The Additivity Of Genera Of Two Classes Punctured Sphere Sum Of Complicated Three-dimensional Manifold

Topology is the foundation of higher mathematics,the low-dimension topology is an important branch of topology.There is no doubt three dimensional manifold theory is a significant part of low-dimensional topology.Since 2000,from the view point that 3-manifolds can be split along incompressible surfaces,many experts and scholars from home and abroad indepth and systematically study the genus of the surface sum and bounded surface sum of complicated 3-manifolds and discuss the properties and the classification of incompressible surfaces in the surface sum and bounded surface sum of complicated 3-manifolds.Under their hard work,a lot of meaningful results are obtained.From the five and six punctured sphere sum of thickened orientable closed surfaces,by the number of the boundary components of five and six punctured sphere sum of thickened orientable closed surfaces,we deeply discussing whether the genera of five and six punctured sphere sum of thickened orientable closed surfaces and complicated 3-manifolds are additive and giving some sufficient conditions for the additivity of five and six punctured sphere sum of thickened orientable closed surfaces and complicated 3-manifolds.In addition,we discuss and proof that the genera of the n-punctured sphere sum of thickened closed surfaces and some complicated 3-manifolds are additive.In fact,by the same method,we can study the additivity of genera of the any bounded surface sum of thickened orientable closed surface and some complicated 3-manifolds.