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^ …

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 isomorphis …

This question is motivated by http://mathoverflow.net/questions/8829/what-manifolds-are-bounded-by-rpodd (as well as a question a fellow grad student asked me) but I can't seem to …

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

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 l …

This question is inspired by http://mathoverflow.net/questions/4361/cohomology-of-fibrations-over-the-circle Moreover, it can be considered a subquestion of the above, but somehow …

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 …

Let $M,N$ be real smooth manifolds and $p\colon M\to N$ a smooth map. Then smooth functions on $M$ form a module over the ring of smooth functions on $N$ (via pullback). Is it know …

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 severa …

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 inc …

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 co …

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 subspac …

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 …

Similar question as for the torus-bundles. i.e. what $N_3$-bundles over the circle are those which are constructed with a reducible mapping class (in the sense of Nielsen-Thursto …

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 sl …