Universal surgery problems with trivial Lagrangian
Related to the link-slice problem, a \(k\)-component link \(L\) is a \textit{boundary link} if the components bound disjoint Seifert surfaces, equivalently, there is a homomorphism \(\phi\) from \(\pi_1(S^3 \backslash L)\) to the free group of rank \(k\). Furthermore, \(L\) is a \textit{good boundary link} if \(\ker\phi\) is perfect for some \(\phi\). On a free group, an elementary \textit{Nielsen transformation} is a replacement of one of the free generators \(g_i\) with \(g_i g_j\) or \(g_i g_j^{-1}\) (\(i\not=j\)). Geometrically, it corresponds to a handle slide. Given a link \(L\) with Seifert form, in some symplectic basis of simple closed curves on a Seifert surface \(S\), there is a half-rank Lagrangian subspace of \(H_1(S; \mathbb{Z})\), on which linking and self-linking vanish. A condition \textit{Lagrangian-trivial} is that the subspace is spanned by the curves which constitute a homotopically trivial link. In the paper under review, it is observed that some links (e.g. the Whitehead doubling of a parallelized Whitehead link) can get the Lagrangian-trivial property by Nielsen moves, contrast to a slightly stronger condition Lagrangian-trivial\(^+\). A correction of a Kirby calculus identity in another paper by the authors [Forum Math. Sigma 4, Paper No.e22, 57p. (2016; Zbl 1366.57010)] is also given. Throughout the paper under review, questions and lemmas in [\textit{J. C. Cha} et al., Math. Ann. 376, No. 3--4, 1009--1030 (2020; Zbl 1437.57001)] are referred to many times.
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