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Questions tagged [classifying-spaces]

The classifying space BG of a group G classifies principal G-bundles, in that homotopy classes of maps [X, BG] are naturally identified with isomorphism classes of principal G-bundles P ⭢ X.

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Bar construction of a commutative monoid

Let $M$ be a commutative monoid. Define the bar construction $BM$ as the thin geometric realization of $[p] \mapsto M^p$. I am looking for a reference for the fact that $BM$ is again a commutative ...
qqqqqqw's user avatar
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Group completion of a monoid (braid groups)

Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups. For a discrete group $G$, we let $BG$ to be the classifying space of $G$. After reading this question, I was ...
May's user avatar
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Homotopy equivalence between certain loop spaces

I've been reading some papers carefully, with their proofs (Notations are given at the end). The following comes from "Braids, mapping class groups and categorical delooping" by Song & ...
wind's user avatar
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When is the classifying space of a group/H-space rationally equivalent to a product of Eilenberg-MacLane spaces?

The MO-question asks why the classifying space of a group is not necessarily rationally a product of Eilenberg–MacLane spaces. I am looking for classes of examples of connected topological groups/...
ThorbenK's user avatar
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Bⁿ and coherence

I understand that an internal abelian group in CW-complexes (while it is not the most extensive structure for which this can be done) produces a classifying space which itself has the structure of an ...
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Reference request: cohomology of BTOP with mod $2^m$ coefficients

I am searching for a reference with information pertaining to the $\mathbb{Z}/{2^m}$ cohomology of ${\rm{BTOP}}(n)$, for $n \geq 8$ and $m=1,2$, where $${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
Baylee Schutte's user avatar
2 votes
1 answer
179 views

combinatorical description of classifying map for principal $G$-bundle over Delta set

Let $X$ be topological realization of a (finite) Delta set, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. It's standard that the isom' classes of such principal $G$-bundles are ...
JackYo's user avatar
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Natural morphisms between stable unitary, orthogonal, and (compact) symplectic groups

I am a physicist knowing a bit of algebraic topology, and trying to answer the following question. This is perhaps not appropriate as a question on MO, in which case I apologize. I posted this ...
Hyeongmuk LIM's user avatar
2 votes
0 answers
145 views

Local-to-global philosophy for crossed modules

In the survey Groupoids and crossed objects in algebraic topology Ronald Brown made after Corollary 5.17 (p 30) an very interesting remark I not fully understand. He stated that this Corollary 5.17 ...
user267839's user avatar
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10 votes
1 answer
227 views

Classifying space of centralizer

$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let $$H^{\h G}:=\Map_G(\mathrm{E}G, H)$$ be the homotopy ...
Thomas's user avatar
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Reference for isomorphism between group cohomology and singular cohomology

Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that $$ H^i(G, ...
Aidan's user avatar
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1 answer
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Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gamma_q$?

Recall that Bott's obstruction for integrability [Bott70] asserts that: Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every ...
Ken's user avatar
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2 answers
364 views

The classifying space of any topological group is paracompact and locally contractible

I read somewhere that the classifying space $B_{G}$ for any topological group $G$ is paracompact and locally contractible. How can I prove this or can you give me a reference? Another question that I ...
Mehmet Onat's user avatar
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Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?

What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower? Namely, how do we know $$ K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)? $$ Naively -- in each step ...
zeta's user avatar
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2 answers
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Pullbacks of classifying spaces

In what follows all the groups will be discrete, not necessarly finite. Let $f:G\to H$ be a morphism of groups and $H'\to H$ be the inclusion of a subgroup. It seems to me (but correct me if I am ...
Tommaso Rossi's user avatar
6 votes
1 answer
183 views

Odd integral Stiefel–Whitney classes in terms of even ones

As computed in various places (e.g. in Brown - The Cohomology of $B\mathrm{SO}_n$ and $B\mathrm O_n$ with Integer Coefficients), the integral cohomology ring of $B\mathrm{O}(n)$ (and similarly $B\...
Matthias Ludewig's user avatar
1 vote
0 answers
96 views

Homotopy colimit construction for internal group actions of $\Omega X$ in spaces over $X$

Let $\mathbf{CW}$ be the category of based connected CW-complexes. Let $D(\mathbf{CW}/X)$ be the homotopy category of based connected CW-complexes over (based overcategory!) a based connected CW-...
user avatar
5 votes
0 answers
95 views

Classifying spaces of crossed modules

Let $\mathcal{G}$ be a strict topological $2$-group, i.e. a strict $2$-category with a single object, a space of invertible $1$-morphisms, a space of invertible $2$-morphisms and continuous structure ...
Ulrich Pennig's user avatar
1 vote
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276 views

G-local systems via the classifying stack BG

First, let $BG$ be the classifying stack of a Lie group $G$ in either Top or Diff (compactly generated topological spaces or differentiable manifolds). A map $f: X \to BG$ determines a principal $G$-...
Emerson Hemley's user avatar
8 votes
1 answer
476 views

Trivial group cohomology induces trivial cohomology of subgroups

From the answer to another question I asked (Projective representations of a finite abelian group) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian ...
Andrea Antinucci's user avatar
6 votes
1 answer
393 views

Is there a clear pattern for the degree $2n$ cohomology group of the $n$'th Eilenberg-MacLane space?

Let $G$ be a finite abelian group, and its higher classifying space is $B^nG=K(G,n)$. For $n=1$ it is well known that $H^2(B G, \mathbb{R}/\mathbb{Z}) \cong H^2(G,\mathbb{R}/\mathbb{Z})$ is isomorphic ...
Andrea Antinucci's user avatar
4 votes
0 answers
118 views

Homotopy type / Homology of the free loop space of aspherical manifolds

Let $X$ be a (connected, smooth) closed aspherical manifold. Let $LX:=Map(S^1,X)$ be the free loop space of $X$. Pick $x_0\in X$ and let $\Omega_{x_0}(X)$ be the based loop space of $X$ (based at $x_0$...
Yeah's user avatar
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181 views

Singularity category of a hypersurface associated to $M_{11}$

For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...
Dave Benson's user avatar
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2 votes
1 answer
214 views

Explicit 3-cocycle of group cohomology of dihedral group and generalization to other semidirect products

The dihedral group $D_8$ can be presented as $(\mathbb{Z}_2\times \mathbb{Z}_2)\rtimes _{\rho}\mathbb{Z}_2$, where the last factor acts on $\mathbb{Z}_2\times \mathbb{Z}_2$ as $$ \rho_1(a,b)=(b,a) \ . ...
Andrea Antinucci's user avatar
4 votes
2 answers
371 views

Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups

I have a very soft question which might be very standard in textbooks or literature but I haven't seen it. To a fixed group $G$ we may attach different topologies to make it different topological ...
XYC's user avatar
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1 vote
0 answers
125 views

Cohomology of the classifying space of a semidirect product, and some specific examples with cyclic groups

Let $G$ and $A$ be finite abelian groups and $\rho :G \rightarrow \text{Aut}(G)$ a representation of $G$. We can form the semidirect product $A\rtimes _{\rho} G$. Just to agree on the notation this is ...
Andrea Antinucci's user avatar
4 votes
0 answers
106 views

Monodromy action on homogeneous spaces

If $H$ is a Lie subgroup of $G$, then there is a fibration sequence $$ G/H\to BH\to BG. $$ By choosing a model for $EG$ we can promote this into a fibre bundle. My question is about how to understand ...
Mark Grant's user avatar
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0 votes
0 answers
83 views

Cellular structure of BSU(n)

I read somewhere that $BSU(n)$ has a cellular decomposition that consists of one 4-cell and higher dimensional cells. Can someone tell me why this is the case? In fact I am not sure if this statement ...
user48975's user avatar
3 votes
1 answer
158 views

Geometric vs cohomological dimension with families - on a proof of Lueck and Meintrup

Let $G$ be a discrete group, and let $\mathcal{F}$ be a family of subgroups of $G$ (closed under conjugation and taking subgroups). Then we may define the geometric and cohomological dimensions of $G$ ...
Mark Grant's user avatar
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3 votes
1 answer
190 views

Defining the classifying space of a group acting on a set

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts on $n+1$-...
user494312's user avatar
3 votes
0 answers
187 views

When is $BG \rightarrow BH \rightarrow BK$ a principal fibration?

Let $1 \rightarrow G \rightarrow H \rightarrow K \rightarrow 1$ be a short exact sequence of groups. Assume for simplicity that $G$ is finite, with the discrete topology (so $BG$ is a $K(G,1)$). ...
Tanny Sieben's user avatar
1 vote
1 answer
216 views

The simplicial set with a unique non-degenerate simplex in each dimension

There is a unique simplicial set with a unique non-degenerate simplex in each dimension, (updated) and such that all faces of the non-degenerate simplex are non-degenerate. Does it have a name, and ...
user494312's user avatar
5 votes
2 answers
326 views

Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$

Let G be a discrete or say for sake of simplicity a finite group. In Hatcher's book Algebraic Topology on p 89 the construction of universal bundle $EG$ carries structure of a $\Delta$-complex whose $...
user267839's user avatar
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2 votes
0 answers
399 views

About infinite loop space and $\Omega$ spectrum

Let $A$ is an topological abelian monoid. Also $\pi_0(A)$ is a group and $A$ has $CW$ structure. $BA$ is a classifying space of the topological abelian monoid. My purpose is to construct an infinite ...
Victory's user avatar
  • 121
6 votes
1 answer
438 views

Completion of the classifying stack $BG$ at a point

With the classifying stack $BG$ I have come across "the formal completion of $BG$ at point", which is denoted $\widehat{BG}$, for instance on page 7 of https://arxiv.org/pdf/1703.08578.pdf, ...
Robert Hanson's user avatar
4 votes
1 answer
138 views

CW structure for $\mathrm{BSp}(n,\mathbb{C})$ and $\mathrm{BPSp}(n,\mathbb{C})$ in degrees $4i$

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\USp{USp}\DeclareMathOperator\BSp{BSp}\DeclareMathOperator\BUSp{BUSp}\DeclareMathOperator\BPSp{BPSp}$Let $\USp(n,\mathbb{C})...
Faye3's user avatar
  • 317
1 vote
0 answers
136 views

Representablility of maps between classifying spaces

Assume that $G,H$ are two sheaves of groups (say in fpqc topology on the scheme $X$) and there is a map $G\to H$ which is representable by a closed immersion. Let us also assume that the quotient is ...
ali's user avatar
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3 votes
0 answers
122 views

Bar constructions of $A_\infty$-algebras and rectifications

Let $\mathscr{C}_1$ be the little 1-cubes operad. If $X$ is an algebra over $\mathscr{C}_1$, I can think of (at least) two ways how to deloop it: I can consider its two-sided bar construction $B_\...
FKranhold's user avatar
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7 votes
2 answers
501 views

Simplicial nerve of a topological group

Given a group $G$, one way to construct the classifying (topological) space $BG$ is to regard $G$ as a category with one object and morphisms $G$, take its nerve, and then apply the geometric ...
Ken's user avatar
  • 1,857
11 votes
1 answer
449 views

Characteristic classes of non-linear sphere bundles

It is well known that the diffeomorphism group of there sphere $\operatorname{Diff}(S^n)$ has the homotopy type of a product $X:=O(n+1)\times \operatorname{Diff}_{\partial D^n}(D^n)$ of the orthogonal ...
Jason DeVito - on hiatus's user avatar
6 votes
0 answers
229 views

"Whenever we have some interesting invariant of spaces, we try to cook up a space that represents this invariant"

In his essay Classifying Spaces Made Easy Baez writes: We've seen this trick a couple of times lately, and it's actually a big theme in homotopy theory: whenever we have some interesting invariant of ...
Arshak Aivazian's user avatar
0 votes
0 answers
108 views

Is there a classifying space for transitive Lie algebroids? If so, what is it?

Let $M$ be a manifold. The data of a Lie groupoid over $M$ is equivalent to the data of a singular foliation $M=\sqcup\mathcal{F}_i$ and, for each $i$, a map (mod homotopy) $f_i:F_i\to BG_i$ (where $...
Doron Grossman-Naples's user avatar
1 vote
0 answers
131 views

What are the obstacles for a complex to be a space of loops?

It is known that any space of loops is an H-space. So my question has two parts: What are the obstacles for a complex to be an H-space? Is there any hope to somehow reasonably classify/characterize ...
Arshak Aivazian's user avatar
2 votes
0 answers
142 views

Chain-level representability of simplicial cohomology

There's a simple construction for a classifying space $K(G,1)$ of a finite group $G$ as an infinite simplicial complex consisting of one $i$-simplex for each possible simplicial $1$-cocycle restricted ...
Andi Bauer's user avatar
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5 votes
0 answers
204 views

Do classifying spaces determine categories of principal bundles?

If $X$ is a topological space, $G$ a topological group and $E G \to BG$ a universal bundle, isomorphism classes of numerable principal $G$-bundles over $X$ are in one-to-one correspondence with ...
Blazej's user avatar
  • 334
3 votes
0 answers
205 views

What characteristic classes are there?

Can someone concisely list all characteristic classes (i.e., the cohomology classes $H^*(BX,A)$ of the corresponding classifying spaces) for the most relevant structure groups $X$ such as $O(n)$, $SO(...
Andi Bauer's user avatar
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2 votes
1 answer
162 views

Explicitly calculating the homotopy fiber of a 3-cocycle in the category of simplicial sets

I am stuck at a critical step in my master's thesis. If someome can help me out here, I will give appropriate credit. We know that the data of a 2-group $G$ can be given by a group $\tau_0(G)$ and a 3-...
Alexander Praehauser's user avatar
7 votes
2 answers
321 views

If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?

We know that if $G$ is a topological group that contains a torsion element and $G$ satisfies additional conditions such as $G$ discrete or $G$ finite-dimensional, then the classifying space $BG$ is ...
wonderich's user avatar
  • 10.4k
6 votes
2 answers
574 views

Classifying space $\text{BU}(n)$ from the differential-geometric point of view?

The classifying space of a topological group $G$ is the quotient of $EG$ (a topological space with vanishing homotopy groups) by a proper free action of $G$. The standard notation for the classifying ...
Kaleb's user avatar
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3 votes
0 answers
219 views

Cohomology ring $H^*(BG,\mathbb{Z}_2)$ for $G=\mathbb{Z}_2\ltimes B^2\mathbb{Z}^2$

$$ \newcommand{\Z}{\mathbb{Z}} $$ Consider the group $G=\Z_2\ltimes B^2\Z^2$ where $\Z_2$ acts on $\Z^2$ by interchanging the two factors of $\Z^2$, hence $\Z_2$ also acts on the Eilenberg-MacLane ...
wonderich's user avatar
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