Questions tagged [fibre-bundles]
for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.
273
questions
3
votes
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
votes
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 \...
7
votes
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
votes
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
votes
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
votes
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 ...
1
vote
0answers
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
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
votes
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$-...
0
votes
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$ ...
1
vote
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 ${...
1
vote
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
votes
0answers
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
0answers
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
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$ ...
4
votes
0answers
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
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
votes
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 ...
2
votes
0answers
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
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
votes
0answers
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
0answers
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
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
votes
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
votes
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
votes
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 \...
1
vote
0answers
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
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
votes
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
votes
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]$ ...
1
vote
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 ...
1
vote
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\...
3
votes
0answers
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
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
votes
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
votes
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
votes
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
votes
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 $...
1
vote
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 $\...
2
votes
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\...
1
vote
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 ...
5
votes
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$...
3
votes
0answers
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
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
0answers
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
0answers
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?
