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12
votes
7answers
976 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
4answers
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 ...
2
votes
3answers
298 views

Can there exist two non-equivalent equivariant actions of a group on vector bundle?

Can there exist two non-equivalent equivariant actions of a group $G$ on vector bundle over a $G$ space?
15
votes
2answers
723 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. ...
5
votes
3answers
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
3answers
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 ...
5
votes
1answer
336 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
2answers
414 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
3answers
590 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 ...
14
votes
4answers
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 ...
18
votes
1answer
2k 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 ...
11
votes
3answers
1k 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 ...
5
votes
2answers
824 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: ...
2
votes
2answers
1k 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 ...
11
votes
2answers
994 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 ...
24
votes
9answers
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 ...
31
votes
4answers
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 ...
5
votes
4answers
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 ...
6
votes
2answers
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 ...