Periodic automorphisms of surfaces and cobordism
Let \(F\) be an oriented closed surface equipped with an automorphism \(f\) (= an orientation preserving self-diffeomorphism of \(F\)). In particular, \(f\) is said to be periodic if there are positive integers \(n\) such that \(f^n=\text{id}_F\). Suppose that \((F,f)\) is periodic and is periodic null-cobordant, i.e. there exists a periodic map \((M,\widehat{f})\) of a 3-manifold such that \(\partial(M,\widehat{f})=(F,f)\). In this paper the author makes a study of an explicit construction of such a manifold \(M\). The first step consists in showing that a periodic map \((F,f)\) is cobordant to a trivalent map with an automorphism of a compression body and introducing a graph which corresponds to a null-cobordant trivalent map. By using the above graphs the author defines a certain manifold, called a trivalent manifold, which is a null-cobordism of a null-cobordant trivalent map, and constructs a hyperbolic structure on this manifold. In conclusion, the author proves the following theorem: If \((F,f)\) is a null-cobordant automorphism, it bounds an automorphism of an irreducible 3-manifold whose Seifert factor consists of an orientable \(I\)-bundle over a surface and whose simple factor is a trivalent manifold. In addition, by applying trivalent maps and graphs the author determines the group \(\Delta^P_{2+}(n)\) of cobordism classes of periodic maps \((F,f)\) with period \(n\). The result is \(\Delta^P_{2+}(n)\cong \mathbb{Z}^{[(n-1)/2]}\) as already determined by \textit{F. Bonahon} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 237-270 (1983; Zbl 0535.57016)]. The author shows this fact explicitly by giving the basis of this abelian group in terms of trivalent maps.
scientific article; zbMATH DE number 1018461