Generalized M-V Sequence And The Simplicial Complex Representation Of The Cech Cohomology

Algebraic topology is an important subject in basic mathematics,and Homotopy theory and homology theory are two branches of algebraic topology,which contain many important ideas and tools.This article is based on the existing Mayer-Vietoris sequence and Cech cohomology theory,further explores the application and expansion of Mayer-Vietoris sequence,and uses the simplicial method to reinterpret Cech cohomology from an algebraic perspective,It is convenient to solve the cohomology group.The first chapter mainly introduces the relevant background of the research content of this article,analyzes the research status of the Mayer-Vietoris sequence and Cech cohomology theory in domestic and foreign coherence theory,and briefly introduces the main research content of this article.The second chapter introduces the relevant theoretical knowledge required to study Mayer-Vietoris sequence and coherence on Cech from four aspects:abstract algebra,topology,homotopy and homology theory of algebraic topology,and layer theory.In the third chapter,the double complex C*(U,A*)is defined according to the structure in the Mayer-Vietoris sequence,and it is proved that the de Rham homology group is isomorphic to the double complex C*(U,A*).At the same time,according to the differential form and the Mayer-Vietoris sequence under the coverage of two open sets,the Mayer-Vietoris sequence on the manifold is extended to the case of countable multiple open sets.The fourth chapter introduces the definition of the common geometric direction of Cech cohomology and the simplicial method,and according to the structure of Cech cohomology,related functors and surface operators are constructed from the perspective of algebra,and the Cech cohomology is given.The simple complex indicates that the Cech on-chain complex is a special kind of simple complex.He also proves that Cech homology is the corresponding simple homology.The fifth chapter summarizes the research results of this article and explains the follow-up research directions.