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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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Is it true that $K(G \ast H, 1) = K(G,1)\vee K(H,1) $?

I know that, unlike the case of the fundamental group (where $\pi_1(X \vee Y) \cong \pi_1(X)\ast \pi_1(Y)$ at least for CW complexes, which are the spaces I care about for the purpose of this ...
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Cohomology of spectra as free modules over Steenrod Algebra

I'm studying the construction of Adams spectral sequence in Hatcher's "Algebraic Topology" chapter five; I'm in trouble with an "algebraic statement". Let $X$ be a connective CW-spectrum of finite ...
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1answer
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Is the complex cobordism spectrum, $MU$, a finite spectrum?

Is the complex cobordism spectrum, $MU$, a finite spectrum? If yes, what other examples of finite spectrums there are? Is the Eilenberg-MacLane spectrum finite? What about the connective $K$-theory $...
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Does trivial cohomology of spectra imply trivial homology?

It is known that for any spectrum $X$, $H\mathbb{Z}^*(X)=0$ implies that $H\mathbb{Z} \wedge X =0$. Also, for the case $HF_p,$ if we consider $ HF_p^*(X) =0.$ This gives $[HF _ p \wedge X , \sum^i ...
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1answer
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Homology of $K(\mathbb{Z}/n,1)$ via Fibrations.

I would like to calculate the $\mathbb{Z}$-Homology of $K(\mathbb{Z}/n,1)$ via the Fibration $$K(\mathbb{Z},1)\hookrightarrow K(\mathbb{Z},1)\twoheadrightarrow K(\mathbb{Z}/n,1)$$ and the Serre-...
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Structure of module over Eilenberg MacLane spectrum

Let $HR$ be the Eilenberg-Maclane spectrum for a commutative ring $R$ and $M$ be a module over $HR.$ Then I want to prove that $M$ is a product of Eilenberg-Mac Lane spectra. Construction: Let $\...
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1answer
87 views

How to visualize the String(n) group?

I am trying to get some more intuitions about the statement: Killing the $\pi_3$ homotopy group in Spin(n), one obtains the infinite-dimensional string group String(n). Formally: I know that here ...
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Reduced mod $p$ homology of a $p$-complete Eilenberg-MacLane space

Let $A$ be a $p$-complete abelian group for some prime $p$. Is it true that $\tilde{H}_*(K(A,2);\mathbb{F}_p)=0$? If so, how can one prove it? Please, also let me know if this hold only if we require ...
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What is the total space of the universal bundle over $B\mathbb{Q}$?

What is the total space of the universal bundle over $B\mathbb{Q}$, i.e. what is $E\mathbb{Q}$ for $B\mathbb{Q}=E\mathbb{Q}/\mathbb{Q}$ where $B\mathbb{Q}$ is the classifying space? Thoughts/Attempt: ...
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1answer
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“2”-group cohomology

In order to define the cohomology of a topological group G, we first have to introduce the concept of a classifying space. A classifying space BG is the base space of a principal G bundle EG. The EG ...
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Eilenberg–MacLane space $K(\mathbb{Z}_2,n)$

We know that the generalized classifying space / Eilenberg–MacLane space $$ B\mathbb{Z}_2=\mathbb{RP}^{\infty} $$ $$ BU(1)=\mathbb{CP}^{\infty} $$ How do one construct/derive the (infinite ...
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Brown representability and based homotopy classes

The following statement is the version of Brown's representability theorem I learned: Let $\mathbf{CW}$ be the category of based, connected CW complexes together with based homotopy classes of ...
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Weak product of Eilemberg MacLane spaces

I'm studying some homotopy theory of topological monoids from the book Algebraic Topology from a Homotopical Viewpoint. I'm trying to understand the corollary below. I'm stuck on the first claim of ...
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Classifying spaces [related to Eilenberg–MacLane] for explicit group examples

I am interested in knowing the generalized classifying spaces (related to Eilenberg–MacLane space $K(G,n)$ when $G$ is discrete) for explicit group examples ($G=$ the entries given at the top row) ...
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Eilenberg–MacLane space for explicit group examples

I am interested in knowing the explicit answers of Eilenberg–MacLane space $K(G,n)$ for explicit group examples ($G=$ the entries given at the top row) given below. Can someone fill in the Table? $$\...
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
653 views

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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1answer
81 views

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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1answer
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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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Why is $H_5(K(\Bbb Z_n,4))$ finite?

I want to see that the cohomology $H^i(K(\Bbb Z_n,4); \Bbb Z)$ starts with $\Bbb Z_n$ in degree 5. How do we know that $\operatorname{hom}(H_5(K(\Bbb Z_n,4);\Bbb Z), \Bbb Z)$ is zero? I.e. why is $H_5(...
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1answer
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Concluding $\Bbb Z$-cohomology from $\Bbb Z_2$-cohomology using Bocksteins

According to a theorem of Serre, the cohomology algebra $H^*(K(\Bbb Z,3); \Bbb Z_2)$ is a polynomial ring on elements $\iota_3, \,\operatorname{Sq}^2(\iota_3), \,\operatorname{Sq}^4\operatorname{Sq}^2(...
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1answer
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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 ...
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Eilenberg-Mac Lane functor being an embedding

So, I'm reading Rognes's paper Galois Extensions of Structured Ring Spectra and was wondering about one of his claims. In the beginning of section 4.2 he says the following: The Eilenberg–Mac Lane ...
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1answer
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When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$.

Let $X$ be a connected CW complex. Let $X_n$ fit into a commutative postnikov diagram for $X$ and let the fibrations $K(\pi_n(X),n) \hookrightarrow X_n \xrightarrow{\mathscr p} X_{n-1}$ be given. ...
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1answer
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$E_{p, 0}^2$ and $E_{0, 1}^2$ terms in sequence, in terms of homology of $K(G, 1)$, homology of $K(R, 1)$, and action of $G$ on $R$ by conjugation?

This is a followup to my previous question, reproduced here. Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R, ...
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1answer
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Extend a map over a $n+1$-cell IFF $f_nϕ$ is nullhomotopic.

Prove: If the map $f_n$ is defined on the $n$-skeleton $X_n$ and you want to define it on an $n+1$-cell with attaching map $ϕ:S_n→X_n$, then you can do so if and only if $f_nϕ$ is nullhomotopic. I ...
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1answer
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Cohomology of Eilenberg Maclane space

In a book on spectral sequences that I am reading, it is stated, without proof, that $H^i(K(\mathbb{Z},2);H^0(K(\mathbb{Z},1);\mathbb{Z}))$ is isomorphic to $\mathbb{Z}$ for even $i$ and $0$ for odd $...
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1answer
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Eilenberg-MacLane Spaces $K(G,n)$: Construction!

I'm looking for an easy construction of $K(G,n)$, Eilenberg-MacLane spaces. I know I can use the Postnikov Towers for the upper part $\pi_i(X)=0$ for $i > n$. For the lower part $\pi_i(X)=0$ for $...
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1answer
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Why is the cohomological $\delta$-functor for $H^\ast(K(\pi,1),–)$ effaceable?

Let $\pi$ be an abelian group and $\mathbf Z[\pi]$ its group ring. One obtains a cohomological $\delta$-functor from the short exact sequence of $\mathbf Z[\pi]$-modules $$0\rightarrow A\rightarrow B\...
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Homology of Eilenberg-MacLane $K(\pi,1)$ in terms of group homology and Tor [duplicate]

Let $\pi$ be a group and $K(\pi,1)$ be the associated Eilenberg-MacLane space (it has a contractible universal cover). I have proved the following isomorphism for the the chain complex with local ...
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1answer
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$[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)$

For Abelian groups $\pi$ and $\rho$, what is the easiest way to see that $$[K(\pi, n), K(\rho, n)] \cong \text{Hom}(\pi, \rho)?$$My idea is the use the natural isomorphism$$[X, K(\rho, n)] \cong \...
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1answer
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Why is $H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)$?

As the question title suggests, what is the easiest way to see that there is an isomorphism$$H^*(K(\pi, 1); A) \cong \text{Ext}^*_{\mathbb{Z}[\pi]}(\mathbb{Z}, A)?$$
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Is there an analogue of Eilenberg-Maclane spaces for homology?

Let $G$ be a group and $n$ a positive integer. A connected topological space $Y$ is called an Eilenberg–MacLane space of type $K(G, n)$, if $\pi_n(Y) \cong G$ and all other homotopy groups of $Y$ are ...
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1answer
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Eilenberg–Maclane space: nonvanishing cohomology

I would like to prove (self study) that $H^{np}(K(\mathbb{Z},n),\mathbb{Z})\neq 0$, where $n$ is even , $p\geq1$ and $K(\mathbb{Z},n)$ is the Eilenberg–Maclane space. I used the adjunction between $[ ...
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1answer
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Relation between cohomology of Eilenberg- MacLane space and product of projective space

In an article, it says that "Consider the map $\mathbf{RP}^\infty\times\cdots\times\mathbf{RP}^\infty$(n copies) $\to$ $K(\mathbf{Z}_2,n)$", I think this map is the map related to killing homotopy to ...
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1answer
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“Cohomology classes correspond to homotopy classes of maps to Eilenberg Maclane spaces” and cup product?

I read this in Hatcher. I am especially interested in knowing if the cup product can be understood from this perspective? I would appreciate a reference.
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Proof of the isomorphism $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2).$

I want to prove the following: $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2)$ via the map $[f] \mapsto f^* w_1(\gamma^1)$ where $\gamma^1$ denotes the tautological line bundle over $\mathbb{...
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2answers
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How to construct $K(\mathbb{Z}/5\mathbb{Z},1)$?

From the Wikipedia article on Eilenberg-MacLane spaces: A $K(G, n)$ can be constructed stage-by-stage, as a CW complex, starting with a wedge of n-spheres, one for each generator of the group $G$,...
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2answers
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Explicit construction of Eilenberg-Maclane spaces with n=1

Is there any examples of explicit construction of Eilenberg-Maclane spaces $K(G,1)$ for concrete groups except for G=$\mathbb Z$ and $\mathbb Z_n$? I know about general simplicial bar construction, ...
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1answer
69 views

Enumerating fiber bundles with fiber and base first Eilenberg-Maclane spaces

How to enumerate fiber bundles (maybe, only as spaces, not bundles) with fiber $K(A, 1)$ and base $K(B, 1)$? It seems to be connected with enumerating short exact sequences of form $0 \to A \to ... \...
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2answers
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$\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 ...
2
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1answer
558 views

homology of smash product of Eilenberg-Maclane spaces

Let $K_n=K(\mathbb{Z},n)$ be the Eilenberg-Maclane space. Prove: (1). $K_m\wedge K_n$ is $(m+n-1)$-connected. (2). $H_{m+n}(K_m\wedge k_n;\mathbb{Z})= H_{m+n}(K_m\times k_n;\mathbb{Z})$. How to ...
2
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1answer
204 views

Theorem on space of maps to Eilenberg-Maclane space

I was reading the classic paper of Atiyah-Bott on Yang-Mills equations on Riemann surfaces. They mention a theorem attributed to Thom saying that if $X$ is a finite CW complex, then \begin{equation} \...