Singular Cohomology And RO(G)-graded Cohomology Theory

In the study of Algebraic Topology, topology methods are used to solve some algebraic questions, or some classical algebraic methods are used to solve related difficult topology questions. This paper discusses the singular cohomology theory in derived category. By defining a cochain complex , a quasi-isomorphism relation between the cochain complex and the reduced singular cochain complex of n-sphere is obtained. Also an isomorphism relation between two sets, the singular cohomology groups, and the set of homomorphisms from singular cochain complex to the reduced singular cochain complex of n-sphere in the derived category is gotten. In addition, a sheaf explanation to the reduced singular cochain complex of is given by this paper. Based on the singular cohomology theory, further discussion to -graded cohomology theory is conducted, an equivariant Eilenberg-Mac Lane spectrum with finite group actions is advanced, and a module of Eilenberg-Mac Lane spectrum is constructed.Chapter 1 mainly introduces the research background of singular(co)homology theory. Besides, the main contents of this paper is given out.Chapter 2 mainly introduces some necessary basic knowledge, including the basic definitions and related knowledge in courses of topology space, homotopy theory and homology theory in algebraic topology. It plays a foundation for theoretical research in subsequent chapters.Chapter 3 discusses the singular cohomology theory from the view of category. Furthermore, an important relation between a set of homomorphisms from derived category and the singular cohomology is studied. With the presheaf functor actions on the category of singular cochain complexes, a sheaf explanation to the reduced singular cochain complex of is obtained.In chapter 4, on the basis of singular cohomology theory, the -graded cohomology theory with finite group actions is investigated. There have been some results for the trivial group actions. The results to the finite group actions are generalized, and the Eilenberg-Mac Lane spectrum is defined. In addition, a model of the Eilenberg-Mac Lane spectrum is discovered.In chapter 5, all conclusions of the paper are summarized and the directions of following study are illustrated.