Families of similar simplices inscribed in most smoothly embedded spheres

Abstract

Let $\Delta$ denote a non-degenerate $k$-simplex in $\mathbb{R}^k$. The set $\text{Sim}(\Delta)$ of simplices in $\mathbb{R}^k$ similar to $\Delta$ is diffeomorphic to $O(k)\times [0,\infty)\times \mathbb{R}^k$, where the factor in $O(k)$ is a matrix called the {\em pose}. Among $(k-1)$-spheres smoothly embedded in $\mathbb{R}^k$ and isotopic to the identity, there is a dense family of spheres, for which the subset of $\text{Sim}(\Delta)$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in O(k)$. Further, the intersection of $\text{Sim}(\Delta)$ with the configuration space of $k+1$ distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in $O(k)$ via the pose map. This gives a high dimensional generalization of classical results on inscribing families of triangles in plane curves. We use techniques established in our previous paper on the square-peg problem where we viewed inscribed simplices in spheres as transverse intersections of submanifolds of compactified configuration spaces.