Injectivity of the Seifert linking form for even-dimensional manifolds

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Let NN be a closed connected orientable nn-manifold with H1(N)H_1(N) torsion-free, n4n\geq 4, and nn even, and let N0N_0 denote a punctured version of NN. For an embedding f ⁣:N0R2n1f\colon N_0\to\mathbb{R}^{2n-1}, let L(f)L(f) be its Seifert linking form. Injectivity conjecture. The Seifert linking form is injective: for any two embeddings f,g ⁣:N0R2n1f,g\colon N_0\to\mathbb{R}^{2n-1}, if L(f)=L(g)L(f)=L(g), then ff and gg are isotopic. This would strengthen the classification of embeddings in the specified even-dimensional setting; the source presents the assertion as an open conjecture, while related surjectivity results are known.

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Primary source

Mikhail Fedorov, “A description of values of Seifert form for punctured n-manifolds in (2n-1)-space”, arXiv:2107.02541 (2021).

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