ON HEEGAARD GENUS, BRIDGE GENUS AND BRAID GENUS FOR A 3-MANIFOLD

We introduce the bridge genus and the braid genus as two kinds of genera of a closed connected orientable 3-manifold, and compare them with the Heegaard genus. We show that for any closed connected orientable 3-manifold, the Heegaard genus is less than or equal to the bridge genus and the bridge genus is less than or equal to the braid genus. We consider four inequalities between these three genera and we construct a 3-manifold which satisfies each one of inequalities. Moreover, we show that there exist infinite many 3-manifolds which satisfy each one of inequalities.