The Heegaard genera of surface sums

Let M be a compact orientable 3-manifold, and let F be a separating (resp. non-separating) incompressible surface in M which cuts M into two 3-manifolds M 1 and M 2 (resp. a manifold M 1 ). Then M is called the surface sum (resp. self surface sum) of M 1 and M 2 (resp. M 1 ) along F, denoted by M = M 1 ∪ F M 2 (resp. M = M 1 ∪ F ). In this paper, we will study how g ( M ) is related to χ ( F ) , g ( M 1 ) and g ( M 2 ) when both M 1 and M 2 have high distance Heegaard splittings.