Articles of eilenberg maclane spaces

Higher homology group of Eilenberg-Maclane space is trivial

I’m trying to solve the following exercise from Algebraic Topology by Hatcher (self-study): Show that $ H_{n+1}(K(G,n);\mathbb{Z}) = 0 $ if $ n > 1 $. $ K(G,n) $ is the Eilenberg-Maclane space. I’m following a hint suggesting starting with a Moore space $ M(G,n) $. The idea I have is to kill homotopy groups […]

Maps between Eilenberg–MacLane spaces

I was re-reading an algebraic topology book the other day, and I came across the following problem: Suppose that $\pi$ and $\rho$ are abelian groups and $n\geq 1$. Determine $[K(\pi,n),K(\rho,n)]$, the set of (based) homotopy classes of maps between the corresponding Eilenberg-MacLane spaces. I believe that the following is a solution: We have two functors […]

Is the classifying space $B^nG$ the Eilenberg-MacLane space $K(G, n)$?

Question: How should we interpret and understand the classifying space $B^nG$? Is that Eilenberg-MacLane space $K(G,n)$? What one can learn about $BG$ follows the basic: A classifying space $BG$ of a topological group $G$ is the quotient of a weakly contractible space $EG$ (i.e. a topological space for which all its homotopy groups are trivial) […]

Proof that classifying spaces for discrete groups are the Eilenberg-MacLane spaces

A classifying space $BG$ of a topological group $G$ is the quotient of a weakly contractible space $EG$ by a free action of $G$. The claim is that if $G$ is a discrete group then $EG/G$ is an Eilenberg-MacLane space, so that the fundamental group of $EG/G$ is $G$ and all higher homotopy groups are […]

$\pi_n(X^n)$ free Abelian?

I have encountered a problem which states that denote $X$ as an Eilenberg-MacLane space $K(G,1)$ and is a CW complex, show that $\pi_n(X^n)$ is free Abelian for $n \geqslant 2$. However, I think I have a conceptual misunderstanding in this problem. According to cellular approximation, the map \begin{equation} f: (S^n, s_0) \rightarrow (X,x_0) \end{equation} can […]

Group structure on Eilenberg-MacLane spaces

How do we put a group structure on $K(G,n)$ that makes it a topological group? I know that $\Omega K(G,n+1)=K(G,n)$ and since we have a product of loops this makes $K(G,n)$ into a H-space. But what about being a topological group?

Why is the cohomology of a $K(G,1)$ group cohomology?

Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular cohomology of $K(G,1)$, and it is a theorem that this is isomorphic to the group cohomologies $H^*(G, \mathbb{Z})$. According to one of […]