Questions tagged [fibre-bundles]

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

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3
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1answer
161 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
75 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
54 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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0answers
31 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 ...
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0answers
71 views

General homomorphism of bundles with different structure group

I there any notion of homomorphism of smooth G-bundles with different structure group? If so, is it equivalent to the particular cases of vector bundles and principal bundles?
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39 views

Subbundles & slice charts

Let $F\hookrightarrow E \overset{\pi}{\longrightarrow} M$ and $F'\hookrightarrow E' \overset{\pi'}{\longrightarrow} M'$ be two smooth fibre bundles. If $F'\subset F$, $M'\subset M$, $E'\subset E$ ...
17
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1answer
552 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
209 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
43 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
163 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
121 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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22 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
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0answers
153 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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91 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
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1answer
217 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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0answers
114 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
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0answers
80 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
250 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
564 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 ...
2
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2answers
222 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 ...
7
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1answer
813 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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217 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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0answers
98 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?
1
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1answer
210 views

Existence of horizontal lifts in $G$-bundles

I wanted to show that for any smooth principal $G$-bundle $E\xrightarrow\pi B$ any smooth curve $\gamma\colon I\to B$ has a unique horizontal lift from a fixed starting point $u_0\in\pi^{-1}\left(\...
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0answers
98 views

Symmetric products of varieties and projective bundles

Given a smooth projective geometrically connected curve $C$, a symmetric product of $C$ has the structure of a projective bundle over the Jacobian of $C$ (e.g. see Symmetric powers of a curve = ...
2
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0answers
77 views

Is there a notion of “graph of bundles” analogous to a graph of groups?

Since the notion of a graph of groups relies mostly on the pushout, can we construct graphs of objects in some other category, say, vector bundles? If this is the case and we have a "fundamental ...
2
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1answer
240 views

When is the cohomology of a fiber bundle a tensor product?

Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{...
5
votes
1answer
173 views

Extension of Hopf fiber bundle to (an equivariant) 2 dimensional vector bundle

Let $p:S^3 \to S^2$ be the Hopf fibration which is a result of the standard action of $S^1$ on $S^3$. Is there a $2$ dimensional vector bundle $\tilde{p}:E \to S^2$ such that $S^3\subset E$ ...
2
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0answers
144 views

Volume form on unit normal bundle via moving frames

Let $M$ be an $m$-dimensional Riemannian submanifold of $\mathbb{R}^{m+n}$. Let $B_1$ denote the unit normal bundle of $M$, whose fiber at $p \in M$ is the $(n-1)$-sphere $\mathbb{S}^{n-1}$ in the the ...
4
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1answer
115 views

Orientable surface bundle

Is it true that every orientable surface bundle can be made into a symplectic fibration?If yes, why? What about the particular case that $M$ is a connected compact 4-manifold?
6
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2answers
266 views

In the real analytic category, are the fibers of a proper submersion isomorphic?

Ehresmann's theorem says that a proper smooth submersion is a fiber bundle. The proofs I know rely on the existence of connections locally on the base, and this is furnished by partitions of unity. ...
3
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1answer
148 views

Reduction of the structure group of $\mathbb{R}^n$-fiber bundles to a special subgroup of $\mathrm{Homeo}(\mathbb{R}^n)$

Let $G$ be the group of all self-homeomorphisms $f$ of $\mathbb{R}^n$ which satisfy $$f(x+m)=f(x)+m,\quad \forall m\in \mathbb{Z}^n.$$ In other words, $G$ is the group of all equivariant self-...
5
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0answers
71 views

Is there a representation theoretic way to define the pullback of densities and differential forms?

I find it convenient to define the bundle of densities of weight $\alpha$,say $\Omega_\alpha(M)$ over a smooth manifold $M$ as the associated vector bundle of the frame bundle $F(M)$ with the ...
7
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0answers
79 views

Circle foliations not induced by circle actions on an compact orientable manifold

It is known that if we have an orientable fiber bundle $E\to B$, with fiber a circle $\mathbb{S}^1$, then it is a principal $SO(2)$-bundle. In other words, the fibers are spanned by the orbits of a ...
10
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1answer
525 views

Classification of bundles, Postnikov towers, obstruction theory, local coefficients

RECAP on classification of bundles We want to classify $G$-principal bundles over $X$ (smooth manifold, G compact Lie). These are in 1-1 correspondence with homotopy classes of maps $[X,BG]$ (where $...
14
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6answers
818 views

Is the concept of a “numerable” fiber bundle really useful or an empty generalization?

Numerable fiber bundles are defined by Dold (DOLD 1962 - Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space : the trivialization cover of the ...
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0answers
112 views

Fibre transfer of $\mathbb{S}^1$-bundles

Let $p:E\to X$ and $p':E'\to X'$ be two orientable $\mathbb{S}^1$-bundles. Denote their homological transfers by $p_!:H_*(X)\to H_{*+1}(E)$ resp. $p'_!$. Now let $(u,f)$ be a bundle morphism ($u:E\to ...
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0answers
50 views

Does the total space of a bundle satisfy the Tietze extension property when the fiber and base space do satisfy this property?

We say that a Topological space $Y$ satisfies the Tietze extension propery, TE property, if in the formulation of Tietze extension theorem "$\mathbb{R}$" can be replaced by $Y$. Obvioysly the ...
3
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1answer
207 views

Locally trivial fibration over a suspension

For $X$ a paracompact space, I am trying to classify all locally trivial fibration with base the suspension $SX = X \times [-1,1]\, /\, (X \times \{-1\} \cup X \times \{1\})$, and fiber-type a space $...
1
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1answer
187 views

Principal bundles and fibre bundles

Let $\pi_P:P\rightarrow M$ a principal $G$ (right action) bundle. Let $F$ be a manifold with a left action of $G$. Then we have the notion of associated fibre bundle over $M$ whose fibre is $F$. I do ...
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0answers
93 views

Classifying map of a simple circle bundle

Let $\mathbb{K}_0 \subset \mathbb{K}$ be two tori (subtori of $(S^1)^n$). We suppose that $\mathbb{K}_0$ is obtained from $\mathbb{K}$ by the following procedure: consider, on the lie algebra $\text{...
5
votes
2answers
257 views

Existence transverse sections in $\mathbb{CP}^1$-bundles over compact Riemann surfaces

We have that every holomorphic $\mathbb{CP}^1$-bundle on a compact Riemann surface admits a holomorphic section, due Tsen and as found in Compact Compact Surfaces of Barth, Peters and Van de Ven, for ...
5
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1answer
325 views

Classifying space of semidirect product of groups

Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...
4
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0answers
132 views

Torus bundle over spheres

I was wondering what is the classification of all torus bundles over spheres? That is, to classify the fibration $$ T^m \hookrightarrow M \to S^n. $$ It is well known that if $n=1$, all fibrations ...
2
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0answers
120 views

Fiber-bundle : continuity of transition maps and inverse in general

Let $(E,\pi,B)$ be a locally trivial fibration, with fiber a topological space $F$, $\Phi_i$ and $\Phi_j$ two trivializations over $U_i$ and $U_j$. The transition map from $i$ to $j$ is the ...
4
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1answer
208 views

Smooth structure on the space of sections of a fiber bundle and gauge group

Let $\xi$ be a fiber bundle $F\hookrightarrow E\to B$ (where every space is smooth, T2 and second countable), let $\Gamma(\xi)$ be the space of smooth sections. We can complete $\Gamma(\xi)$ with ...
4
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1answer
153 views

The existence of the extension of a non-trivial line bundle

In Three Dimensional Gravity Revisted, Witten studied the Abelian Chern-Simons theory in three dimensions. Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over ...
2
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0answers
31 views

Smoothings over a real interval

I am asking if somebody knows if the following kind of objects are studied somewhere or if there is some kind of obvious obstruction for them to appear. Let $(X,0) \subset \mathbb{C}^n$ be a germ of ...
5
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0answers
97 views

Induced new structures on Poincare dual manifolds

"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...
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2answers
229 views

Lifting a diffeomorphism into a spinor bundle automorphism

I know several papers that treat this, but it seems that most of these papers do things very differently with quite different conclusions, so I am confused. Basically, when one tries to do classical ...

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