Round twin groups on few strands

Abstract

We study the space $Q_n$ of all configurations of n ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for $n<6$ and describe its homology for $n=6,7$. For arbitrary n, we compute its first homology and its Euler characteristic.

We use three geometric approaches. On one hand, $Q_n$ is naturally defined as the complement of an arrangement of codimension-2 subtori in a real torus. On the other hand, $Q_n$ is homotopy equivalent to an explicit nonpositively curved cubical complex. Finally, $Q_n$ can also be assembled from no-3-equal manifolds of the real line.

We also observe that, up to homotopy, $Q_n$ may be identified with a subspace of the oriented double cover of the moduli space ${\bar{M}}_{0,n}\left(R\right)$ of stable real rational curves with n marked points. This gives an embedding of ${\pi }_1{Q}_n$ into the pure cactus group. As a corollary, we see that ${\pi }_1{Q}_n$ is residually nilpotent.