Questions tagged [fibre-bundles]
for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.
336
questions
6
votes
0
answers
273
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
49
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
199
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
115
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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
403
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
121
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 ...
5
votes
0
answers
144
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
328
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 ...
9
votes
3
answers
971
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 ...
7
votes
2
answers
1k
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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
1k
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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
346
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
174
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
1
answer
1k
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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
0
answers
227
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
0
answers
88
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 ...
5
votes
1
answer
810
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
1
answer
257
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
0
answers
285
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
1
answer
133
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?
7
votes
2
answers
425
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
1
answer
268
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
0
answers
159
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
0
answers
97
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 ...
11
votes
1
answer
1k
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 $...
17
votes
6
answers
1k
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
0
answers
128
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
0
answers
57
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
1
answer
235
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 $...
3
votes
1
answer
599
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
0
answers
104
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
2
answers
310
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 ...
7
votes
1
answer
1k
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
1
answer
264
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 ...
5
votes
1
answer
649
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 ...
5
votes
1
answer
254
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
0
answers
34
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
0
answers
107
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 $\text{Pin}^-$...
8
votes
2
answers
704
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
vote
0
answers
119
views
Explicit description of the scheme obtained by relative gluing data over a base scheme
I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
3
votes
1
answer
122
views
Naturality of minimal model of a fibre bundle
$\require{AMScd}$
For rational fibrations $F \rightarrow E \rightarrow B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's
$$
\begin{CD}
...
0
votes
1
answer
89
views
Is $TS^2\setminus Z$ a $S^2$- fibre bundle on the puntured plane?(Swapping the role of fibre points and base space)
Let $X=TS^2\setminus Z$ where $Z$ is the zero section of the tangent bundle of $S^2$.
Is there a $S^2$- fiber bundle structure on $(X,\mathbb{R}^2\setminus\{0\},q)$ for some continuous fibre map $q$?
4
votes
0
answers
788
views
When is the Chern class of a principal $G$-bundle same to the Chern class of the associated complex vector bundle?
Let $P\rightarrow M$ be a principal bundle, with structure group $G$. If $G$ has a representation $\rho:G\rightarrow GL(n,\mathbb{C})$, then we can define its associated vector bundle $E=P\times_{\rho}...
4
votes
0
answers
112
views
Representation on square integrable sections of a principal bundle
Let $X\rightarrow Y$ be a smooth principal $G$-bundle for some Lie group $G$. Then $L^2(X)$ has a natural $G$-action determined by fibrewise action of $G$ on $X$.
We have an abstract isomorphism of ...
2
votes
0
answers
138
views
Extension of a given section and obstruction cocyles
Let $p:E \to X$ be a fibration with the fiber $F$ where $X$ is a CW-complex. Denote by $U$ the set $U:=D^n \times \{0\} \cup S^{n-1} \times I$ (part of a cylinder) and let $\tilde{f}:U \to E$ be a ...
3
votes
3
answers
978
views
Lie algebra bundle associated to a Lie group bundle
I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google ...
4
votes
1
answer
212
views
Proper locally trivial bundle is injective on Chow groups
If $X\to Y$ is a map of varieties that is Zariski-locally isomorphic to a projection $U\times P\to U$ with $P$ (smooth) proper, I think the pullback $A_{\bullet}(Y)\to A_{\bullet}(X)$ is supposed to ...
3
votes
0
answers
300
views
Is a Difference of Fiber Bundles a Fiber Bundle?
I have a seemingly very basic question in differential topology, but I could not find the answer by a short google search.
Let $M,N$ be smooth manifolds, and let $f:M\to N$ be a smooth fiber-bundle,
...
4
votes
1
answer
144
views
Homological class of the fiber in the total space of the one circle bundle
It is a well known fact that (isomorphism classes of) princial $\mathbb{S
}^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z}$), by the Euler class....
4
votes
0
answers
262
views
Instanton configurations of self-dual and anti-self-dual instantons interplay
Yang-Mills gauge theory is given by the action
$$ S_\text{YM}[A] = \int_M\mathrm{Tr}_\mathfrak{g}(F\wedge \star F)$$
whose Euler-Lagrange equations are the classical equations of motion. The classical ...