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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1answer
32 views

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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59 views

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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Inflation of Equivariant Eilenberg-Mac Lane spectra

Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Set $Q=G/N$ -- the quotient group. Consider the quotient homomorphism $\pi: G \to G/N$. It induces a functor $\pi^\ast\colon Q\text{-...
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1answer
48 views

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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64 views

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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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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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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$[X,K(G,n)] = H^n(X;G)$ 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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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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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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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
158 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
157 views

"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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556 views

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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59 views

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
164 views

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
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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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1answer
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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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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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2answers
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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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255 views

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 ...