Links and submersions to the plane on an open 3-manifold (Q2828062)

The present paper arose from a fault in a paper by the author [Topology 34, No. 2, 383--387 (1995; Zbl 0845.57024)], detected by \textit{G. Hector} et al. [Am. J. Math. 134, No. 3, 773--825 (2012; Zbl 1257.57032)]. It deals with the question of when an oriented link \(L\) in an open \(3\)-manifold \(M\) is the fiber of some submersion to the Euclidean plane. The error is corrected by Theorems A and B, proved in the appendix. Theorem A gives a necessary and sufficient condition (1) for the above realizability, taking into account transverse orientations, namely that the locally finite homology class \([L] \in H^\infty_1(M;{\mathbb Z})\) is zero, together with a certain framing condition. Theorem B asserts that any link in an orientable open \(3\)-manifold \(M\) is the union of compact components of a fiber of a suitable submersion \(M \to {\mathbb R}^2\). The Main Theorem deals with the case of a knot \(K\) satisfying \([K]=0\) in \(H^\infty_1(M;{\mathbb Z})\): The above-mentioned condition (1) is then equivalent to \([K]=0\) in \(H_1(M;{\mathbb Z}_2)\). The classification of submersions by \textit{A. Phillips} is used in an essential way [Topology 6, 171--206 (1967; Zbl 0204.23701)].