Questions tagged [fibre-bundles]

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

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1answer
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
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1answer
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 \...
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0answers
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
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1answer
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
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0answers
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
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1answer
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 ...
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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
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3answers
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
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2answers
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 $p$-...
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1answer
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 $E'\to E\to B$ ...
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0answers
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 ${...
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1answer
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
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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 ...
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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.
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1answer
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 $E$ ...
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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 ...
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0answers
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
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1answer
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 ...
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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 ...
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0answers
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
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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 ...
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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
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1answer
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
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1answer
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
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1answer
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
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1answer
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 \...
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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 ...
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2answers
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
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1answer
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
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1answer
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 $[X,B'G]$ ...
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0answers
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 ...
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0answers
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\...
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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
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1answer
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
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1answer
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
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0answers
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
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2answers
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
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0answers
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 $...
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0answers
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 $\...
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0answers
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\...
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0answers
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 ...
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1answer
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 $U \to M$...
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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 ...
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0answers
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
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2answers
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
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3answers
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
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2answers
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
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1answer
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 ...
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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 (...
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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?

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