Non-orientable surfaces in 4-manifolds and inscribed rectangles in the Euclidean plane
Friday, 26 November, 2021 - 10:30
Does every Jordan curve in the Euclidean plane contain 4 points that form the corners of a square? What about the corners of a rectangle with a prescribed aspect ratio? The latter was recently answered by Greene-Lobb for smooth Jordan curves, but even the original question remains open in general. We follow a strategy of Hugelmeyer to relate Toeplitz's question to the following variant of a classical knot concordance question: For a given knot K in the 3-sphere S3 (or S^2xS1), what is the smallest integer among the first Betti numbers of non-orientable surface in the 4-ball B4 (or B^3xS1) with boundary K? Based on joint work with Marco Golla.
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