# Questions tagged [eilenberg-maclane-spaces]

A topological space with a single nonzero homotopy group. Use in conjunction with (algebraic-topology), (homotopy-theory), (homology-cohomology) or (classifying-spaces).

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### group cohomology and singular cohomology for $K(G,1)$

For $K(G,1)$ space, I know that there is an isomorphism between group cohomology and singular cohomology. Is there an example for which this fact is useful (for example, it is hard to calculate group ...
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### K(G,1) space is unique up to homotopy equivalence

I am reading Hatcher's Proposition 1B.9 page 90. He is trying to prove that if $X$ is a connected CW complex, $Y$ is $K(G,1)$ then every homomorphism $\pi_1(X,x_0) \rightarrow \pi_1(Y,y_0)$ is induced ...
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### understanding construction and definition of classifying space BG

Let $G$ be a discrete group. $EG$ is defined as the $\triangle$-complex (Hatcher p.102) whose $n$-simplices are given by $[g_0,g_1,...,g_n]$ glued together in the obvious way. Then define $BG=EG/G$. ...
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### Low-degree integral cohomology of $K(\mathbb{Z}/n,2)$

Consider the Serre spectral sequence for the fibration $K(\mathbb{Z}/n,1)\rightarrow * \rightarrow K(\mathbb{Z}/n,2)$,  E^{pq}_{2}=H^{p}\bigl(K(\mathbb{Z}/n,2);H^{q}(K(\mathbb{Z}/n,1);\mathbb{Z})\...
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### First cohomology of topological spaces with non abelian coefficients

I would like to have a reference about the construction and properties of $H^1(X;G)$ for $X$ a topological space and $G$ a non-abelian group (in the spirit of expanding and clarifying the first rows ...
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### Fibration with fibre $\mathbb{C}P^\infty$

I have a fibration $F\to E\to B$ where $B$ is a nice space (a compact manifold) and the fibre is $\mathbb{C}P^\infty$, i. e. an Eilenberg–MacLane space $K(\mathbb{Z},2)$. Are there good criteria to ...
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### Is the Hurewicz isomorphism the identity for $K(G,n)$?

Let $X=K(G,n)$ be the Eilenberg-MacLane space with $\pi_n(X)=G$ and $\pi_i(X)=0$ for $i \neq n$. Is the Hurewicz isomorphism $h: \pi_n(X) \rightarrow H_n(X)$ the identity for such spaces? I mean, if ...
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### On the homology of $K(G,n)$

I am wondering (but I may be wrong) if one can say that the homology of a K(G,n) is finite dimensional for G finitely generated abelian group. I looked up on the internet for some counterexample and I ...
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### Which $BG$s are also $K(\pi,n)$s?

As a motivation for the question, note that $\mathbb{C}P^\infty$ is at the same time a $BU(1)$ and a $K(\mathbb{Z},2)$; therefore, $H^2(X,\mathbb{Z})$ classifies complex line bundles on a space $X$. ...
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### Nonabelian second relative homotopy group

Can anyone demonstrate a pair of spaces $(X,A)$ such that $\pi_1(A)$ is abelian but $\pi_2(X,A)$ is nonabelian? I have tried to consider a pair of form $(K(A,2),K(C,1))$ such that (the interesting ...
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### 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 ...
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### Is There a Term for a Manifold K(G,1)?

Is there a generally-accepted term (which has appeared "in print", in a peer-refereed, published paper) for a finitely-presented group G which has an aspherical, closed, connected, finite-dimensional ...
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### Calculating torsion in $\pi_i(S^{2n})$

Let $p$ be an odd prime and $n>1$. I want to prove that $\pi_i(S^n)$ has no $p$-torsion for $i<n+2p-3$. For odd $n$ this is Proposition 6.26 in McCleary's book (p.206). He mentions afterward ...
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### 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 ...
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### Equivalence of $[X,K(\mathbb{Z},n)]$ and $\langle X,K(\mathbb{Z},n)\rangle$ when $X$ is a CW complex but not connected
So Hatcher remarks that $[X,K(\mathbb{Z},n)]\cong\langle X,K(\mathbb{Z},n)\rangle$ when $X$ is a connected CW-complex and $n>0$. I was wondering if the result holds even if $X$ is not connected. If ...