Configuration spaces, multijet transversality, and the square-peg problem

We prove a transversality “lifting property” for compactified configuration spaces as an application of the multijet theorem: given submanifold configurations points on embedding compact manifold M in Euclidean space, we can find dense set smooth embeddings which corresponding space is transverse to any long two submanifolds are boundary-disjoint. use this setup provide attractive proof square-peg problem: there family smoothly embedded circles plane where each simple closed curve has odd number inscribed squares, and Rn square-like quadrilaterals.