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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Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?
For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...
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If $X\times Y$ is homotopy equivalent to a finite-dimensional CW Complex, are $X$ and $Y$ as well?
Is there a space $X$ that is not homotopy equivalent to a finite-dimensional CW complex for which there exists a space $Y$ such that the product space $X\times Y$ is homotopy equivalent to a finite-...
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Folk Functorial Figuring
In the CRM Proceedings & Lecture Notes Volume 50 "A Celebration of the Mathematical Legacy of Raoul Bott" Herbert Shulman writes (p. 48):
"[Bott] taught many of us to think functorially, like ...
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Can the intermediate Chern classes be expressed as Euler classes?
General question: We know that the top Chern class $c_n(\xi)$ of an $n$-dimensional complex vector bundle $\xi$ is its Euler class, while the first Chern class, $c_1(\xi)$, is the Euler class of its ...
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A step in Toda's computation of a Cotor
I am trying to understand a proof from Toda's paper Cohomology of classifying spaces. The step I am stuck on is at page 96. Here is the setup.
We work with cohomology with $\mathbb{F}_2$ coefficients. ...
13
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Is every simply connected finite complex the classifying space of a finite monoid
On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy ...
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Roadmap to homotopical group theory
I have been lurking here for a long time just enjoying the scenery from my beginner's viewpoint. I have a math.SE account but I think this question is appropriate here based on the nature of the ...
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Baum Connes conjecture and abstract isomorphism
Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...
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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{...
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Milnor's universal bundle as a colimit?
I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the ...
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Is every space a classifying space?
Despite a pretty thorough look (I think) I can’t find the answer to the following question: Is every (reasonable?) path connected space weakly equivalent to the classifying space of some topological ...
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Are the unwound thin realization and fat realization homotopy equivalent?
This is a question about a theorem (proposition 2) in the article--On the homotopy type of classifying spaces
Recall some definitions first:
Given a category $\mathcal{C}$ internal in $\...
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Tangent space, metrics etc. on simplicial sets
Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
...
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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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Classifying spaces of monoidal categories and deloopings
$\newcommand{\abs}[1]{|#1|}$The classifying space $\abs{\mathcal{C}}$ of a category $\mathcal{C}$ is the geometric realisation $\abs{\mathrm{N}_{\bullet}(\mathcal{C})}$ of its nerve $\mathrm{N}_\...
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Positive instances of the Eilenberg-Ganea conjecture with families
The original Eilenberg-Ganea conjecture, which remains unsettled, can be formulated as: any (discrete) group $G$ of cohomological dimension $\operatorname{cd}(G)=2$ has geometric dimension $\...
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Manifold approximations to $BO(3)$
We know that $BO(1) =\mathbb{R}P^\infty$ has closed, finite-dimensional manifold approximations $\mathbb{R}P^1\subset \mathbb{R}P^2\subset\cdots.$
Similarly $BO(2)$ can be approximated by closed, ...
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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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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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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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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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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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Čech nerve $C(U)$ corresponds to $BG$ in same manner as a hypercover $\mathcal{H}(U)$ to
We can via the bar construction canonically associate to a monoid $A$ the nerve $N(B A)$, a simplicial set with $N(\mathbf{B}A)_k := \times^{k+1} A $ and canonical face maps and degeneracy maps ...
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Applications of Chow rings of classifying spaces in algebraic geometry
For an algebraic group $G$, the Chow ring of its classifying space $BG$, in the sense of
Totaro, The Chow ring of a classifying space
has been computed in many cases. Is there any interesting ...
5
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Filling points to a simplex in models for EG
I have a question which is related to higher Dehn functions of groups.
I also have a group $G$ with a finite $K(G,1)$. Let us denote by $EG$ the universal cover of this complex. We choose a path-...
5
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rational cohomology of classifying spaces of complex reductive Lie groups
I am looking for a reference or an ad-hoc proof of the following fact, which seems to be known to experts: Let $\mathbf{G}$ be a complex algebraic group with maximal (algebraic) torus $\mathbf{T}$ and ...
5
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Adding morphisms to a category without changing homotopy type
I have a really tame category $C$: there are only finitely many objects $C_0$, each hom-set $C(x,y)$ for $x,y \in C_0$ has at most one element, and (aside from identity morphisms) if $C(x,y)$ is non-...
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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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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 ...
4
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Different definitions of a structure on principal bundles
Let $P\to B$ be a principal $G$-bundle and $\psi:H\to G$ a homomorphism of topological groups. A $\psi$-structure for $P$ can be defined in two different ways. I am trying to prove their equivalence.
...
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Why is any $G$-resolution a principal $G$-bundle?
In the article The Cohomology of Classifying Spaces of H-Spaces by M. Rothenberg and N. Steenrod (https://projecteuclid.org/euclid.bams/1183527356) it is stated as a theorem that if $G$ is a ...
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Two definitions of central extensions of simplicial groups
This is a cross-post from MSE.
Let $\overline W$ be a classifying space functor on $\mathrm{sGrp}$ with $G$ be a corresponding left adjoint (Kan's loop group).
Def 1 : a sequence of maps $A\to E\...
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The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group
Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...
4
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Cohomology of BG, algebraically
Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient ...
4
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Bar construction for spectra
Suppose we have a group $G$ then one can construct $BG$ and one of the essential part of the construction is the co-unit map. Now suppose we have a ring spectrum $R$, then having a co-unit splits the ...
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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 ...
3
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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)$). ...
3
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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_\...
3
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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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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 ...
3
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Classifying spaces of amalgamated topological monoids
Let $\mathsf{Top}_*$ be the category of well-based spaces and $\mathsf{TopMon}$ the category of topological monoids. Recall the James construction $\mathcal{J}:\mathsf{Top}_*\to \mathsf{TopMon}$ which ...
3
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Displaying displayed categories
Displayed categories provide a natural categorification from classifying functions to the world of functors. The spirit of the idea is to encode a functor $ F: D \to C $ using a suitable 2-functor (...
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Cobordant of 5d manifolds, and the generalization of bordisms
Some of the 5-dimensional manifolds are (co)bordant via oriented cobordism.
For example, if I understand correctly, 5-dimensional Dold manifold and Wu manifold are manifolds which are cobordant to ...
3
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Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(2)$ or $BO(2)$
Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying ...
3
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Classifying space for singular foliations
I apologize if this question is not appropriate for this site. I am aware of the n'th Haefliger groupoid, which acts as a classifying space for codimension-n foliations. Is there something similar for ...
3
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fibre sequence of classifying space
I read Steve Mitchell's Notes on principal bundles and classifying spaces (pdf).
There is a theorem: Let $G$ be any topological group, $H$ an admissible
normal subgroup. Then there is a homotopy-fibre ...
3
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Classifying spaces of algebraic groups
I am working on Fabien Morel's paper : "A1-Algebraic topology over a field" and I am a bit confused about certains properties of classifying spaces.
For example : How to show that $BSL_r \...
2
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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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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 ...
2
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0
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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 ...