The fibre-bundles tag has no usage guidance.

**31**

votes

**4**answers

2k 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 ...

**23**

votes

**9**answers

4k 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 ...

**18**

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 ...

**17**

votes

**3**answers

1k 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

### 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 ...

**17**

votes

**1**answer

756 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 ...

**15**

votes

**3**answers

579 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 ...

**15**

votes

**2**answers

707 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.
...

**14**

votes

**4**answers

1k 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 ...

**14**

votes

**3**answers

322 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 ...

**13**

votes

**1**answer

298 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 ...

**12**

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 ...

**12**

votes

**7**answers

955 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

579 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 ...

**11**

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 ...

**11**

votes

**2**answers

978 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

1k 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

3k 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 ...

**10**

votes

**2**answers

484 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

382 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**

votes

**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

**1**answer

450 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 ...

**9**

votes

**0**answers

137 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 ...

**8**

votes

**3**answers

883 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

**2**answers

711 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 ...

**8**

votes

**1**answer

349 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

377 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 ...

**8**

votes

**2**answers

441 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: ...

**8**

votes

**1**answer

545 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

**1**answer

136 views

### first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles [on hold]

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 ...

**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 ...

**6**

votes

**3**answers

326 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 ...

**6**

votes

**2**answers

530 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

1k views

### 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

288 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

311 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 ...

**6**

votes

**0**answers

251 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**

votes

**4**answers

1k 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

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

**5**

votes

**2**answers

815 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

321 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 ...

**5**

votes

**1**answer

331 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 $G\to E$ ...

**5**

votes

**0**answers

53 views

### characteristic classes of a covering space with symmetric group action

Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space
$$
S_n\to M\to M/S_n
$$
where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...

**5**

votes

**0**answers

132 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

**2**answers

327 views

### Cup product of cohomology in a Serre spectral sequence

How to use Serre spectral sequence to compute cup product structures?
Let $F\to E\to B$ be a fibration. Suppose all the differentials of the corresponding Serre spectral sequence of cohomology are ...

**4**

votes

**2**answers

205 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

331 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

119 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

386 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**

votes

**3**answers

203 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 ...