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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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/...
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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
$${\rm{TOP}}(n) = \{f \colon \mathbb{R}^n \to \...
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combinatorical description of classifying map for principal $G$-bundle over Delta set
Let $X$ be topological realization of a (finite) Delta set, $G$ a finite group and $p: P \to X$ a principal $G$-bundle. It's standard that the isom' classes of such principal $G$-bundles are ...
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Natural morphisms between stable unitary, orthogonal, and (compact) symplectic groups
I am a physicist knowing a bit of algebraic topology, and trying to answer the following question.
This is perhaps not appropriate as a question on MO, in which case I apologize.
I posted this ...
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Local-to-global philosophy for crossed modules
In the survey Groupoids and crossed objects in algebraic topology Ronald Brown made after Corollary 5.17 (p 30) an very interesting remark I not fully understand. He stated that this
Corollary 5.17 ...
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Classifying space of centralizer
$\DeclareMathOperator\Map{Map}\newcommand{\B}{\mathrm{B}}\newcommand{\h}{\mathrm{h}}$Let $f:G\to H$ be a morphism of topological groups and let
$$H^{\h G}:=\Map_G(\mathrm{E}G, H)$$
be the homotopy ...
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Reference for isomorphism between group cohomology and singular cohomology
Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that
$$ H^i(G, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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-...
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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 ...
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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 $G$-...
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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 ...
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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 ...
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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 $x_0$...
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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 $|...
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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
$$
\rho_1(a,b)=(b,a) \ .
...
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Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
I have a very soft question which might be very standard in textbooks or literature but I haven't seen it.
To a fixed group $G$ we may attach different topologies to make it different topological ...
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Cohomology of the classifying space of a semidirect product, and some specific examples with cyclic groups
Let $G$ and $A$ be finite abelian groups and $\rho :G \rightarrow \text{Aut}(G)$ a representation of $G$. We can form the semidirect product $A\rtimes _{\rho} G$. Just to agree on the notation this is ...
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Monodromy action on homogeneous spaces
If $H$ is a Lie subgroup of $G$, then there is a fibration sequence
$$
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 ...
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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 ...
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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$ ...
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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$-...
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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)$). ...
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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 ...
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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 $...
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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 ...
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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, ...
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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})...
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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 ...
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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_\...
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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 ...
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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 ...
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"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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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(...
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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-...
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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 ...
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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 ...
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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 ...
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Classifying space of bundles over bundles
Consider a sufficiently nice topological space $X$ as well as topological groups $G$ and $H$. Consider the functor $F$ that associates to $X$ the set of all isomorphism classes of all principal $H$-...
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Geometric models for the classifying spaces of the spin and string covers of the orthogonal, symplectic, and symmetric groups
$\newcommand{\oUConf}{\widehat{\mathrm{UConf}}}\newcommand{\UConf}{\mathrm{UConf}}\newcommand{\oGr}{\widehat{\mathrm{Gr}}}\newcommand{\Gr}{\mathrm{Gr}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\Spin}{\...
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Stable homotopy type of $BG^{\wedge}_p$ in algebraic terms
In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
$\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{...