Questions tagged [fibre-bundles]

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

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53 votes
12 answers
9k views

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 ...
Themaninthebox's user avatar
41 votes
5 answers
3k views

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 ...
Michael Bächtold's user avatar
34 votes
4 answers
11k views

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 views

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 ...
Jason DeVito - on hiatus's user avatar
24 votes
4 answers
5k views

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 ...
Qiaochu Yuan's user avatar
24 votes
1 answer
1k views

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 $\mathbb{C}P^\infty$ (...
ZZY's user avatar
  • 707
24 votes
1 answer
2k views

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 ...
Igor Belegradek's user avatar
24 votes
1 answer
3k views

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 ...
Jason DeVito - on hiatus's user avatar
21 votes
3 answers
2k views

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 ...
algori's user avatar
  • 23.2k
19 votes
1 answer
1k 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 ...
annie marie cœur's user avatar
18 votes
2 answers
2k views

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.
asv's user avatar
  • 21.1k
18 votes
3 answers
904 views

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 ...
Yaniv Ganor's user avatar
  • 1,873
18 votes
2 answers
1k views

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 ...
მამუკა ჯიბლაძე's user avatar
18 votes
2 answers
1k views

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. ...
Jason DeVito - on hiatus's user avatar
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 ...
ychemama's user avatar
  • 1,326
16 votes
3 answers
525 views

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 $...
Oldřich Spáčil's user avatar
16 votes
2 answers
1k views

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 ...
Bilateral's user avatar
  • 3,064
15 votes
4 answers
1k views

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 ...
Xiao-Gang Wen's user avatar
15 votes
3 answers
2k views

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 ...
Roberto Frigerio's user avatar
14 votes
4 answers
2k views

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 ...
Ilya Grigoriev's user avatar
14 votes
3 answers
2k views

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 ...
Kirill Levin's user avatar
13 votes
4 answers
2k views

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 \...
Dima Sustretov's user avatar
13 votes
2 answers
2k views

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 ...
Romeo's user avatar
  • 2,714
13 votes
2 answers
2k views

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'\...
user46652's user avatar
  • 655
12 votes
4 answers
815 views

$S^n \to S^m \to B$ 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 ...
Romeo's user avatar
  • 2,714
12 votes
4 answers
3k views

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 ...
Fernando Galaz-Garcia's user avatar
12 votes
2 answers
797 views

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 ...
ychemama's user avatar
  • 1,326
12 votes
7 answers
1k views

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 $a \in H^3(B; \mathbb{Q})$ ...
Petya's user avatar
  • 4,686
12 votes
1 answer
783 views

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 ...
QSR's user avatar
  • 2,213
11 votes
2 answers
611 views

$ \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....
Ian Gershon Teixeira's user avatar
11 votes
1 answer
692 views

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 ...
Mauricio's user avatar
  • 1,415
11 votes
2 answers
954 views

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(...
QSR's user avatar
  • 2,213
11 votes
1 answer
2k views

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 ...
Asaf Shachar's user avatar
  • 6,611
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 $...
Overflowian's user avatar
  • 2,523
11 votes
1 answer
864 views

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):...
Mohammad Ghomi's user avatar
10 votes
3 answers
1k views

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 ...
Peter Crooks's user avatar
  • 4,870
10 votes
1 answer
763 views

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 ...
Ariyan Javanpeykar's user avatar
10 votes
3 answers
618 views

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 &...
Samuel Monnier's user avatar
10 votes
1 answer
1k views

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 ...
Roberto Frigerio's user avatar
10 votes
2 answers
796 views

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,...
Marco Gualtieri's user avatar
10 votes
1 answer
498 views

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 ...
David Spivak's user avatar
  • 8,549
9 votes
3 answers
4k views

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 $Gr(k,E_x)$. ...
Evgeny Shinder's user avatar
9 votes
3 answers
512 views

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}...
Bilateral's user avatar
  • 3,064
9 votes
2 answers
1k views

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 $\Imb(S, M)$...
Romeo's user avatar
  • 2,714
9 votes
2 answers
1k views

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 ...
Ryan Budney's user avatar
  • 42.8k
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 ...
aglearner's user avatar
  • 14k
9 votes
2 answers
1k views

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 ...
Theo Johnson-Freyd's user avatar
9 votes
1 answer
780 views

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 ...
Tyler Holden's user avatar
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 ...
Jens Reinhold's user avatar
9 votes
0 answers
646 views

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)...
user46652's user avatar
  • 655

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