Questions tagged [fibre-bundles]
for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.
336
questions
53
votes
12
answers
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Looking for an introduction to orbifolds
Is there any source where the basic facts about orbifolds are written and proved in full detail?
I found the article by Satake "The Gauss-Bonnet Theorem for V-manifolds", but I'd like to have a more ...
41
votes
5
answers
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Are submersions of differentiable manifolds flat morphisms?
Let $\pi \colon M\to N$ be a smooth map between real smooth manifolds. Then $C^\infty(M)$ forms a module over $C^\infty(N)$ (via pullback). Is this module flat when $\pi$ is a submersion?
Recall that ...
34
votes
4
answers
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Vector bundles vs principal $G$-bundles
It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres
$$F=\pi^{-1}(x), \ \ \ x\in B $$
over any $x\in B$, are ...
25
votes
1
answer
996
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When are fiber bundles reversible?
My question, in its most general form is this:
Given a fiber bundle $F\rightarrow E\rightarrow B$, when is there a fiber bundle $B\rightarrow E\rightarrow F$?
Here, F,E, and B can lie in whichever ...
24
votes
4
answers
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What is a section?
This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
24
votes
1
answer
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All fiber bundles over $S^2$ extendable to $\mathbb{C}P^\infty$?
I ran into the following sanity check. Is the following statement true?
Every smooth fiber bundle (with compact fiber) over $S^2$ can be extended to a smooth fiber bundle over $\mathbb{C}P^\infty$ (...
24
votes
1
answer
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Which principlal bundles are locally trivial?
If $H$ is a closed subgroup of a topological group $G$, then the orbit map $G\to G/H$ is a principal bundle, yet somewhat surprisingly, it need not be locally trivial.
In the wikipedia article on ...
24
votes
1
answer
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Characteristic classes of sphere bundles over spheres in terms of clutching functions
I'm trying to understand Milnor's proof of the existence of exotic 7-spheres.
Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be ...
21
votes
3
answers
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Cohomology of fibrations over the circle: how to compute the ring structure?
This question is inspired by Cohomology of fibrations over the circle Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points ...
19
votes
1
answer
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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 ...
18
votes
2
answers
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Serre fibration vs Hurewicz fibration
What are the simplest/ typical examples of a Serre fibration which is not a Hurewicz fibration? Is it something pathological?
Sorry if the question is too elementary for MO.
18
votes
3
answers
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When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$
For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$
In my case ...
18
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2
answers
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Anything between vector bundles and sphere bundles?
There are two extremities: on the "easy end" one has vector bundles which are classified by maps to the (more or less) well understood spaces like Grassmanians; on the "hard end" there are spherical ...
18
votes
2
answers
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What manifold has $\mathbb{H}P^{odd}$ as a boundary?
This question is motivated by What manifolds are bounded by RP^odd? (as well as a question a fellow grad student asked me) but I can't seem to generalize any of the provided answers to this setting.
...
17
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6
answers
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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 ...
16
votes
3
answers
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Existence of sections of the evaluation map for the diffeomorphism group
Let $M$ be a closed connected oriented smooth manifold and $\mathrm{Diff}_{+}(M)$ the group of orientation preserving diffeomorphisms of $M$ endowed with the compact-open topology. Pick a base point $...
16
votes
2
answers
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Classification of $O(2)$-bundles in terms of characteristic classes
I had asked this question in stackexchange but there seems to be no consensus in the answer
It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by ...
15
votes
4
answers
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Cohomology ring of mapping torus
A mapping torus, $M \rtimes_\varphi
S^1$, is a fiber bundle over $S^1$ with fiber $M$, where $\varphi$ is an element of mapping class group of $M$, describing the twist around $S^1$.
For $M=S^1\times ...
15
votes
3
answers
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smooth sections of smooth fiber bundles
A maybe trivial question about fiber bundles (I'm not an expert, and I didn't find quickly an answer looking here and there). Suppose you are given fiber bundle $p\colon E\to M$, where
$E,M$ are ...
14
votes
4
answers
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When do two holonomy maps determine flat bundles that are isomorphic as just bundles (w/o regard to the flat connections)?
Suppose we have a surface S (although the question might make as much sense in higher dimensions) and a topological group G. The data of a flat vector bundle on S (up to isomorphism) is the same as a ...
14
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3
answers
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In how many ways can an iterated tangent bundle (T^k)M be viewed as a fibre bundle over (T^(k-1))M?
Let M be a smooth manifold. The double tangent bundle, TTM,can be viewed as a fibre bundle over TM in two ways, with the projection maps given by T_πM (i.e. the derivative of the projection from TM ...
13
votes
4
answers
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non-locally trivial A^n bundles
Let $f: X \to Y$ be a morphism of varieties such that its fibres are isomorphic to $\mathbb{A}^n$. Since the definition of a vector bundle stipulates that $f$ be locally the projection $U \times \...
13
votes
2
answers
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Surface bundles over a surface
What can be used to distinguish two $\Sigma_g$-bundles over $\Sigma_h$ up to
(1) homotopy?
(2) homeomorphism?
(3) fiberwise homeomorphism?
(4) bundle isomorphism?
And can these always be computed ...
13
votes
2
answers
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Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections
I previously asked this on Math.SE but didn't receive a satisfactory answer.
Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\...
12
votes
4
answers
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$S^n \to S^m \to B$ bundle: possible?
Sphere bundles and bundles over spheres are everywhere and are excellent things to get one's hands dirty with.
(1a) But when can we have a bundle $S^n \to S^m \to B?$ It seems like requiring the ...
12
votes
4
answers
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Circle bundles over $RP^2$
Does anybody know if orientable, closed $3$-manifolds that are circle bundles over $RP^2$ have been classified?
One can determine the isomorphism classes of bundles using obstruction theory, but I am ...
12
votes
2
answers
797
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Is a topological fiber-bundle, whose total space admits a retraction onto a fiber, trivial?
Let $\xi = \pi \colon E \to B$ a topological fiber bundle with connected base $B$, $E_x = \pi^{-1}(x)$ the fiber at $x \in B$, $j \colon E_x \hookrightarrow E$ the canonical injection, and let suppose ...
12
votes
7
answers
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Cohomology classes annihilated by pullbacks
A friend of mine is interested in examples of the following situation:
an oriented smooth fiber bundle $\pi \colon M \to B$ with $M$ and $B$ compact
and a non-zero class $a \in H^3(B; \mathbb{Q})$
...
12
votes
1
answer
783
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Stiefel-Whitney class of fibre bundles
Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...
11
votes
2
answers
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$ \mathbb{R}P^n $ bundles over the circle
Is every $ \mathbb{R}P^{2n} $ bundle over the circle trivial?
Are there exactly two $ \mathbb{R}P^{2n+1} $ bundles over the circle?
This is a cross-post of (part of) my MSE question
https://math....
11
votes
1
answer
692
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representatives of the group of homotopy 7-spheres
In Milnor's paper "On manifolds homeomorphic to the 7-sphere" it is proven that there are manifolds homeomorphic but not diffeomorphic to the standard 7-sphere. His construction involves sphere ...
11
votes
2
answers
954
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first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles
Let $\xi$ be a (real) vector bundle of dimension $n$. Then the first Stiefel-Whitney class
$$
w_1(\xi)=0
$$
if and only if $\xi$ is orientable, i.e. the structure group of $\xi$ can be reduced to $SO(...
11
votes
1
answer
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Does pullback in the category of smooth manifolds always exists?
I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist.
Remarks:
1) A pullback in a certain category is defined as ...
11
votes
1
answer
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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 $...
11
votes
1
answer
864
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Equivariant sections of fiber bundles
One of the fundamental facts in fiber bundle theory is the following result for existence and extension of sections (see Thm. 9 in this paper of Palais, and compare with Thm. 12.2 in Steenrod's book):...
10
votes
3
answers
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Serre Spectral Sequence of Representations
Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure ...
10
votes
1
answer
763
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Can one bound the Todd class of a 3-dimensional variety polynomially in c_3
This question is on bounding the degree of the Todd class on a complex threefold.
Let $X$ be a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern class. Recall the ...
10
votes
3
answers
618
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Spin 4-manifold bounded by a mapping torus of tori
Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 &...
10
votes
1
answer
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Two questions about sphere bundles
I would like to better understand the relationship between different notions of orientable sphere bundle. Let me say that a locally trivial fiber bundle $\pi\colon E\to M$ with fiber $S^n$ and ...
10
votes
2
answers
796
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Classification of holomorphic disc bundles
I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces. Note that by "bundle", I mean a holomorphic fiber bundle,...
10
votes
1
answer
498
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Bundle-to-function correspondence
To a morphism of sets $f\colon E\to B$ with finite fibers, one may assign a function $$|f^{-1}|\colon B\to{\mathbb N}$$ sending an element $b\in B$ to the cardinality of the fiber $f^{-1}(b)$.
To a ...
9
votes
3
answers
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Grassmannian bundle theorem
Let's consider a vector bundle $E$ of rank $n$ over a compact manifold $X$. Consider the associated Grassmannian bundle $G$ for some $k < n$, obtained by replacing each fiber $E_x$ by $Gr(k,E_x)$.
...
9
votes
3
answers
512
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A conjecture about parallelizable generalized spheres
Let $S^{d}$ denote the standard $d$-dimensional sphere. I heard from a physicist that from physical arguments they have been able to show that the vector bundle:
$E_{d} = TS^{d}\oplus \Lambda ^{d-2}...
9
votes
2
answers
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Restrictions of diffeomorphisms
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Imb{Imb}$Notation: Let $M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $\Diff(M)$ the group of diffeomorphisms of $M$ and $\Imb(S, M)$...
9
votes
2
answers
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Fibrewise homotopy-equivalence of unit sphere bundles vs isomorphism of tangent bundles
Let $M$ be a smooth $m$-dimensional manifold, $TM$ its tangent bundle and $SM$ its unit sphere bundle.
Are there some simple examples where $SM$ is fibrewise homotopy-equivalent to the trivial ...
9
votes
3
answers
971
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$\mathbb CP^k$ 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 ...
9
votes
2
answers
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Critical points on a fiber bundle
Consider a (smooth) bundle $E\to B$, and a (smooth) function $f: E\to\mathbf{R}$ on the total space. Then it makes sense to talk about the derivatives of $f$ along the fibers. Let $C$ be the ...
9
votes
1
answer
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Local trivializations of the non-trivial $SU(2)$-bundle over $S^5$
It is well known that $SU(3)$ is the unique, non-trivial, principal $SU(2)$- bundle over $S^5$. To my knowledge the way this is proven is by using the following fact:
If $G$ is a Lie group and ...
9
votes
0
answers
286
views
Rational cobordism classes of manifolds fibered over spheres
Let us fix positive integers $k, m$. Let $A^k_{4m} \subset \Omega^{\text{SO}}_{4m} \otimes \mathbb Q$ be the subgroup generated by oriented cobordism classes of manifolds fibered over $S^k$.
The ...
9
votes
0
answers
646
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Models for Eilenberg-MacLane space K(Z,3)
Denote by $U(H)$ and $PU(H)$ the unitary and projective unitary groups on an infinite-dimensional Hilbert space $H$. Recall that $U(H)$ is contractible by Kuiper's theorem and that $PU(H)$ is a $K(Z,2)...