The Bounded Surface Sum Of 3-manifolds And Incompressible Surfaces

Combinatorial topology of three manifold is one important branch of the low dimen-sional topology. In the present paper, we mainly use the combinatorial method to analyze and discuss some hot topics in three manifold theory.Incompressible surfaces in 3-manifolds have long been an important subject and di-rection of research in the study of 3-manifolds. Three central problems in the study of incompressible surfaces in 3-manifolds are the existence, properties and classification of incompressible surfaces. By the prime decomposition theorem, Kneser-Haken finiteness theorem and JSJ decomposition theorem, closed incompressible surfaces play an impor-tant role in understanding the structure and the properties of three manifoldsIn recent years, the relation among the following three objects, the genera of the bounded surface sum of complicated 3-manifolds, the genera of the summands, and the Euler characteristic of the attaching surface, has become a very interesting problem. During the study of the additivity of genus of the bounded surface sum of complicated three manifolds, many topologists view the bounded surface sum of 3-manifolds as first take the bounded surface sum of productâ… -bundle of closed surfaces, then take the amalgamation of 3-manifolds and the bounded surface sum of productâ… -bundle of closed surfaces. By this point of view, the following three problems are closely related together, the additivity of genus of the bounded surface sum of complicated three manifolds, the additivity of genus of the bounded surface sum of product I-bundle of closed surfaces, and the classification of essential closed surfaces in the bounded surface sum of productâ… -bundle of closed surfaces. And topologists have obtained many amazing results.In the present paper, we discuss the classification of essential closed surfaces in the bounded surface sum of productâ… -bundle of closed surfaces, give some sufficient condi-tions for the additivity of the bounded surface sum of complicated three manifolds, and prove the genus is additive under the pants sum of complicated three manifolds.