Reducibility of flow-spines (Q1103229)

The notion of flow-spines was introduced by the author in Tokyo J. Math. 9, 505-525 (1986). A flow-spine is a standard spine of a closed 3- manifold M and is generated by a normal pair which is a pair of a non- singular flow on M and its compact local section. In this paper, we consider methods for constructing a simpler flow-spine than a given one. In general, a spine \(P_ 1\) (not necessarily a flow-spine) is thought to be simpler than \(P_ 2\) when \(P_ 1\) has fewer third singularities than \(P_ 2\). And, for example in Topology 10, 9-36 (1971; Zbl 0214.504), several methods for obtaining a spine with fewer third singularities were discovered by H. Ikeda, Yamashita and Yokoyama. However a spine obtained by applying those methods to a flow-spine is not always a flow-spine. Hence, in order to leave our discussion within an extent of flow-spines, we must consider other ``reducibility'' of flow-spines. In {\S} 4 we will give one reasonable definition of the reducibility of flow-spines. In {\S} 3 a ``simply reduced flow-spine'' is defined, and our reducibility will be considered within this sub-class of simply reduced flow-spines. And in {\S}{\S} 5-6 we will give some conditions for a flow-spine to be reducible in our sense.