Orientations of spines of homology balls (Q1404778)

Special spine theory, developed by Casler and Matveev (see \textit{B.G. Casler}, [Proc. Am. Math. Soc. 16, 559--566 (1965; Zbl 0129.15801)]; and \textit{S.V. Matveev}, Math. USSR, Izv. 31, 423--434 (1988); translation from Izv. Akad. Nauk SSSR, 51,1104--1116 (1987; Zbl 0676.57002)]), is a well-known method for representing 3-manifolds. It is known that any compact 3-manifold possesses a special spine and is uniquely recovered from it. In the class of special spines there is an \((M,L)\)-equivalence relation: two special spines \(P\) and \(Q\) are \((M,L)\)-equivalent if we can pass from \(P\) to \(Q\) via the moves \(M^{\pm 1}\) and \(L^{\pm 1}\). In a previous article (see \textit{A. Yu. Makovetskij}, [ Math. Notes 65, 295--301 (1999); translation from Mat. Zametki 65, 354--361(1999; Zbl 0966.57020)], the autor of this paper proved that for any two special spines \(P\) and \(Q\) of a 3-manifold \(\mathcal M^3\) there exists a special spine \(S\) of \(\mathcal M^3\) such that one can pass from \(P\) to \(S\) and from \(Q\) to \(S\) using only the moves \(M^{+1}\) and \(L^{+1}\) and remaining all the time in the class of special spines of the manifold \(\mathcal M^3\). Here the author considers special spines of 3-manifolds with an additional structure: an orientation of a spine. He shows that the moves \(M^{+1}\) and \(L^{+1}\) preserve orientability of a spine, while \(M^{-1}\) and \(L^{-1}\) do not. For spines of homology balls, a class of moves is described, allowing one to pass from a given orientation of a spine to any other orientation of the spine.