| In the low dimensional topology,the study of the three-dimensional manifold is the most important,and the classification problem is one of the basic directions.In many methods of three-dimensional manifolds,this paper mainly adopts the combination scheme,that is,the three-dimensional manifold is analyzed from the fusion angle of the three-dimensional manifold.The combination scheme is one of the main methods to study the three-dimensional manifold.Its characteristic is th at it can easily identify the attributes of manifold,which is a convenient research project.Heegaard structure is a very important structure of three-dimensional manifold,so it is very necessary to study it.The main idea of Heegaard decomposition theor y is: first find some suitable surface in the target manifold,then cut along them,so that the target manifold can be parsed into an indefinite number of relatively easy to identify the "simple",finally put these "simple blocks" bonding operation,restor e the original manifold.The properties and structures of target manifolds can be analyzed by studying the attributes of each compressed body and the connected cues between submanifolds.More than a decade ago,Hempel defined the Heegaard distance,and then a lot of relevant research results were put forward,which made Heegaard decomposition theory grow more space.As an important index of Heegaard decomposition theory,Heegaard genus has always been focused on it.In recent years,the change of genus in the process of three-dimensional manifold fusion has become a hot issue.In this paper,we conduct research on the nature of the surface in a 3-manifold with boundary,through the analysis of the Heegaard structure and the surface manifold manifold,using refined decomposition of Heegaard and fusion theory to provide a plurality of 3-manifolds implemented before and after the integration of Heegaard deficiency and the sufficient condition of non-degenerate.It is worth mentioning that the essential surface of a manifold plays an important role in it because its Euler number is uniquely related to the conditions we provide. |
