Spectral Homology and Cohomology Theories

Abstract

This chapter continues to study homology and cohomology theories through the concept of a spectrum and constructs its associated homology and cohomology theories, called spectral homology and cohomology theories. It also introduces the concept of generalized (or extraordinary) homology and cohomology theories. Moreover, this chapter conveys the concept of an \(\Omega \)-spectrum and constructs a new \(\Omega \)-spectrum \(\underline{A}\), generalizing the Eilenberg–MacLane spectrum K(G, n). It constructs a new generalized cohomology theory \(h^*(~ ~ ; \underline{A})\) associated with this spectrum \(\underline{A}\), which generalizes the ordinary cohomology theory of Eilenberg and Steenrod. This chapter works in the category \( {\mathcal {C}}\) whose objects are pairs of spaces having the homotopy type of finite CW-complex pairs and morphisms are continuous maps of such pairs. This is a full subcategory of the category of pairs of topological spaces and maps of pairs, and this admits the construction of mapping cones. Let \({\mathcal {C}_0}\) be the category whose objects are pointed topological spaces having the homotopy type of pointed finite CW-complexes and morphisms are continuous maps of such spaces. There exist the (reduced) suspension functor \(\Sigma : {\mathcal {C}_0} \rightarrow {\mathcal {C}_0}\) and its adjoint functor \(\Omega : {\mathcal {C}_0} \rightarrow {\mathcal {C}_0}\) which is the loop functor.