# Tagged Questions

A fiber bundle is the most general kind of bundle. Special cases are often described by replacing the word "fiber" with a word that describes the fiber being used, e.g., vector bundles and principal bundles.

256 views

### Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
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 ...
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 ...
350 views

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: $E_{d} = TS^{d}\oplus \Lambda ^{d-2}... 0answers 95 views ### characteristic classes of a covering space with symmetric group action Let$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 ... 2answers 388 views ### first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles 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$SO(...
Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. ...
Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct ...