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 […]

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 […]

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) […]

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 […]

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 […]

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?

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 […]

Intereting Posts

Can we conclude that $u^{-1}+iu$ is constant?
Can the sum $1+2+3+\cdots$ be something else than $-1/12$?
“Proof” that $1-1+1-1+\cdots=\frac{1}{2}$ and related conclusion that $\zeta(2)=\frac{\pi^2}{6}.$
surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism
$H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required
Detail of the proof that the cardinality of a $\sigma$-algebra containing an infinite number of sets is uncountable
How to prove that a set of logical connectives is functionally complete(incomplete)?
Show that $k/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
Why do the elements of finite order in a nilpotent group form a subgroup?
Entropy of sum of random variables
Solving a difference equation with several parameters
Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis
Why is the Cauchy product of two convergent (but not absolutely) series either convergent or indeterminate (but does not converge to infinity)?
How to prove “eigenvalues of polynomial of matrix $A$ = polynomial of eigenvalues of matrix $A$ ”
Why is an orthogonal matrix called orthogonal?