The Homology Intersecting Kernels And The Homology Splitting Homomorphisms Of The Heegaard Splitting

The theory of Heegaard splitting was developed rapidly since it has been proposed. In recent years, many classic problems have been solved base on the establishment of the generalized theory of Heegaard splitting. The theory of Heegaard splitting has become a powerful tool in the study of3-manifold. The splitting homomorphism of fundamental group about the closed orientable3-manifold was put forward by Stallings. He achieved the equivalent algebraic description of three dimensional Poincare conjecture. Fengchun Lei and Jie Wu introduced the concept of intersecting kernel of fundamental group about the closed orientable3-manifold. Through the research of the intersecting kernel of fundamental group, they obtained some significative results about the Heegaard splitting of the closed orientable3-manifold.In this paper, we mainly popularize the splitting homomorphism and intersecting kernel of fundamental group to homology group. We defined the homology intersecting kernel and the homology splitting homomorphism of the Heegaard splitting. Two main theorems have been obtained as follows:Theorem1. Suppose that (M;U,V;S) is a Heegaard splitting of the closed orientable3-manifold M. Inclusion mappings i:S→U and j:S→V induce the homology homomorphisms i#:H1(S)→H1(U) andj#:H1(S)→H1(V), then H2(M)=keri#∩kerj#.Theorem2. Let S be the closed surface with genus g>0and (φ1,φ2) be the homol-ogy splitting homomorphism from H1(S) to H1×H2, then there exists the closed orientable3-manifold M with the Heegaard splitting (M;U,V;S) such that the homology splitting homo-morphism about H1(S) induced by (U, V) is equivalent to (φ1,φ2).Theorem1gives a detailed description of the relationship between two dimensional ho-mology group of M and the homology intersecting kernel of the Heegaard splitting about M. Theorem2describes that there exists the Heegaard splitting (M; U, V; S) of the closed orientable3-manifold M, such that the homology splitting homomorphism induced by (U, V) is equivalent to the homology splitting homomorphism about H1(S) by the given closed surface S.