Volume 12, issue 3 (2012)

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Vincent Blanlœil and Osamu Saeki

Algebraic & Geometric Topology 12 (2012) 1443–1455
Abstract

A $\left(2n-1\right)$–dimensional $\left(n-2\right)$–connected closed oriented manifold smoothly embedded in the sphere ${S}^{2n+1}$ is called a $\left(2n-1\right)$–link. We introduce the notion of exact links, which admit Seifert surfaces with good homological conditions. We prove that for $n\ge 3$, two exact $\left(2n-1\right)$–links are cobordant if they have such Seifert surfaces with algebraically cobordant Seifert forms. In particular, two fibered $\left(2n-1\right)$–links are cobordant if and only if their Seifert forms with respect to their fibers are algebraically cobordant. With this broad class of exact links, we thus clarify the results of Blanlœil [Ann. Fac. Sci. Toulouse Math. 7 (1998) 185–205] concerning cobordisms of odd dimensional nonspherical links.

Keywords
high dimensional knot, knot cobordism, Seifert form, algebraic cobordism, nonspherical link, fibered link
Mathematical Subject Classification 2010
Primary: 57Q45
Secondary: 57Q60, 57R65, 57R40