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

TL;DR

This paper computes low-degree cohomology groups of Eilenberg-MacLane spaces $K( ext{Z},4)$ and $K( ext{Z},5)$ using spectral sequences, providing methods and proofs relevant to topology and geometry.

Contribution

It offers a detailed method for calculating cohomology groups of $K( ext{Z},n)$ spaces, including proofs for specific cases of $n=4$ and $n=5$, enhancing understanding of their structure.

Findings

Cohomology groups of $K( ext{Z},4)$ and $K( ext{Z},5)$ are computed for degrees less than 11.

A general method for finding cohomology groups of $K( ext{Z},n)$ is provided.

Basic facts and proofs about these cohomology groups are established.

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 $K(\mathbb{Z},4).$