The fibre-bundles tag has no wiki summary.

**30**

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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 ...

**22**

votes

**9**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 ...

**17**

votes

**3**answers

901 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 ...

**17**

votes

**1**answer

1k 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 ...

**16**

votes

**1**answer

711 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 ...

**14**

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**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 ...

**14**

votes

**3**answers

309 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 ...

**14**

votes

**1**answer

1k 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 ...

**13**

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**2**answers

674 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.
...

**12**

votes

**7**answers

900 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 $a \in H^3(B; \mathbb{Q})$
...

**11**

votes

**4**answers

1k 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 ...

**11**

votes

**2**answers

887 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 ...

**11**

votes

**3**answers

980 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 ...

**10**

votes

**4**answers

1k 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 ...

**9**

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**4**answers

561 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 ...

**9**

votes

**1**answer

368 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 ...

**9**

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**2**answers

1k 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 ...

**9**

votes

**2**answers

460 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 ...

**9**

votes

**1**answer

438 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 ...

**8**

votes

**3**answers

792 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 ...

**8**

votes

**1**answer

345 views

### A sphere bundle map

I think this may all be classical bundle-theory. But I'm trying to read some old papers on classifications of bundles and the following came up as questions I couldn't immediately answer:
Consider ...

**8**

votes

**1**answer

509 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 ...

**7**

votes

**2**answers

672 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 ...

**7**

votes

**1**answer

336 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 ...

**6**

votes

**2**answers

1k views

### Totally geodesic surfaces in fibered 3-manifolds

Is there an easy example of a (closed) hyperbolic 3-manifold that fibers over the circle but contains some totally geodesic surface?
(Of course such manifolds exist if the 'Virtually Fibered ...

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votes

**2**answers

465 views

### Restrictions of Diffeomorphisms

Notation: Let$M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $Diff(M)$ the group of diffeomorphisms of $M$ and $Imb(S, M)$ the group of smooth imbeddings of $S$ into $M$.
A classical ...

**6**

votes

**2**answers

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### Critical points on a fiber bundle

Consider a (smooth) bundle E→_B_, and a (smooth) function f: E → R on the total space. Then it makes sense to talk about the derivatives of f along the fibers. Let C be the subspace of E consisting ...

**6**

votes

**1**answer

260 views

### how to obtain a generalized Morse function out of a fiber bundle?

I already asked this question in MSE but did not get any answer/comment yet.
Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, ...

**6**

votes

**1**answer

302 views

### Homomorphisms of Topological Groups which are Automatically Fiber Bundles?

Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will ...

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216 views

### Flat morphisms whose fibers are affine spaces

Let $f:X \to Y$ be a flat morphism, such that each fiber is isomorphic to the affine space $\mathbb{A}^n$. Then is is true that $f$ is a Zariski affine bundle? If not, is it at least an ètale affine ...

**5**

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**4**answers

942 views

### How to partition R^3 into pairwise non-parallel lines?

Problem. How to partition R^3 into pairwise non-parallel lines?
A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't ...

**5**

votes

**2**answers

795 views

### Singular K3 — mathematical meaning?

There's a very interesting text by Cumrun Vafa called Geometric Physics.
Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration:
...

**5**

votes

**1**answer

323 views

### Killing Chern classes

Let $G$ be a compact connected Lie group and let $E\to B$ be a principal $G$-bundle. Suppose $a$ is a rational cohomology class of $E$ such that its pullback $b$ under an orbit inclusion map $G\to E$ ...

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**0**answers

130 views

### Joins and classifying spaces in the category of compactly generated spaces

In Milnor's Construction of Universal Bundles, II, he defines $E_nG$ by repeated
joins of $G$ with itself, but he has to use the `strong topology' on the join instead
of the everyday topology that ...

**4**

votes

**3**answers

2k 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 ...

**4**

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**3**answers

2k 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 $Gr(k,E_x)$.
...

**4**

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**2**answers

185 views

### The unit tangent bundle of 2- or 4-manifolds as a principal $S^{1}$- or $S^{3}$-bundle

What type of obstructions have been studied so that the unit tangent bundle of a Riemannian 2-(4-)manifold have a structure of a principal $S^{1}$-($S^{3}$-)bundle?

**4**

votes

**2**answers

319 views

### Name for bundle of algebraic varieties over a smooth manifold

Consider a smooth manifold $M$ and a bundle $\pi\colon E\to M$ over it, where each fibre of $E$ is an algebraic variety. Is there a special name for this kind of bundle? The idea I have is that ...

**4**

votes

**1**answer

114 views

### connections on principal bundles over $S^1$

Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My ...

**4**

votes

**2**answers

356 views

### Getting rid of exceptional fibers by passing to finite covers?

Consider a Seifert fiber space. Is it always possible to find a finite cover that is a circle bundle and the preimage of any fiber is a finite union of circles?

**4**

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**3**answers

200 views

### Homology of bundles over a triangulated base and $A_\infty$-algebras

Let $p:E \to B$ be a fiber bundle over a triangulated base $B$ with fiber $F$, $\sigma$ simplex in $B$, $\sigma \mapsto H_{*}(p^{-1}(\sigma)) \simeq H_{*}(F)$ the obvious map and let $\mathcal{S}$ be ...

**4**

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**1**answer

190 views

### A generalization of covering spaces to fiber bundles with totally path-disconnected fibers

There is a classical theorem about covering spaces and the actions of the fundamental group.
Theorem 1: Let $B$ be a non-empty locally path-connected and path-connected space. The category of ...

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votes

**1**answer

197 views

### isomorphism of extensions by abelian varieties

Let $A$ and $G$ be abelian varieties over $\mathbb{C}$. An element $P$ of $\text{Ext}(A, G)$ is an exact sequence
$0 \to G \to P \to A \to 0$,
here one can give $P$ the structure of an abelian ...

**4**

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**0**answers

358 views

### What is the spectrum of the commutative C*-algebra I have constructed here?

Let $B$ and $F$ be compact Hausdorff spaces.
Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$.
I think this induces a fiber ...

**3**

votes

**4**answers

950 views

### Nontrivial examples of non-trivial principal circle bundles

It is a well known fact that (isomorphism classes of) principal $S^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z})$.
There is always the ...

**3**

votes

**2**answers

133 views

### Differentiable structure on the Gauge group of a principal bundle?

I am currently reading this paper in which we have a map $g:I\rightarrow Gauge(P)$ for some principal bundle $P$ which is differentiated. I am looking for a reference or explanation what the "most ...

**3**

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**1**answer

123 views

### Which varieties of general type admit fibrations with non-general type fibres

Disclaimer. I don't know much about the things I'm asking. This is why my other question pencils on varieties of general type was a bit unclear. I believe the following question makes up for this.
...

**3**

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**1**answer

430 views

### How was this Lie algebra found?

In a paper the author lists, without justification, generators for a Lie algebra. I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself.
...

**3**

votes

**1**answer

145 views

### How to classify von Neumann algebra bundles?

If we consider algebra bundles over X where the fiber is an algebra of bounded operators in a separable Hilbert space H over the complex numbers. I learn from "Isomorphism Classification of Operator ...

**3**

votes

**1**answer

300 views

### degeneration of quadric bundles

Suppose I have a smooth 2-dimensional quadric bundle $f:X\to S$ over a surface $S$. Suppose furthermore that the discriminant locus $\Delta \subset S$ is smooth. Can I immedately conclude that the ...