The Pants Sum of High Distance Heegaard Splittings

Abstract:

      Let $M_i$ be a compact orientable 3-manifold, and $F_i$ be an incompressible surface on $\partial M_i$, $i=1,2$. Let $f: F_1\rightarrow F_2$ be a homeomorphism, and $M=M_{1}\cup_{f} M_{2}$. In this paper, under certain assumptions for the attaching surface $F_i$, we show that if both $M_1$ and $M_2$ have Heegaard splittings with distance at least $2(g(M_1)+g(M_2))+1$, then $g(M)=g(M_1)+g(M_2)$.