Sigma\) is a connected oriented 2-manifold and \(Q_{n+1}\) are \(n+1\) fixed points. This paper gives the presentation of \(X_{n+1}(R^2)\) and \(X_{n+1}(S^2).\) It introduces two groups, namely As\((X_{n+1}(\Sigma))\) and Op\((X_{n+1}(\Sigma)),\) where \(\Sigma\) is \(R^2\) or \(S^2.\) It discusses the centers of the associated groups, and the relationship among the two groups, the centers, the braid groups \(B_{n+1}(\Sigma)\) and the mapping class groups \(M_{n+1}(\Sigma).\)NEWLINENEWLINEFor the entire collection see [Zbl 0959.00034].