Positive flow-spines and contact 3-manifolds, II

This paper corresponds to Section 8 of arXiv:1912.05774v3 [math.GT]. The contents until Section 7 are published in Annali di Matematica Pura ed Applicata as a separate paper. In that paper, it is proved that for any positive flow-spine P of a closed, oriented 3-manifold M, there exists a unique contact structure supported by P up to isotopy. In particular, this defines a map from the set of isotopy classes of positive flow-spines of M to the set of isotopy classes of contact structures on M. In this paper, we show that this map is surjective. As a corollary, we show that any flow-spine can be deformed to a positive flow-spine by applying first and second regular moves successively.