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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Naturality of Puppe sequence

$\require{AMScd}$ On page 398 Hatcher states that the Puppe sequence has a naturality property, namely that a map $f : (X,A)\to (Y,B)$ of CW-pairs induces maps between the Puppe sequences of these ...
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If $M$ is an $R$-module, how can I show that the Eilenberg-Maclane spectrum $HM$ is an $HR$-module spectrum

Let $R$ be a commutative ring with identity, and let $M$ be an $R$-module. I want to show that the Eilenberg-Maclane spectrum $HM$ is an $HR$-module spectrum. Specifically, I want to know how to ...
3 votes
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Principal $PU(\mathcal H)$-bundle from a principal $\mathbb Z$-bundle and a principal $S^1$-bundle.

Since there is a cup product map $H^1(X;\mathbb Z) \times H^2(X;\mathbb Z) \to H^3(X;\mathbb Z)$, one should be able to produce from a principal $\mathbb Z$-bundle and a principal $S^1$-bundle a ...
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$\mathbb{C} \mathbb{P}^1$-bundles and $\mathbb{C}^\times$ bundles

The first Chern class can be obtained from the exponential sequence: $$0 \rightarrow \mathbb{Z} \rightarrow \mathbb{C} \stackrel{\text{exp}}{\rightarrow }\mathbb{C}^\times \rightarrow 0$$ Note that $\...
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How did Thom calculate $MSO(k)$ using Silber's polyhedron in his 1954's paper?

I am now reading Thom's famous paper Quelques propriétés globales des variétés différentiables. In page 48, Thom used an auxiliary space $K$, which is a principal fiber bundle with base space $K(\...
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Functorial descripiton of Low Degree Cohomology of Eilenberg-MacLane spaces

I remember stumbling across a paper in which the author(s) compute the cohomology groups $$H^{n+k}(K(a,n);\mathbb{Z})$$ of the Eilenberg-MacLane space $K(a,n)$ on the finite abelian group $A$, for low ...
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If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one.

If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one. Definition. A path-connected space whose fundamental group is isomorphic to a given group $G$ and which has ...
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Construction of Eilenberg-Maclane Space

I have just learned about the construction of the Eilenberg-Maclane space $K(A,n)$ ($n \geq 1$). I am using Hatcher's text. He briefly describes the construction on p.365. I will recap the ...
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$H_n(X)/h(\pi(X))\cong H_n(K(\pi_1(X),1))$, Hatcher section 4.2 ex. 25

Let $X$ be a connected CW-complex, with $\pi_i(X)=0$ for $1<i<n$. I want to show that $H_n( K(\pi_1(X),1) ) \cong H_n(X)/h(\pi_n(X))$, where $h$ denotes the Hurewicz map. This is exercise 25, ...
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Can we use the algebraic definition of group cohomology for $H(BG, A)$ for compact Lie groups $G$?

For any discrete group $G$, the classifying space $BG$ is a $K(G, 1)$ and can be constructed as a simplicial set with $|G|^i$ $i$-simplices. Accordingly, elements of $H^i(BG, A)$ can be represented by ...
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Twisting Products in Hatcher's Algebraic Topology

In Allen Hatcher's book Algebraic Topology, in several places is used the terminology 'twisted product'; eg on page 338: Among other things, fibrations allow one to describe, in theory at least, how ...
6 votes
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Torsion in the integral (co)homology of Eilenberg-MacLane spaces

I've had trouble finding a reference. I know that the classifying space of a group $G$ is an example of an Eilenberg-MacLane space $K(G,1)$, so that the cohomology of $K(G,1)$ is group cohomology. If ...
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Relationship among $BG, K(G,1)$ and Čech cohomology

(1) If $G$ is an abelian group. We know Čech cohomology equals singular cohomology, $\check{H^1}(X,G)=H^1_{sing}(X,G)$. In addition, $\check{H^1}(X,G)$ classifies the isomorphism classes of principal $...
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Group cohomology as cohomology of a K(G,1) CW-complex

I am trying to understand the topologycal interpretation of group chomomology. I am familiar with the algebraic definition of cohomology for a group $G$, i.e. $H^i(G; A) := \text{Ext}^i_{\mathbb{Z}G}(\...
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Two proper maps inducing same map at the level of fundamental groups are properly homotopic

I am dealing with the following problem: Let $X$ and $Y$ be two connected non-compact surfaces, possibly with boundary. Let $f_0,f_1:(X,x_0)\to (Y,y_0)$ be two proper maps with $f_{0*}=f_{1*}:\pi_1(X,...
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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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Explicit realization of Eilenberg-MacLane spaces and TOP/PL [closed]

What is the explicit realization of following Eilenberg-MacLane spaces? There are some examples I know but not for the higher $${\displaystyle K(\mathbb {Z} ,1)}=S^1$$ $${\displaystyle K(\mathbb {Z} ...
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$K(G,n) \otimes K(G',n) \to K(G \times G',n)$

Let $G,G'$ be two abelian groups and $X$ a topological space, we know that there is a natural bijection $$H^n(X,G) \cong [X, K(G,n)]$$ where $K(G,n)$ is an Eilenberg-Maclane space. Therefore, from the ...
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Homology and the reduced A-linearization of a space.

On page 16 of his book on symmetric spectra, Stefan Schwede defines the $n$-th Eilenberg-Mac Lane space $(HA)_n$ of an abelian group $A$ by means of a construction called the reduced $A$-...
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Wedge product of Eilenberg-MacLane spaces.

It is the case that the wedge product of aspherical spaces is aspherical, so in particular $K(G, 1) \vee K(H, 1) \simeq K(G*H, 1)$. I know homotopy groups are notoriously poorly behaved with respect ...
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What is the "natural homomorphism" in the definition of an *essential manifold*?

The following definition of "essential manifold" is in this wiki page: A closed $n$-manifold $M$ is called essential if its fundamental class $[M]$ defines a nonzero element in the homology ...
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Problem 22.39(a) in " Modern classical homotopy Theory " by Jeffery Strom on pg.511.

Here is the problem: Suppose $R$ is a field. (a) Show that $h^{n}(?) = Hom_{R}(H_{n}(?; R), R)$ is a cohomology theory defined on (at least) the category of finite CW complexes. I got a hint to ...
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Why Eilenberg Maclane spaces $K(G,n)$ are $(n-1)$ connected?

Why Eilenberg Maclane spaces $K(G,n)$ are $(n-1)$ connected? could anyone explain this for me please?
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1 vote
1 answer
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Embedding a pointed topological space into a Eilenberg-MacLane space of its homology group

We let $X$ be a CW complex with a unique 0-cell, given by the basepoint $x_0$, and no $k$-cells for $0 < k < n$. I want to show that $X$ can be embedded into a $K(\pi_{n}(X,x_0),n)$ space. I am ...
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An Question related to calculating $H^4(K(\mathbb{Z}/2,2))$

This is an assignment question with $3$ steps as follows: $1.$ Prove that $H^2(K(\mathbb{Z},2);\mathbb{Z}/2)\cong \mathbb{Z}/2.$ Use this generator to obtain a map $K(\mathbb{Z},2)\rightarrow K(\...
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5 votes
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Free resolution of a group $G$ and the chain complex of the universal cover of $K(G,1)$

Consider a group $G$ having a finite, free resolution $C_*(G)\to \mathbb Z$ over the group ring of $G$. I want to understand why this resolution may be viewed as the equivariant chain complex of the ...
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2 votes
1 answer
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$[X,K(G,n)] = H^n(X;G)$ for non-CW-complex X?

It is a standard fact that if $X$ is a path-connected CW-complex, then: $$[X,K(G,n)] = H^n(X;G)$$ where: $G$ is an abelian group; $n>1$ is an integer; $K(G,n)$ is the Eilenberg-Maclane space; $[X,...
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11 votes
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Specific examples of Eilenberg-Maclane spaces?

Given an integer $n$ and a group $G$ (abelian if $n \geq 2$), it's always possible to construct a $K(G,n)$ as a cell complex. The standard procedure is to choose a presentation $\langle S | R \rangle$ ...
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Confusion about Hurewicz isomorphism

I'm currently studying Algebraic Topology from Hatcher's book and from Mosher and Tangora. However, when I try to compute the homology of Eilenberg-Maclane spaces using the Hurewicz theorem, there ...
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4 votes
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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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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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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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3 votes
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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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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: ...
1 vote
1 answer
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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 ...
1 vote
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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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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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