In the paper, we give a lower bound on the genus of an amalgamation of two Heegaard splittings V1, ∪S1, W1 for M1 and V2, ∪S2, W2 for M2 along boundary components F1 and F2 under some conditions on the distances of the factor Heegaard splittings. A direct consequence is that under some circumstances, g(M) = g(M1) + g(M2) − g(F) holds, and another is that under some circumstances we have a better lower bound.

A Haken sphere $S$ in a 3-manifold $M$ with a Heegaard splitting $V\cup_F W$ is a separating 2-sphere in $M$ which intersects the Heegaard surface $F$ only in an essential circle in $F$. We describe how any two Haken spheres in the genus 2 Heegaard splitting of connected sum of two lens spaces are related, which extends Scharlemann–Thompson's corresponding result on Haken spheres in genus 2 Heegaard splitting for $S^3$.

It is a theorem of Reidemeister [4] and Singer [6] that any two Heegaard splittings of a closed 3-manifold are stably equivalent. In this short note, we give this theorem a simple proof.

A complete system (CS) [Jscr]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr] is a CS of surfaces for M, then any CS equivalent to [Jscr] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds: