S_n Action On (?)_{0, N}

We discussed the natural action of the symmetry group on the modulispace m_0,n of n-marked Riemannian sphere, where the n ordered symmetry group actas permutation the marked points.This is not a freely action, The fixed points and the local group of this action reflexthe corresponding symmetric Riemannian sphere and it's symmtry.We can present m_0,ncoordinately by fixed three marked points, Bas on this we analyze the cycle structureof the member of S_n and find the cycle type of the member with fixed points, This isour theorem 2.2.Based on theorem 2.2., we found that the fixed point set of the this action is theperiodic points of some special fractional transformation.We translate this situation toa problem about the n-periodic fractional transformation of three distinct points. Afterbuilt the correspondence recursion relation, bas on the degeneration function methodof combination, We calculated out all the n-periodic fractional transformation. Based onall the above work we give the description of the fixed points of this action, This is ourtheorem 3.2.and it's corollary.The moduli space M_n of n-punctured Riemannian sphere is the quotient spaceof this action, it's singular points naturally correspondence to the fixed points of thisaction. So we can obtain the description the sigularity of M_n, this is our theorem 4.1.We can see that m_0,n is a relatively configuration space. So we can describe thetopology of m_0,n bas on the well know theory of configuration space. When expressthe generator of H₁(m_0,n), We did not apply the fixed method, but freely make relatechanges in the same homotopy class. With the help of de Rahm theory, we are ableto establish the critical lemma6.1., which greatly simplified the calculation of relatedKronecker product. We believe this method is equally effective when calculate the affec-tion of other types of groups in H^*(m_0,n). Base on above, utilizing the duality betweenH¹(m_0,n) and H₁(m_0,n), we can calculate. the transformation law for H₁(m_0,n) and H¹ (m_0,n). At the same time, due to the well known and critical Yang-Baxter relation inthe ring structure of H^* (m_0,n), in fact we have figured out the universal transformationlaw on H^*(m_0,n), which is the theorem of 6.6A and 6.6B.Applying the transformation law of S_n on H^*(m_0,n) that we've calculated out,adding well known production structure of H^*(m_0,n), We can calculate out all theS_n invariant of H^*(m_0,n) in theory. Through considering the function of generatorof S_n on the base of H¹(m_0,n), we found out that there is no S_n invariant class inH¹(m_0,n). Which is our theorem 7.1. In addition, applying the transformation law ofS_n on H^*(m_0,n) that we calculated out, in the computer we calculated out that thereis no S_n invariant class in H^*(m_0,n) until n=8. So we conjecture this result is true toany n.