Algorithmic compression of surface automorphisms
An automorphism \(\theta\) : \(F\to F\) of a closed oriented surface is said to be null-cobordant if there is a compact 3-manifold M and an automorphism \(\Theta\) : \(M\to M\) such that \(\partial M=F\) and \(\Theta | F=\theta\). This definition gives rise to a cobordism group of surface automorphisms \(\Delta_ 2\). The algebraic structure of \(\Delta_ 2\) has been determined by \textit{A. Edmonds} and \textit{J. Ewing} [Math. Ann. 259, 497-504 (1982; Zbl 0468.57023)] and the reviewer [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 237-270 (1983; Zbl 0535.57016)]. However, this determination is not explicit, in the sense that it does not give a way to decide whether or not a given surface automorphism is null-cobordant. In the present paper, the authors give an algorithm to decide this. The core of the argument is to decide if a given surface automorphism extends to an automorphism of a compression body, i.e. of a 3-manifold obtained from a thickening of this surface by gluing 2-handles on one of its two sides.
cobordism group of surface automorphisms
scientific article; zbMATH DE number 3947220