High-dimensional knots with 𝜋₁≅𝑍 are determined by their complements in one more dimension than Farber’s range

Richter, William

doi:10.1090/S0002-9939-1994-1195730-8

High-dimensional knots with $\pi _ 1\cong \textbf {Z}$ are determined by their complements in one more dimension than Farber’s range
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by William Richter

Proc. Amer. Math. Soc. 120 (1994), 285-294

DOI: https://doi.org/10.1090/S0002-9939-1994-1195730-8

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Abstract:

The surgery theory of Browder, Lashof, and Shaneson reduces the study of high-dimensional smooth knots ${\Sigma ^n} \to {S^{n + 2}}$ with ${\pi _1} \cong \mathbb {Z}$ to homotopy theory. We apply Williams’s Poincaré embedding theorem to a highly connected Seifert surface. Then such knots are determined by their complements if the $\mathbb {Z}$-cover of the complement is $[(n + 2)/3]$-connected; we improve Farber’s work by one dimension.

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