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Configuration spaces, multijet transversality, and the square-peg problem. (English) Zbl 1498.51016

The aim of this paper is to prove a special variant of the square-peg problem, which goes back to Toeplitz.
The variant proved in this paper can be stated as: “There is, for all \(m\), a \(C^m\)-dense set of smooth embeddings of a circle in \({\mathbb R}^k\), each of which has an odd number of square-like quadrilaterals (a quadrilateral \(abcd\) is said to be square-like if it has equal sides, \(|ab|=|bc|=|cd|=|da|\), and equal diagonals, \(|ac|=|bd|\)).”
To prove this, the authors use “a transversality ‘lifting property’ for compactified configuration spaces as an application of the multijet transversality theorem: given a submanifold of configurations of points on an embedding of a compact manifold \(M\) in Euclidean space, we can find a dense set of smooth embeddings of \(M\) for which the corresponding configuration space of points is transverse to any submanifold of the configuration space of points in Euclidean space, as long as the two submanifolds of compactified configuration space are boundary- disjoint.”

MSC:

51M05 Euclidean geometries (general) and generalizations
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
57Q65 General position and transversality
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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