MathOverflow

Questions tagged [fibre-bundles]

Ask Question

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

247 questions
Newest
Active
Bountied
Unanswered
Filter by
Sorted by
Tagged with
3
votes
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 $[X,B'G]$...
1
vote
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 ...
1
vote
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\...
2
votes
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 ...
-1
votes
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?
0
votes
0answers
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
votes
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
votes
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
votes
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
votes
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
votes
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 $...
0
votes
0answers
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
votes
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\...
1
vote
0answers
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
votes
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 $U \to M$...
3
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
0answers
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 (...
1
vote
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
vote
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(\...
0
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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
vote
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 ...
1
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 $\text{Pin}^-$...
6
votes
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 ...

1
2 3 4 5