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.

20 pages, 2 figures. arXiv admin note: text overlap with arXiv:2103.07506 New version has correct term for $k$-simplex and other minor corrections

References in corpus (4)

• Inscribing a regular crosspolytope
• Configuration Spaces, Multijet Transversality, and the Square-Peg Problem
• Inscribed triangles of Jordan curves in ${\mathbb{R}}^n$
• Square-like quadrilaterals inscribed in embedded space curves

Cited by in corpus (1)

• Configuration Spaces, Multijet Transversality, and the Square-Peg Problem