The Study Of Some Questions With Orbifold Fundamental Group And Chen-Ruan Cohomology

Orbifolds lie at the intersection of many different arears of mathematics, including algebraic and differential geometry,topology,algebra,and string theory,among others. The study with orbifolds is one of the most interesting topic tiday. The main work of this paper is to study some questions with orbifold fundamental group and Chen-Ruan cohomology.First,we consider the relation between orbifold fundamental group and the weak Morita equivalence of symplectic orbifold groupoids.As well known, two symplectic mainfolds are Morita equivalent if and only if their fundamental groups are isomorphic. In this paper,we generalize the notions, and then prove that two symplectic orbifold groupoids are weak Morita equivalent if and only if they have isomorphic orbifold fundamental groups.The second topic in this paper is to blow up the weighted projective spaces and to seek the change of Chen-Ruan cohomology. Because weighted projective spaces are toric varieties, we figure out the universal transformation low of the toric structure while blowing up,and then the one of twisted sectors.We calculate the cohomology of each sector with traditional topological methods,while Chen-Ruan cup with deRham model, the modern method. Finally,we prove that the Chen-Ruan cohomology of blowup is the direct sum of the Chen-Ruan cohomology of the original weighted projective spaces and the one of the exceptional divisor.