Linear Configurations of Complete Graphs $K_4$ and $K_5$ in ${\mathbb{R}}^3$, and Higher Dimensional Analogs

We investigate the space $C(X)$ of images of linearly embedded skeleta of simplices $X$ in ${\mathbb{R}}^n$, for two families of codimension 2 complexes, each ranging over $n$. In the first family, $X=K$ is the $(n-2)$-skeleton of the $n$-simplex. In the second family, $X=L$ is the $(n-2)$-skeleton of the $(n+1)$-simplex. The main result is that for $n>2$, $C(X)$ (for either $X=K,L$) deformation retracts to a subspace homeomorphic to the double mapping cylinder $$SO(n)\mathrm{/}{A}_{n+1}\leftarrow SO(n)\mathrm{/}{A}_n\to SO(n)\mathrm{/}{S}_n,$$ where $A_n$ is the alternating group and $S_n$ the symmetric group. The resulting fundamental group provides an example of a generalization of the braid group, which is the fundamental group of a configuration of points in the plane. This group is presented, for the case $n=3$, and its action on $F_3$ is presented.