Families of similar simplices inscribed in most smoothly embedded spheres

Jason Cantarella ; Elizabeth Denne ; John McCleary

Forum of Mathematics, Sigma, Tome 10 (2022)

Résumé

Let $\Delta $ denote a nondegenerate k-simplex in $\mathbb {R}^k$. The set $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices in $\mathbb {R}^k$ similar to $\Delta $ is diffeomorphic to $\operatorname {O}(k)\times [0,\infty )\times \mathbb {R}^k$, where the factor in $\operatorname {O}(k)$ is a matrix called the 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 $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in \operatorname {O}(k)$. Further, the intersection of $\operatorname {\mathrm {Sim}}(\Delta )$ 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 $\operatorname {O}(k)$ via the pose map. This gives a high-dimensional generalisation 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.

Détail

DOI : 10.1017/fms.2022.88