All Questions
20
questions
1
vote
1
answer
169
views
Lie group framing and framed bordism
What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
- 447
4
votes
0
answers
213
views
Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?
Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...
- 3,991
4
votes
1
answer
405
views
Faithful locally free circle actions on a torus must be free?
Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is free?
I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$.
Another related question is: ...
- 375
6
votes
1
answer
579
views
Torus bundles and compact solvmanifolds
I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m ...
- 3,991
5
votes
0
answers
129
views
geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
- 3,991
9
votes
1
answer
407
views
Compact flat orientable 3 manifolds and mapping tori
There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones.
The six orientable ones are ...
- 3,991
2
votes
1
answer
428
views
Mapping torus of orientation reversing isometry of the sphere
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f_n $ be an orientation reversing isometry of the round ...
- 3,991
3
votes
0
answers
162
views
Equivalent condition for compact group to act transitively
Let $ M $ be a connected manifold. Let $ \pi_1(M) $ be the fundamental group of $ M $.
Suppose there exists a compact group $ K $ that acts transitively on $ M $. Then $ \pi_1(M) $ must have a finite ...
- 3,991
5
votes
1
answer
369
views
$\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible for $n>1$?
Question: Is the element $\alpha$ in $\pi_{2n-1}(\operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?
My understanding so far —
An $\...
- 10.4k
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
7
votes
1
answer
482
views
Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$
Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\...
- 12.8k
2
votes
0
answers
108
views
Cohomology of colored braid groupoids
Consider braids on $n$ strands and pick $n$ distinct labels $1, \dots, n$. There is a groupoid $\mathcal P_n$ whose objects are tuples $(l_1, \dots, l_n)$ of labels and whose morphisms are braids, ...
- 2,493
5
votes
1
answer
319
views
Homotopy classes of maps between special unitary Lie groups
I am sorry to mislead the notations: $SU(n)$ should be replaced by $PSU(n)$. I will reformulate it now.
We consider the special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and we ...
- 268
35
votes
3
answers
1k
views
Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
- 62.3k
6
votes
0
answers
611
views
Quotient space, a fundamental group, and higher homotopy groups 2
Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...
- 10.4k