Transversality for Configuration Spaces and the "Square-Peg" Theorem

TL;DR

This paper develops a transversality lifting property for configuration spaces using multijet transversality, enabling proofs of inscribed configurations like squares and regular simplices in embedded spheres.

Contribution

It introduces a new transversality lifting technique for compactified configuration spaces and applies it to prove existence of inscribed geometric configurations.

Findings

Existence of inscribed squares in dense families of circles in the plane.

Existence of inscribed regular simplices in dense families of spheres in Rn.

The transversality method simplifies proofs of geometric configuration theorems.

Abstract

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclidean space by an arbitrarily $C^1$-small variation of the initial submanifold, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide attractive proofs of the existence of a number of "special inscribed configurations" inside families of spheres embedded in $\mathbb{R}^n$ using differential topology. For instance, there is a $C^1$-dense family of smooth embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a $C^1$-dense family of smooth embedded $(n-1)$-spheres in $\mathbb{R}^n$ where each sphere has a family of inscribed regular $n$-simplices with the homology of $O(n)$.