Critical Heegaard Surface

Heegaard splitting is a classical way to study the topology of 3-manifolds.Heegaard splitting decompose a closed orientable 3-manifold M into two handlebodies which intersect exactly along their boundaries.There have been numerous developments relating topological and geometrical properties of a Heegaard splitting but there persist abundant open questions about surfaces in 3-manifolds among them one of the most significant surfaces is critical Heegaard surface which are a natural extension of incompressible surfaces and strongly irreducible surfaces.In this dissertation we defined a notion of critical surface,which can be regarded as a topological index 2 minimal surface,we define critical Heegaard surface via 1-complex whose definition is reminiscent of the curve complex.We give a condition to obtain critical Heegaard surfaces by amalgamation of a pair of strongly irreducible Heegaard splittings of compact orientable 3-manifold along their boundary components.Let Mi=Vi ∪si Wi(i=1,2)be strongly irreducible Heegaard splittings of compact orientable 3-manifold,M=V USW is an amalgamation of V1 Us1W1and V2US2 W2 along homeomorphic boundary components of(?)_W1 and(?)_W2.Suppose W2 is a compression body which contains only one non-separating disk E.If M=VUs W is not reducible and there exist essential disks Di(?)Vi(i=1,2)such that D1 is disj oint form D2 in V and W,then S is critical.