All Questions
20
questions
4
votes
2
answers
383
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 ...
- 389
10
votes
0
answers
291
views
Compact Lie groups are rational homotopy equivalent to a product of spheres
According to [1] and [2], it is “well-known” that a compact Lie group $G$ has the same rational homology, and according to [2] is even rational homotopy equivalent, to the product $\mathbb{S}^{2m_1+1} ...
- 30.8k
4
votes
0
answers
311
views
Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$
Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified.
Is there any characterization of $\Gamma$ such that $\Gamma$...
- 1,401
4
votes
2
answers
303
views
Low dimensional integral cohomology of $BPSO(4n)$
Toda has calculated the $\mathbb{Z}/2$‐cohomology ring of $BPSO(4n+2)$, and also gave the simple exceptional calculation of the $\mathbb{Z}/2$‐cohomology of $BPSO(4)$, in
Hiroshi Toda, Cohomology of ...
- 34.8k
9
votes
1
answer
340
views
Homotopy groups $\pi_{4n-1}(SO(4n))$
There is a very natural way to define generators of $\pi_{4n-1}(SO(4n))\cong \mathbb{Z}\oplus \mathbb{Z}$ in terms of quaternions when $n=1$ and octonions when $n=2$ (see for example Tamura, On ...
- 641
4
votes
2
answers
261
views
Finite models for torsion-free lattices
Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$?
I know this to be true in many instances (e.g. ...
- 1,245
8
votes
1
answer
262
views
When does $BG \to BA$ loop to a homomorphism?
If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) ...
- 34.8k
11
votes
1
answer
219
views
Reference requests: Integral cohomology of $G_2$-homogeneous spaces
Do you know a place where the integral cohomology of $G_2$-homogeneous spaces is computed?
Great computational efforts using representation theory in order to determine the ...
- 2,620
5
votes
1
answer
240
views
Equivariant cohomology ring is an integer domain
Let $G$ be a connected compact Lie group and let $V$ be a complex $G$-representation. Denote by $\mathbb{P}(V)$ the projectivization of the vector space $V$. I would like to ask a couple of questions ...
6
votes
0
answers
157
views
Are there versions of highly connected covers of Lie groups with highly periodic homotopy groups?
There is much activity around the study of highly connected covers of Lie groups (well, of their "infinite rank" versions like $\displaystyle{\lim_{N\to\infty}} \ O(N)$, say).
Looking at the ...
- 17.8k
14
votes
1
answer
749
views
The fifth k-invariant of BSO(3)
From work of Pontryagin and Whitney, as I understand it, the homotopy 4-type of $BSO(3)$ is $K(\mathbb{Z}/2,2) \times_{K(\mathbb{Z}/4,4)} K(\mathbb{Z},4)$, where the pullback is along the maps $\...
- 34.8k
12
votes
3
answers
841
views
$A_{\infty}$-structure on closed manifold
Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...
- 1,974
3
votes
0
answers
108
views
$Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]
This is some sort of "follow-up" to the (unanswered) question posted here.
Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$
Then $\varphi $ is an automorphism of $O(2n)$, and ...
- 3,051
2
votes
0
answers
296
views
A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group
Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$.
The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence
$$
\...
- 13.8k
13
votes
0
answers
457
views
Singular cohomology of $BG$ and Borel cohomology of $G$
Stasheff, in "Continuous Cohomology of Groups and Classifying Spaces", attributes the following result to Wigner.
For $A$ a discrete abelian group and $G$ a finite dimensional locally compact, $\...
- 9,473