Cohomology Group of $K(\mathbb{Z},4)$ and $K(\mathbb{Z},5)$

Abstract

We are familiar with properties and structure of topological spaces. One of the powerful tools, which help us to figure out the structure of topological spaces is (Leray- Serre) spectral sequence. Although Eilenberg-Maclane space plays important roles in topology, and respectively geometry. Actually finding cohomology groups of this space can be useful for classifying spaces, and also homotopy groups structure of these groups. This paper discusses how to compute cohomology groups of Eilenberg-Maclane spaces $K(\mathbb{Z}, 4)$ and $K(\mathbb{Z}, 5)$ (cohomology degree less than $11$). Furthermore we give the method to find cohomology groups of $K(\mathbb{Z}, n)$. Some proofs are given to the basic facts about cohomology group of $K(\mathbb{Z}, 5)$ and $K(\mathbb{Z},4).$