# Questions tagged [fibre-bundles]

for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.

298
questions

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### Pull-back of factor of automorphy

Let $M=\mathbb C^g/ \Gamma$ be a complex tori and $E$ a be a holomorphic vector bundle of rank $r$ over $M$. Then $E$ is characterised by factor of automorphy, i.e. a holomorphic map $J:\Gamma\times\...

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0
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133
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### Fiber bundle orientability vs manifold orientability

This question seems like a pretty straightforward generalization of a result from vector bundles but its been on MSE for over a week with no answers so I'm reposting https://math.stackexchange.com/...

1
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1
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### Projective bundle is stable under twisting by a line bundle [closed]

I want to prove that "Given a bundle $E$, for any line bundle $L$ the projectivizations of $E$ and $E$ tensor $L$ are isomorphic i.e $P(E)≅P(E⊗L)$".
The statement can also be seen on the ...

15
votes

4
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912
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### Cohomology ring of mapping torus

A mapping torus, $M \rtimes_\varphi
S^1$, is a fiber bundle over $S^1$ with fiber $M$, where $\varphi$ is an element of mapping class group of $M$, describing the twist around $S^1$.
For $M=S^1\times ...

2
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0
answers

139
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### Stiefel Whitney number of a fiber bundle

I was going through this paper, and the author rights the following
The Stiefel-Whitney class of $E$ is given by $$w(E)=(1+\alpha)^{2m+1}\left\{(1+c)^{2n+1}+u_1(1+c)^{2n}+\dots+u_{2n}(1+c)+u_{2n+1}\...

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221
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### Linear $S^{2k}$-bundles over $S^{4k}$

By the classification of Dold and Whitney, linear $S^2$-bundles over $S^4$ are classified by their first Pontryagin class $p_1$, which takes the value $4\lambda$ for the bundle corresponding to $\...

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1
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145
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### Patching up two trivial fibre bundles induces homology equivalence

I was wondering to ask this question may be it's a silly one. I could not prove or disprove it.
Let $X,Y$ be smooth connected manifolds. Let $X=X_1\cup X_2$ ($X_i$'s sub-manifold of $X$) and $X_1 \cap ...

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1
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280
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### 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 ...

0
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1
answer

85
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### Non existence of a preferred Horizontal subspace on a bundle. Why not ? (Basics) [closed]

If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=S^1 \times S^1$ and $B=S^1$. Can I choose to put my finger on the identity element of the group over a ...

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130
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### Isometries of the complex projective space for the Fubini Study metric

$\DeclareMathOperator\SU{SU}$I am trying to understand a geometric proof in our mathematical quantum mechanics lecture regarding Wigner's theorem in finite dimensions. We have already shown that it ...

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184
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### What conditions are sufficient for the Leray-Hirsch theorem to be a Künneth formula?

This was originally posted on MSE, and since it didn't receive much attention, I'll try here. Let me know if this is not the appropriate place.
Given a fiber bundle $F \to E \to B$ over a paracompact ...

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2
answers

401
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### $ \mathbb{R}P^n $ bundles over the circle

Is every $ \mathbb{R}P^{2n} $ bundle over the circle trivial?
Are there exactly two $ \mathbb{R}P^{2n+1} $ bundles over the circle?
This is a cross-post of (part of) my MSE question
https://math....

3
votes

0
answers

107
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### Combinatorial fiber bundles

Triangulations (as simplicial complexes) and bi-stellar flips are a combinatorial analogue of (piece-wise linear) topological manifolds. I'm looking for a similar combinatorial analogue for fiber ...

4
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answer

171
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### Classification of functorial smooth vector fiber bundles

Let $\mathrm{Bundle}$ be the category whose objects are smooth vector fiber bundles over $\mathbb{R}$, and morphisms are fiberwise smooth linear map (that is, the base is not assumed to be fixed).
Let ...

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229
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### Fibering cobordant to projectivization of a vector bundle

I was going through this Stong's paper, I am stuck in the proof of the proposition 8.4 (given below)
I understand the proof till he derives the expression for the Steenrod square operation of the ...

4
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1
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164
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### Cobordism class of projectivization of a bundle

I was reading the book "Differentiable Periodic Maps" by P.E. Conner (1979). I am stuck at the following problem given at the end of section 21:
Let $\xi\to V^n$ be a $k$-plane bundle over a ...

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0
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108
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### $P^1$ bundle over complex tori

Let $M$ be a fiber bundle over $\mathbb C^2/ \Lambda$ whose fibers are $P^1$. $M$ is a complex manifold of dimensional 3. Is there a classfication about such $M$. And can we deduce that $M$ is a ...

4
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224
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### Is there a notion of „flatness” in point-set topology?

In algebraic geometry, flat morphisms are usually associated with the intuition that they have „continuously varying fibers”. Is there a notion in topology formalizing the same intuition? Consider for ...

3
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207
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### Reference request: functoriality of $\underline{E}$ and $\underline{B}$

For any group $G$, the universal example for proper $G$-actions, $\underline{E}G$, is a proper $G$-space such that for any other proper $G$-space $X$, there exists a map (unique up to $G$-equivariant ...

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### construction of open subsets in classifying space $BG$

Let $G$ be an arbitrary group and we
construct the classifying space $BG$ as quotient
of $EG$ where the latter one is considered in
this discussion to be constructed in natural way as $\Delta$-complex ...

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141
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### genus one curves bundle over complex torus

Let $M$ be a holomorphic fiber bundle over torus $\mathbb C^2/\Lambda$ whose fibers are curves of genus one. Is there any classification about such $M$ ?
When the fibers are given by $\mathbb P^1$ ...

4
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1
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98
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### Relationship between two bundles approaches of spontaneous symmetry breaking

I am trying understand if there is a relation between two formulations of the spontaneous symmetry breaking.
The first is provide by Derdzinski in his book "Geometry of the standard model of ...

2
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1
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234
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### Classification of disk bundle over surfaces

Are there any reference for the classification of orientable disk bundle over a closed surface? I am particularly interested in the case if the surface is $S^2,RP^2,T^2$ or the Klein bottle.
Many ...

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165
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### Trivialization of fibration by etale base change

Let $f:Y \to X$ be a smooth fibration over $\mathbb{C}$ in the sense that $X$ is a smooth, quasi-projective, connected variety and $f$ is a smooth, projective (surjective) morphism. Suppose that every ...

3
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1
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275
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### Curvature of principal bundle

Let $(P,M,G)$ be a principal bundle with connection 1-form $\omega$. In all books I have seen so far, the curvature is defined by
\begin{equation}
F:=D_{\omega}\omega \in \Omega({P,\mathfrak{g}})
\end{...

4
votes

1
answer

293
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### Surface bundles associated to a short exact sequence of groups

Suppose $S$ is a closed, connected, oriented surface of genus at least two and $G$ is any group. Suppose further that $\Gamma$ is any group that fits into the following short exact sequence:
$$ 1 \to \...

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0
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209
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### Is there a reasonable definition of an octonionic manifold?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\End{End}$
Todorov and Dubois-Violette have recently shown how to understand the structural gauge group of the standard model via octonions.
Q. Is there ...

8
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1
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247
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### Isotopies, Fiber Bundles and Selection Theorems

The following problem is a culmination of a few questions I've asked the last two months, and it's still giving me some issues. I think I know the right way to solve it, but I'm having trouble with ...

2
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0
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94
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### Minimal symmetry of a fibre bundle

Let $F \to E \to B$ be a topological fibre bundle with fibre $F$ and base $B$. It can be characterized by a map $B \to BAut(F)$. If it can also be characterized as a map $B \to BG$ (or say $G$ is a ...

5
votes

1
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305
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### $1$-cocycle associated to universal $G$-bundle $EG \to BG$

Let $G$ be a (topological) group whose identity element $e_G$ is
a nondegenerated basepoint (e.g. if $G$ is a Lie group). Then that's a
known fact that there is for every 'nice' enough topological ...

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0
answers

64
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### Existence of global section, étale map and totally disconnected space

I am trying to show the following result :
Let $Y$ be a totally disconnected space and compact space, $X$ a locally compact space and $p:Y\to X$ a surjective local homeomorphism. Then, there exist ...

8
votes

3
answers

696
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### Spectral sequences in algebraic topology [duplicate]

What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a ...

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2
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439
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### Conditions under which the preimage of a submanifold in nontrivial in homology

Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-...

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### Is the composition of a finite branched cover and a non-isotrivial Riemann surface bundle still non-isotrivial

Given $E\to B$ a non-isotrivial (compact) Riemann surface-bundle (of genus $g>1$) between two complex manifolds and $E'\to E$ is a finite branched cover. Then is the composition map $E'\to E\to B$ ...

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0
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127
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### Moduli space of genus $g$ curves ${\mathcal{M}_g}$ irreducible by 'Monodromy argument'

I'm reading this post by Charles Siegel on Monodromy Representations
and there is a short remark on the proof of irreducibility of moduli space of genus $g$ curves ${\mathcal{M}_g}$ :
Just look at ${...

1
vote

1
answer

265
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### Principal G-bundles over the circle

To edify my understanding of fiber bundles with structure groups, I was currently trying to reconcile two classifications (in a particular case). For simplicity, I'm taking the base to be $S^1$ and ...

4
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0
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### Dot product of functions on cosets

Some time ago I asked this same question at Math Stackexchange, because I thought that the question is nearly elementary.
To my surprise, it was never answered. So I am elevating it to MathOverflow.
I ...

2
votes

0
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### Example of compact fiber bundle with noncompact fibers

This is a cross post of MSE post somehow:
Is there any example of compact fiber bundle $E$ with noncompact fibers $F$?
Obviously if the base space $B$ is $T_1$ then there is no such example.

3
votes

1
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244
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### Circle bundle with homotopically trivial fiber in the total space

Consider a smooth circle fiber bundle
$$
S^1 \to E\to B
$$
where $E$ is a smooth 3-manifold and $B$ is a smooth surface. Assuming any $S^1$ fiber in $E$ is homotopically trivial, can we prove that $E$ ...

4
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0
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### Isometries of fiber bundles

Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers.
Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For ...

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0
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### Transversality theorem for maps between fiber bundles

I am looking for a possible generalization of the standard Trasversality Theorem which roughly says that transverse maps are generic. For example, see the version below:
From page 74, Theorem 2.1 in ...

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votes

1
answer

365
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### When is a diffeomorphism a bundle map?

Let $F\rightarrow E_0 \rightarrow B$ and $F\rightarrow E_1\rightarrow B$ be two smooth fiber bundles. Suppose $E_0$ and $E_1$ are diffeomorphic.
What are the obstructions for $E_0$ and $E_1$ to be ...

2
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234
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### Differences between induced vector fields on a smooth manifold and on a principal bundle

In the context of the connections on fibre bundle, I have found some difficulties trying to understand the fundamental vector field (my reference is Nakahara, but I'm having some problems with the ...

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0
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115
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### The compactified Jacobian is birational to a $\mathbb{P}^1$-fibration over the Jacobian of normalization

Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\...

2
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0
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74
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### Fibre metrics on non-linear bundles

Usually what is meant under a fibre metric is that one is given a (smooth) vector bundle $\pi:Y\rightarrow X$, and on each fibre $Y_x$ an algebraic inner product $g_x$ that varies smoothly from point ...

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127
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### A possible kind of $K$ theory via comparison of sphere bundles associated to given vector bundles

Let $E$ be a vector bundle on a topological space $X$.Thanks to Allen Hatcher's book "Vector Bundles and K theory", the construction of sphere bundle $S(E)$ can be done without any inner ...

4
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1
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166
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### Describe $\mathcal{N}_{G(\mathbb{P}^1,\mathbb{P}^k)\mid G(\mathbb{P}^1,\mathbb{P}^n)}$ [from MSE]

Note: This question came from MSE, but since I've received some useful observations I posted it here. Post on MSE
Consider $1 \leq k < n$ positive integers, and denote by $G(\mathbb{P}^k,\mathbb{P}...

3
votes

1
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120
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### Cohomogeneity one action on $S^7$-bundles over $S^8$

Is it known if the total space of an $S^7$-bundles over $S^8$ with structure group $SO(8)$ admits a cohomogeneity one action?

7
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275
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### Intersection form of surface bundle over surface

Let $\Sigma_g$ be a Riemannian surface of genus $g$. Let $M^4$ be a surface bundle over surface: $\Sigma_g \to M^4 \to \Sigma_h$. $\Sigma_g$ is the fiber and $\Sigma_h$ is the base space.
My question: ...

4
votes

1
answer

220
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### Positive scalar curvature on the total space of a circle bundle

Let $(\Sigma_\gamma,g)$ be a closed and orientable Riemannian surface of genus $\gamma \geq 1$, $(M^3,\tilde{g})$ be a closed, connected and orientable Riemannian $3$-manifold, and $\pi : M \to \...