# Questions tagged [fibre-bundles]

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

273
questions

**3**

votes

**1**answer

128 views

### 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

229 views

### 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 \...

**7**

votes

**0**answers

164 views

### 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 ...

**7**

votes

**1**answer

213 views

### 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**

votes

**0**answers

90 views

### 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**answer

265 views

### $1$-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 ...

**1**

vote

**0**answers

53 views

### 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

578 views

### 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 ...

**8**

votes

**2**answers

358 views

### 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 $p$-...

**0**

votes

**1**answer

57 views

### 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 $E'\to E\to B$ ...

**1**

vote

**0**answers

119 views

### 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

143 views

### 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**

votes

**0**answers

67 views

### 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**answers

64 views

### 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**answer

217 views

### 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**

votes

**0**answers

50 views

### 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 ...

**1**

vote

**0**answers

73 views

### 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 ...

**5**

votes

**1**answer

275 views

### 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**

votes

**0**answers

161 views

### 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 ...

**1**

vote

**0**answers

112 views

### 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**

votes

**0**answers

45 views

### 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 ...

**1**

vote

**0**answers

127 views

### 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**

votes

**1**answer

146 views

### 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**answer

106 views

### 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?

**6**

votes

**1**answer

256 views

### 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

194 views

### 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 \...

**1**

vote

**0**answers

97 views

### Connection as a jet section [closed]

Let $\pi:E\longrightarrow M$ a smooth fibre bundle. A connection is a linear bundle homomorphism $\Phi:TE\longrightarrow TE$ such that $\Phi$ is a projection to the vertical bundle $VE\subset TE$.
I ...

**1**

vote

**2**answers

242 views

### On the smoothness of transition functions

Let $p:E \longrightarrow M$ be a smooth fibre bundle, with standard fibre space $F$ and $G$ a Lie group acting effectively on $F$ as a structure group.
Then, are the transition functions always ...

**4**

votes

**1**answer

197 views

### Set of all sections of a fiber bundle up to homotopy equivalence

Let $\pi: E \to B$ be a fiber bundle of (topological or differentiable) manifolds. Denote by $[B, E]_{\pi}$ the set of all homotopy classes of sections of the bundle, i.e
\begin{align}
[B, E]_\pi &...

**3**

votes

**1**answer

233 views

### What can be an appropriate notion of principal bundle over a category (with an appropriate notion of local trivialisation)?

Motivation for my question:
It is a well-known fact that there exists a bijection between the set of isomorphism class of
principal $G$ bundles over a nice topological space $X$ and the set $[X,B'G]$ ...

**1**

vote

**0**answers

97 views

### The group of global sections of the automorphism bundle of the tangent bundle on a Grassmannian

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$.
We regard $X$ as an algebraic variety over $\Bbb C$.
Let ${T_X} \to X$ denote the tangent bundle on $X$. For ...

**1**

vote

**0**answers

65 views

### Regarding singular points of a fiber bundle

I had asked this question at math.stackexchange but couldn't find any answer; so I'm posting it here.
Let $X$, $Y$ be two projective varieties over $\mathbb{C}$, where $Y$ is smooth, and let $f:X\...

**3**

votes

**0**answers

39 views

### Euler class and the real homological class of the fiber in an orientable sphere bundle

In the paper Foliations transverse to the fiber of a bundle, Plante considers the following example. Let $p:E\longrightarrow B$ a orientable fiber bundle with fiber $\mathbb{S}^k$. We have the Gysin ...

**17**

votes

**1**answer

720 views

### Anomaly in QFT physics v.s. determinant line bundle

In a quantum field theory (QFT) lecture, a math-physics professor explains the anomaly in physics, say the non-invariance of the partition function of an anomalous theory under background field ...

**7**

votes

**1**answer

333 views

### A relative version of Ehresmann's theorem

Edited: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.
Let $N\subset M$ be a pair ...

**2**

votes

**0**answers

58 views

### Extend fibre bundle

Let $F\rightarrow E\rightarrow B$ be a smooth fibre bundle. Suppose $W$ is a smooth manifold such that $F=\partial W$.
When is it possible to extend the bundle to a bundle over $B$ with fibre $W$?

**6**

votes

**2**answers

186 views

### Fibre preserving maps of Borel constructions

Let $G$ be a discrete group with universal principal bundle $EG\to BG$, and let $X$ and $Y$ be left $G$-spaces. An equivariant map $\overline{f}:X\to Y$ induces a fibre-preserving map $f:EG\times_G X\...

**6**

votes

**0**answers

149 views

### Foliated circle bundles whose Euler class is torsion

Let $X$ be a closed manifold. By a foliated circle bundle $E \rightarrow X$ we mean a circle bundle over $X$ with total space $E$ and structure group $Diff^+(S^1)$, and a codimension one foliation of $...

**1**

vote

**0**answers

33 views

### A “singular” Tischler theorem

The Tischler theorem says that a compact manifold $M$ admitting a closed nowhere vanishing $1$-form $\alpha$ fibers over the circle. I was wondering if anything could be said about the case where $\...

**2**

votes

**0**answers

164 views

### Elementary questions about vanishing cycles and emerging cycles

Let $X\to D$ be a proper $C^\infty$ map with $D$ an open disk about the origin in some Euclidean space. Suppose $0\in D$ is the only singular value, i.e that over $D^\times=D\setminus \left\{ 0 \right\...

**1**

vote

**0**answers

95 views

### Defining the cospecialization in topology

Below is an excerpt from part V of Deligne's Étale cohomology - starting points.
Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the ...

**5**

votes

**1**answer

312 views

### Shrinking and stretching of vector bundles

Let $M$ be a manifold, $p:E\to M$ a rank $d$ vector bundle. Suppose that $U \subset E$ is an open subset such that $U \cap p^{-1}(x)$ is nonempty and convex for all $x \in M$. Is it true that $U \to M$...

**3**

votes

**0**answers

116 views

### Is a $G$-bundle over $\mathbb{R}$ a $G$-fibre bundle?

Let $G$ be a Lie group with a smooth (non-transitive) action on a connected manifold $M$ (none of them need to be compact). Let further $f\in C^\infty(M,\mathbb{R})$ be $G$-invariant. Suppose that for ...

**4**

votes

**0**answers

93 views

### A fiber bundle of the Euclidean space over an orbifold

Consider a fiber bundle $p: F\hookrightarrow
E \to B$, where $E$ and $F$ are smooth manifolds and $B$ is a smooth orbifold. More precisely, each point $b \in B$ has an orbifold chart $U=\tilde U/\...

**7**

votes

**2**answers

280 views

### Foliation of $\mathbb R^n$ by connected compact manifolds

Does there exist a smooth nontrivial fiber bundle $p: F \hookrightarrow \mathbb R^n \to B$ such that $F$ and $B$ are connected manifolds with $F$ compact? "Nontrivial" here means the fiber $F$ is not ...

**7**

votes

**3**answers

634 views

### $\mathbb CP^k$ bundles over $\mathbb CP^n$ are projectivisations of vector bundles. Any reference?

Statement. Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$ in analtytic topology. It is well known that there exists a rank $k+1$ complex vector ...

**3**

votes

**2**answers

431 views

### Why does a principal G-bundle with a discrete structure group G have a unique flat connection?

I'm reading the Dijkgraaf–Witten paper Topological gauge theories and group cohomology (Comm. Math. Phys. 129 (1990) pp 393–429, doi:10.1007/BF02096988) and on page 395, 2nd paragraph they write
...

**8**

votes

**1**answer

876 views

### The free smooth path space on a manifold

Let $M$ be a closed, smooth manifold and let $PM$ be the space of unbased piecewise smooth paths $[0,1] \to M$. Then restricting a path to its boundary gives a map
$$
PM \to M \times M .
$$
Question ...

**1**

vote

**0**answers

242 views

### Is there something wrong with this definition of principal bundle?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...

**1**

vote

**0**answers

124 views

### Open problems in fiber bundles theory

As the title says, what are some problems in fiber bundles theory (especially principal bundles) that are still open?