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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Presentations of smooth manifolds

A presentation of a affine complex variety consists of finitely many polynomials $f_1,...,f_m$ in $\mathbb{C}[x_1,...,x_n]$. A presentation of a projective complex variety consists of finitely many ...
Ronald J. Zallman's user avatar
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Intuition for killing homology group

I am working through Hatcher's Algebraic Topology section 4.2, and landed on exercise 24, which says that an $M(G,1)$ with fundamental group $G$ exists if and only if $H_2(K(G,1); \mathbb{Z})=0$. The ...
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Hatcher 4.2.19 homotopy group of skeleton of $K(G,1)$

Let $X$ be a $K(G,1)$ which is a CW complex. We want to show that $\pi_n(X^n)$ is free (abelian when $n\geq 2$ where $X^n$ is the $n$ skeleton of $X$. I thought I could choose any model for a $K(G,1)$ ...
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Problem regarding the extension of a map from a closed subset of a locally compact separable metric space to where the target space is $K(G,n).$

$\mathbf {The \ Problem \ is}:$ Let $X$ is a locally compact, separable metric space and $A\subset X$ be closed. Assume all the cohomology groups here are Čech cohomology. Let $G$ be an abelian group ...
Rabi Kumar Chakraborty's user avatar
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Why is a finite field considered an étale Eilenberg-Maclane Space?

The last few days I read up on étale morphisms of rings / affine schemes and the étale fundamental group. I have two closely connected questions. For $K$ a field a ring-homomorphism $K \rightarrow A$ ...
Jonas Linssen's user avatar
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How to prove that $\mathbb {CP}^\infty$ represents $H^2(-; \mathbb Z)$ **without** using a cell decomposition?

Let $\mathbb P^\infty$ be the infinite-dimensional complex projective space, which is a known $K(\mathbb Z, 2)$, and let $F$ be the functor that assigns to each cell complex $X$ the set of homotopy ...
isekaijin's user avatar
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How precisely can one prescribe a map into an Eilenberg-MacLane space

Apologies in advance for the vague question. Suppose I have a homomorphism $\phi$ from the fundamental group of a surface of genus g into a free product of groups, $A\ast B$. I want to represent the ...
nl08's user avatar
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Simple explicit example of higher Eilenberg-MacLane space

Given an Abelian group $G$ and a positive integer $n$, the Eilenberg-MacLane space $K(G,n)$ is a topological space such that $\pi_n(K(G,n))=G$, while $\pi_m(K(G,n))=0$ if $m\neq n$. For $n=1$ this ...
Andrea Antinucci's user avatar
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Classifying Spaces of Matrix Lie Groups

I’m currently studying the classifying spaces of some of the matrix Lie groups. I’ve come across a post here that describes the classifying spaces for $SO(n)$, $SU(n)$, $GL(n)$, and $Sp(n)$. I’ve also ...
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Is it possible to recover the Cartan-Leray Spectral Sequence for Group Cohomology from the Leray Spectral Sequence for Sheaf Cohomology?

Let $G$ be a discrete group acting freely and cellularily on a CW-complex $X$. I am interested in the Cartan-Leray spectral sequence from Eilenberg and Cartan's Homological Algebra, Theorem XVI.8.4, ...
dryope's user avatar
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Cohomology ring as a coefficient ring

For a space $X$ and a spectrum $E$, we define $X\wedge E$ as the spectrum with $n$th space $X\wedge E_n$ and obvious connecting maps. I wonder if the following identification holds in general: $$H^n(X\...
timaeus's user avatar
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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 $f \colon (X,A)\to (Y,B)$ of CW-pairs induces maps between the Puppe sequences of ...
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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 ...
Austin Maison's user avatar
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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 $\...
Ronald J. Zallman's user avatar
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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(\...
Tongtong Liang's user avatar
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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 ...
JeCl's user avatar
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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 ...
pyridoxal_trigeminus's user avatar
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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 ...
michiganbiker898's user avatar
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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 ...
Andi Bauer's user avatar
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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 ...
user267839's user avatar
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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 ...
Irwin's user avatar
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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 $...
jingyey's user avatar
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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,...
Sumanta's user avatar
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1 answer
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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 ...
CuriousAlpaca's user avatar
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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 $BG=EG/G$. ...
CuriousAlpaca's user avatar
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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} ...
annie marie cœur's user avatar
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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 ...
Alexey Do's user avatar
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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 $A$-...
merle's user avatar
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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 ...
William's user avatar
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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 ...
Eduardo Longa's user avatar
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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 ...
Emptymind's user avatar
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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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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(\...
Partha's user avatar
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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 ...
palio's user avatar
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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,...
Kenny Lau's user avatar
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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 $\langle S | R \rangle$ ...
tdilos's user avatar
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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 ...
EBP's user avatar
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4 votes
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174 views

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 ...
Mauro's user avatar
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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 ...
Bargabbiati's user avatar
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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 $...
user438991's user avatar
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117 views

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 ...
Surojit's user avatar
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1 answer
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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-...
Takirion's user avatar
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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 $\...
Surojit's user avatar
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1 vote
1 answer
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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 ...
wonderich's user avatar
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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 ...
user09127's user avatar
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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: ...
Sergio Charles's user avatar