# 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.

221
questions

3
votes

0
answers

53
views

### 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 ...

5
votes

3
answers

379
views

### 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 ...

4
votes

1
answer

168
views

### 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 & ...

5
votes

0
answers

171
views

### 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/...

3
votes

1
answer

163
views

### 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 ...

5
votes

0
answers

331
views

### 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 \...

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 ...

3
votes

0
answers

47
views

### 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 ...

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 ...

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 ...

4
votes

1
answer

350
views

### 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, ...

6
votes

1
answer

250
views

### 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 ...

5
votes

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 ...

1
vote

0
answers

129
views

### 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 ...

8
votes

2
answers

710
views

### 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 ...

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\...

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-...

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 ...

1
vote

0
answers

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 $G$-...

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 ...

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 ...

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 $x_0$...

8
votes

0
answers

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 $|...

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) \ .
...

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 ...

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 ...

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 ...

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 ...

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 $G$ ...

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 $n+1$-...

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 $K(G,1)$). ...

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 ...

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 $...

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 ...

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, ...

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})...

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 ...

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_\...

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 ...

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 ...

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 ...

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 $...

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 ...

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 ...

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 ...

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(...

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-...

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 ...

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 ...

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 ...