The Study Of A Class Of Cohomology Rings With Complex Coefficient And Cohomology Operations

Eilenberg-MacLane space plays an essential role in Obstruction theory in Algebraic topology. And it is an important part of Homotopy theory. The most important content in Homotopy theory is to calculate the homotopy group of a topological space. Now the most useful method to calculate homotopy group is spectral sequence. What impress us is that spectral sequence is constructed by the fibration with Eilenberg-MacLane space as fiber. Cohomology operation belongs to Homotopy theory and can be used to calculate the homotopy group of topological space. Further more it has a bijection relationship with the cohomology group of the corresponding Eilenberg-MacLane space. Thus Eilenberg-MacLane space and cohomology operations are valuable to research themselves. Also they can be tools for the calculation of homtopy groups of topological space.In the fundamental part, some important definition and theorem of Algebraic topology were listed. Especially CW complex and cup product. CW complex is the most important space in Algebraic topology. Furthermore we can say that the space which we study in Algebraic topology is CW complex. Cup product gives a multiplicative structure to the direct sum of cohomology groups to make it be a ring.In this paper, my main objects are to calculate the cohomology rings of two Eilenberg-MacLane spaces K(Q,n) and K(C,n) and to classify a kind of cohomology operations in formal algebraic by using the relationship between the cohomology ring of K(C,n) and the cohomology operations of corresponding type. I calculate the cohomology ring by using the relationship between a class of spaces whose cohomology rings were known and the two Eilenberg-MacLane spaces whose cohomology rings are waiting for calculated, directed limit and inverse limit. During the steps in calculation, I get some important conclusions: The homotopy functor commutes with directed limit, the cohomolgy ring of K(C,n) with complex coefficient, and the ring structure of cohomology operations.