Questions tagged [fibre-bundles]
for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.
345
questions
6
votes
0
answers
170
views
Twisting cochain intuition
I'm currently reading through Ed Brown's paper "Twisted tensor products, I", (MR105687, Zbl 0199.58201) and I couldn't find any simple examples of twisting cochains. I understand all ...
- 161
2
votes
1
answer
183
views
Mathematical explanation for connections on gauge bundles in curved spacetime for spinors
I asked this question https://physics.stackexchange.com/questions/820924/is-tetrad-postualte-independent-of-gauge-field
Here is what I know, $g_{\mu \nu} = e^{a}_{\mu} e^{b}_{\nu} \eta_{ab}$ and the ...
- 23
1
vote
0
answers
35
views
Inequality for function on Spinor bundle
I have a function $H(x,\psi)$ defined on the spinor bundle $\mathbb{S}$ with $H_\psi$ being the continuous derivative in fiber direction having the following properties:
(H-1) There exists $0<\...
- 11
2
votes
1
answer
167
views
Cohomology class of fiber bundle
Suppose I have a fiber bundle $\pi: E\rightarrow B$, with fiber $F$, such that the Serre spectral sequence on cohomology is immediately degenerate. In other words, $H^*(E)=H^*(B)\otimes H^*(F)$.
I ...
- 2,908
4
votes
1
answer
200
views
Riemannian submersions and associated fibre bundles
My question is as follows, it is related to the chapter of Associated Fibre Bundles from [1].
Let $(X, g_X)$ and $(Y, g_Y)$ be two smooth manifolds and let $H$ be a Lie group which acts smoothly on ...
- 35
0
votes
0
answers
124
views
What do associated fibre bundles have in common?
Two fibre bundles are said associated if they have isomorphic associated principal bundles. I understand that this means they are defined by the same transition functions, but still is there some more ...
- 67
1
vote
0
answers
55
views
extendability of fibre bundles on manifolds with same dimensions
Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where
$M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where
$N'$ is also an $n$-manifold.
Suppose there is fibre ...
- 123
0
votes
1
answer
184
views
Is every covering space associated to a "principal" covering space? Can they be classified?
Every fiber bundle is associated to a principal bundle with the same structure group. A covering space is just a fiber bundle with a discrete fiber, and hence a structure group $G$ that is a (subgroup ...
- 631
2
votes
1
answer
218
views
Deriving the definition of vector bundle morphisms from Cartan geometry (a.k.a. why are they linear?)
I'm familiar with the definition of the category of vector bundles, but I'm trying to derive it from some first principles about general fiber bundles. My intuition is that vector bundles should be ...
- 631
0
votes
1
answer
135
views
Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
- 2,712
4
votes
1
answer
203
views
Co-index of a Space
I am going through this paper by Tanaka, but I got stuck at Proposition 2.4, given below.
He does not provide any proof, instead referring to Theorem 6.6 of this paper, given below.
Unfortunately, I ...
1
vote
0
answers
70
views
Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations
Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
- 189
7
votes
1
answer
203
views
Connection of principal fiber bundles — history
I wonder who was the first to discover the notion of principal fiber bundle and its connection (gauge field in the physical language). Wikipedia cites the book by Steenrod (1951). But was he the ...
- 183
4
votes
1
answer
205
views
Non compact Seifert manifolds
A Seifert manifold $M$ is a $3$-dimensional orientable smooth manifold with an effective circle action with no fixed points.
Closed connected Seifert manifolds are classified up to an equivariant ...
- 206
3
votes
1
answer
178
views
Isomorphism between tangent bundle of $S^2$ and the kernel of a bundle homomorphism
Let $S^{4n+3} \to \mathbb{H}P^n$ be the standard projection which is a fiber bundle with fiber $S^3$. By the action of $S^1$ on $S^3$ we get a fiber bundle
$$
\mathbb{C}P^1 \xrightarrow{\iota} \mathbb{...
- 195