24
votes
A K(G,1) that is not homotopy equivalent to a CW complex
The pseudocircle is such an example (see also this answer for a picture that I once made). Indeed, let $X$ be the quotient of $\mathbf R/2\mathbf Z$ where the intervals $(0,1)$ and $(1,2)$ are each ...
- 30.4k
21
votes
Accepted
Sullivan conjecture for compact Lie groups
You were right to single out Lie groups as potentially interesting. In [Topology 5 (1966), 241-243], Brayton Gray showed that the homotopy group of maps $[BS^1, S^3]$ was uncountable. Indeed, he ...
- 11.9k
18
votes
Accepted
Classifying space BG and contractable space EG
The easiest way to construct an explicit contracting homotopy
is to observe that EG is the geometric realization of the nerve of the groupoid G//G,
which has G as its set of objects and exactly one ...
- 39.3k
16
votes
Accepted
For which G is BLG weak homotopy equivalent to LBG?
[UPDATE: There were some mistakes in the first version. Here is a more careful account.]
I'll work everywhere with CGWH spaces, so I have a Cartesian closed category.
Note that $BLG$ is always path-...
- 58k
14
votes
If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?
Here is sort of a canonical example.
Consider $GL(\mathbb H)$ the group of invertible operators on a Hilbert space. By Kuipers theorem it is contractible. But $GL(\mathbb H)$ acts freely and properly ...
- 7,848
13
votes
Accepted
When does $BG \to BA$ loop to a homomorphism?
If $G$ is a compact connected topological group and $A$ is a locally compact abelian topological group, then for any map $f:BG\to BA$ the looped map $\Omega f:\Omega BG\to \Omega BA$ is homotopically ...
- 37.2k
13
votes
Accepted
About the cohomology of $BG^\delta$. Making a Lie group discrete
I will only attempt to answer your first question. The reason there is no contradiction is that it is not true for arbitrary spaces that $H^{\ast}(X;\mathbb Q) = H^{\ast}(X;\mathbb Z) \otimes \mathbb ...
- 12.3k
13
votes
Accepted
Do the two orientations on an orientable manifold $M$ uniquely witness lifts of $\tau_M: M \to B\text{O($n$)}$ to $B\text{SO($n$)}$?
Up to homotopy, there is a fibration
$$
BSO_n \to BO_n \to B\mathbb Z_2.
$$
The space of orientations of $M$ is the (homotopy) fiber of the induced map of mapping spaces
$$
\text{map}(M,BSO_n) \to \...
- 20.2k
12
votes
Accepted
Oriented Bordism Group and Un-Oriented Bordism Group of points $pt$
Unoriented cobordism: can be read off from the structure of the unoriented cobordism ring (calculated in Thom's thesis): $\Omega_6^O = (\mathbb Z/2)^3$, $\Omega_7^O = \mathbb Z/2$, $\Omega_8^O = (\...
- 7,031
12
votes
A question about cohomology of the classifying spaces of compact groups
(Edited per request to add more detail.)
Consider first the case $G = U(n)$. If $S^1 \to U(1)^n \subset U(n)$ is injective, with $H^*(BS^1) = Z[t]$, $H^*(BU(1)^n) = Z[t_1, \dots, t_n]$, $H^*(BU(n)) = ...
- 10.1k
12
votes
Reduction of structure group and classifying spaces
To begin, I should mention that the proof of this equivalence is convincingly sketched in Stephen A. Mitchell's "Notes on principal bundles and classifying spaces", see Theorem 10.1 on page ...
- 381
12
votes
Accepted
Circle action on free loop space of a classifying space
The space $EG \times_{Ad} G$ is the geometric realization of the nerve of a category, the action category of $G$ acting on itself via conjugation. Denote this $G //G$. That is the objects are elements ...
- 28.7k
11
votes
Accepted
Map from a classifying space to a stack
You're almost there! The problem is that, as you've surmised, the group $\mathrm{Aut}(x)$ does not capture enough of the geometric structure of $G$. But that's easily solved:
For every $x\in X$ we ...
- 16.8k
11
votes
Accepted
Homotopy type of a specific discrete monoid
This space is contractible, and so all of its homotopy groups are trivial.
Define two elements in $M$ by:
$$
\begin{align*}
A(x) &=
\begin{cases} 2x &\text{if }x \leq 1/2\\1 &\text{if }x \...
- 54.5k
11
votes
Accepted
Characteristic classes of non-linear sphere bundles
For many values of $n$, the answer to both questions is no. Since the fundamental groups of $BX$ and $B\mathrm{Diff}(S^n)$ are finite for $n\ge5$ (This uses that $\pi_0\mathrm{Diff}_\partial(D^n)$ is ...
- 3,024
11
votes
Accepted
Trivial group cohomology induces trivial cohomology of subgroups
For any abelian group $A$ we have a canonical isomorphism $\bigwedge^2A\to H_2(A,\mathbb{Z})$, given by the (anti-symmetric) Pontrjagin product $H_1(A,\mathbb{Z})\times H_1(A,\mathbb{Z}) \to H_2(A,\...
- 22.4k
10
votes
Formality of classifying spaces
This is an old question. But sometimes old questions get answered!
Benson, Greenlees, Formality of cochains on BG
Here is the abstract:
Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(...
- 6,612
10
votes
Classifying space as the geometric realization of the nerve of $G$ viewed as a small category
You should ignore simplicial objects at first, and just consider groupoids. In the following, you can let $G$ be a topological group such that $e\hookrightarrow G$ is a closed cofibration. All ...
- 36.8k
10
votes
Accepted
Classifying space of a non-discrete group and relationship between group homology and topological homology of Lie groups
You may want to look at the classical paper of Jack Milnor, "On the homology of Lie groups made discrete." The Friedlander-Milnor conjecture states that the map $BG^\delta \to BG$ (where $G$ ...
- 22.4k
10
votes
Pullbacks of classifying spaces
This is not true. Note that $BG$ is only well-defined up to homotopy equivalence, so the only question that makes sense is when the square
$$\begin{array}{ccc} B\big(G \underset H\times H'\big) & \...
- 30.4k
9
votes
Accepted
Interesting properties in $...\to K(\mathbb{Z}_4,1) \overset{f}{\to} K(\mathbb{Z}_2,1)\overset{g}{\to}K(\mathbb{Z}_2,2) \to ...$
Represent $p$ by the identity map $id: \mathbb{Z}_2 \to \mathbb{Z}_2$.
Then $(p\cup p)(a,b) = p(a)p(b)$ is non-zero only on the 2-chain $(1,1)$. Namely, as a polynomial mod 2, $(p\cup p)(a,b) = ab$. ...
- 4,249
9
votes
Accepted
Which objects in a topos such as the étale topos are "classifying spaces" of their fundamental groups?
Assuming the étale homotopy groups of $X$ as an object of the topos agree with the étale homotopy groups of $X$ as a scheme (in the sense of Artin-Mazur), the answer to (1.) is no, the converse badly ...
- 163k
8
votes
Accepted
What does the classifying space of a topological monoid classify?
Section 5 of Segal's Classifying spaces related to foliations shows that for discrete monoids $M$ the space $BM$ still classifies principal $M$-bundles (in a suitable sense). In Moerdijk's Classifying ...
- 12.3k
8
votes
Classification of fibrations for classifying spaces $B^2\mathbb{Z}_2$ and $BSO(3)$ or $BO(3)$
Firstly, yes, your examples are all correct. However, in example~(B) we just have $O(3)=\{\pm I\}\times SO(3)$ as groups, so your fibration is just the product of
$$BSO(3)\xrightarrow{1}BSO(3)\to 1$$...
- 58k
8
votes
Cohomology of $BE_8$ and $BSU(2)$
I believe Appendix 1. in ``Finite H-spaces and Lie Groups" by Frank Adams
shows that BE8 has 2,3 and 5-torsion. The letter from E8 at the end of this paper is also quite amusing:
....Be it therefore ...
- 5,730
8
votes
Accepted
Cohomology of BG, G non-connected Lie group, and spectral sequence relating to classifying space of connected component of the identity
Think about the case where $\pi_0(G)=\mathbb{Z}$, so $B(\pi_0(G))=S^1$, so we have a fibre bundle $BG_0\to BG\to S^1$. In this case $G$ is always a semidirect product formed using an automorphism $\...
- 58k
8
votes
Accepted
Classifying space of bundles over bundles
If I understand the question right, I think the classifying space can be described like this. Let $Map(G,BH)$ be the space of all continuous maps from $G$ to $BH$. Make the group $G$ act continuously ...
- 57.7k
8
votes
If $G$ is a topological group that contains a torsion element, then the classifying space $BG$ is infinite-dimensional?
Let $A$ be a discrete group with a torsion element. Let $EA$ be the geometric realization of the action groupoid of $A$ acting on itself by left multiplication. $EA$ is the geometric realization of a ...
- 11k
8
votes
Accepted
Odd integral Stiefel–Whitney classes in terms of even ones
$\DeclareMathOperator\Sq{Sq}$The odd classes $\beta(w_{2i})$ are part of your list of generators.
More interesting (and maybe what you wanted to ask?) is where the classes $\beta(w_{2i+1})$ are.
For ...
- 1,196
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