Cobordism of exact links (Q441084)

A \((2n-1)\)-link is a \((2n-1)\)-dimensional \((n-2)\)-connected closed oriented manifold smoothly embedded in the \((2n+1)\)-dimensional sphere. \textit{V. Blanlœil} [Ann. Fac. Sci. Toulouse, VI. Sér., Math. 7, No. 2, 185--205 (1998; Zbl 0932.57025)] obtained many important results concerning cobordisms of such odd dimensional nonspherical links. Unfortunately, some statements must be clarified. When studying knot cobordisms it is very difficult to begin with an arbitrary Seifert surface of a given link.NEWLINENEWLINEIn this paper, the authors introduce the notions of exact Seifert surfaces and exact links, and obtain similar results concerning cobordisms of exact links. A Seifert surface is said to be exact if it satisfies certain homological conditions, and a link is said to be exact if it admits an exact Seifert surface.NEWLINENEWLINEThe main results are as follows. Let \(n \geq 3\). For two exact \((2n-1)\)-links, if their Seifert forms with respect to exact Seifert surfaces are algebraically cobordant, then the exact links are cobordant. In particular, two fibered \((2n-1)\)-links are cobordant if and only if their Seifert forms with respect to their fibers are algebraically cobordant. The second result can be regarded as a correction of Théorèmes 2 et A in the above paper.