On invariants of link maps in dimension four (Q2833313)

A link map is a map from a union of spheres into another sphere with pairwise disjoint images. Two link maps are link homotopic if one is deformed to the other by a homotopy through link maps. In this paper, the author considers link maps of the form \(S^2 \cup S^2 \to S^4\), and the link homotopy invariants \(\sigma\), \(\omega\) and \(\tau\). Let \(f: S^2_+ \cup S^2_- \to S^4\) be a link map, and let \(f_{\pm}: S^2_{\pm} \to S^4\backslash f(S^2_{\mp})\) be the restriction of \(f\) to each component. By perturbing \(f\) if necessary, we can assume that \(f_\pm\) are self-transverse immersions, so that the self-intersections of \(f_{\pm}\) consist of transverse double points. Then, the invariant \(\sigma (f)=(\sigma_+(f), \sigma_-(f))\) is defined as a pair of integer polynomials such that \(\sigma_+(f)=\sum_p \mathrm{sign} (p) (t^{|n(p)|}-1) \in \mathbb{Z}[t]\) where \(p\) runs over double points of \(f(S^2_+)\) and \(n(p)\) is an integer representing the homology class in \(H_1(S^4\backslash f(S^2_-))\cong \mathbb{Z}\) of a loop in \(f(S^2_+)\) changing branches at \(p\), and \(\sigma_-(f)\) is defined similarly. It is known that if \(f\) is link homotopic to a link map which embeds one component, then \(\sigma(f)=(0,0)\) [\textit{P. Kirk}, Trans. Amer. Math. Soc. 319, 663--688 (1990; Zbl 0705.57014)]. The invariant \(\omega(f)=(\omega_+(f), \omega_-(f))\) valued in \(\mathbb{Z}_2 \oplus \mathbb{Z}_2\) is defined for a link map \(f\) satisfying \(\sigma(f)=(0,0)\) [\textit{G. S. Li}, Topology 36, 881--897 (1997; Zbl 0870.57034)]. The invariant \(\tau (g)\) is defined for a link map \(g: S^2 \to X^4\) satisfying \(\mu(g)=0\) for Wall's self-intersection number \(\mu\) [\textit{R. Schneiderman, P. Teichner}, Algebr. Geom. Topol. 1, 1--29 (2001; Zbl 0964.57022)]. In the present situation, the invariant \(\tau (f_+)\) is defined for a good link map \(f\) satisfying \(\sigma_+(f)=0\).NEWLINENEWLINEThe main results are as follows. The author fills a gap in the proof of the result due to Li in the paper referred to above that if \(f\) and \(g\) are homotopic link maps such that \(\sigma (f)=(0,0)=\sigma (g)\), then \(\omega(f)=\omega(g)\). Further, the author shows that for a link map \(f\) with \(\sigma_+(f)=0\), \(f\) is link homotopic to a link map \(g\) with well-defined \(\tau(g_+)\) such that \(\tau(g_+)=0\) if and only if \(\omega_+(f)=0\). The first result is shown by proving the following: If \(f\) and \(g\) are regularly homotopic good link maps, then there is a regular homotopy from \(f\) to \(g\) consisting of a sequence of abelian link homotopies that alternately fix one component. Here, a link map \(f\) is said to be abelian if \(\pi_1(S^4\backslash f(S^2_{\pm})) \cong \mathbb{Z}\), and an abelian link homotopy is a link homotopy through abelian link maps. A link map \(f\) is called good if it is abelian and \(f_\pm\) are self-transverse immersions with vanishing self-intersection number.