# Tagged Questions

**7**

votes

**1**answer

282 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 ...

**2**

votes

**0**answers

130 views

### Global sections for torus fiber bundle

Let us consider the following situation: $\pi:X\rightarrow B$ is a locally trivial fibration between smooth manifolds, its fiber being a torus $T$. My question is two-fold:
1) what is the obstruction ...

**6**

votes

**1**answer

247 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, ...

**2**

votes

**2**answers

245 views

### Homotopy Equivalences and Induced Correspondences between Fibre Bundles

Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ...

**8**

votes

**3**answers

724 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 ...

**3**

votes

**4**answers

836 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 ...

**2**

votes

**1**answer

482 views

### Cross sections in bundles and principal G-bundles

A principal $G$-bundle has a cross section iff it is trivial (e.g. Husemoller's Fibre Bundles, 3rd ed., 8.3 in chapter 4).
A principal $G$-bundle is in particular a fiber bundle with fiber $G$.
My ...

**9**

votes

**1**answer

361 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 ...

**3**

votes

**1**answer

252 views

### Reference for monkeying with the topology of a mapping cylinder

In "Construction of Universal Bundles, II", Milnor has to replace the standard topology on the join with what he calls the "strong topology" which is the smallest topology such that certain maps are ...

**5**

votes

**0**answers

126 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 ...

**7**

votes

**2**answers

630 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 ...

**9**

votes

**4**answers

542 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 ...

**8**

votes

**1**answer

336 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 ...

**7**

votes

**2**answers

973 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 ...

**4**

votes

**3**answers

194 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 ...

**6**

votes

**2**answers

416 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 ...

**10**

votes

**3**answers

887 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 ...

**16**

votes

**1**answer

694 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 ...

**12**

votes

**7**answers

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

**13**

votes

**2**answers

633 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.
...

**4**

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

**17**

votes

**3**answers

828 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 ...

**5**

votes

**1**answer

313 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 $G\to E$ ...

**2**

votes

**2**answers

396 views

### Branched coverings over orbifolds with reflector lines

It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via ...

**2**

votes

**3**answers

555 views

### Reducible 3d torus bundles

Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So,
could anyone give me a hint to classify them?
In contrast, do you agree ...

**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 ...

**2**

votes

**2**answers

894 views

### (how) are vector bundles and homotopy groups related?

Hello,
homotopic maps induce isomorphic pullback bundles, and so isomorphism classes of vector bundles over X correspond to homotopy classes of maps from the grassmannian to X. But I think, in the ...