Open Access

Detecting Whitney disks for link maps in the four-sphere

Faculty of Mathematics, National Research University Higher School of Economics, Usacheva str. 6, Moscow 119048, Russia

https://doi.org/10.1142/S0218216517500778Cited by:0 (Source: Crossref)

Abstract

It is an open problem whether Kirk’s $σ$ -invariant is the complete obstruction to a link map $f : S_{+}^{2} \cup S_{-}^{2} \to S^{4}$ being link homotopic to the trivial link. The link homotopy invariant associates to such a link map $f$ a pair $σ (f) = (σ_{+} (f), σ_{-} (f))$ , and we write $σ = (σ_{+}, σ_{-})$ . With the objective of constructing counterexamples, Li proposed a link homotopy invariant $ω = (ω_{+}, ω_{-})$ such that $ω_{\pm}$ is defined on the kernel of $σ_{\pm}$ and which also obstructs link null-homotopy. We show that, when defined, the invariant $ω_{\pm}$ is determined by $σ_{\mp}$ , and is strictly weaker. In particular, this implies that if a link map $f$ has $σ (f) = (0, 0)$ , then after a link homotopy the self-intersections of $f (S_{+}^{2})$ may be equipped with framed, immersed Whitney disks in $S^{4} ∖ f (S_{-}^{2})$ whose interiors are disjoint from $f (S_{+}^{2})$ .

Keywords:

AMSC: 57N35, 57Q45

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