Periodic automorphisms of surfaces and cobordism

Abstract

In this paper, we work in the differential category. Unless otherwise stated, a surface is an oriented closed, possibly disconnected, surface, and an automorphism is an orientation preserving self-homeomorphism. An automorphism of a surface (/%/) is said to be null-cobordant if there is a compact oriented 3-manifold M equipped with an automorphism (M,/), such that d{M,f) = {dM,f\dM) is equal to {F,f). We call this 3-manifold M the null-cobordism for {F,f). Two automorphisms of surfaces {Fufx) and {F2,f2) are cobordant if {F^f^ { — F2,f2) is null-cobordant. The cobordism classes form a group Δ2+ whose group law is induced by disjoint sum II . Bonahon [B], Edmonds and Eving [EE] proved that Δ 2 + is isomorphic to Z z φ(Z/2Z) z . Bonahon asked the following question in his paper [B section 9] Given an automorphism of a surface {for instance presented as a product of Dehn twists), decide whether it is null-cobordant or not. For the sake of characterizing null-cobordant automorphisms, we want to know, for arbitrary null-cobordant automorphism, what kind of 3-manifold can be constructed as its null-cobordism, and we want to get an explicitly constructed family of 3-manifolds in which, for any null-cobordant automorphism, we can find a null-cobordism of this automorphism. For example, if an automorphism of a 2-torus is null-cobordant then it bounds an automorphism of a solid torus ([B]). In this paper, we show that the same kind of things are true for other surfaces: