Disclaimer. I don't know much about the things I'm asking. This is why my other question http://mathoverflow.net/questions/130144/pencils-on-varieties-of-general-type was a bit unc …

In most references, only a principal G-bundle is called universal (ie every other bundle can be pullbacked from this one, or an equivalent definition). Does it make sense to spea …

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 …

Usually, the most general condition for fiber product of manifolds (or vector bundles) to exist is that we require the images cleanly intersects. See e.g. http://mathoverflow.net/q …

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 …

If we consider algebra bundles over X where the fiber is an algebra of bounded operators in a separable Hilbert space H over the complex numbers. I learn from "Isomorphism Classifi …

If $\pi:E\to M$ is a vector bundle then the set of sections $\Gamma(E)$ is naturally a vector space under fibrewise addition and scalar multiplication on the bundle $E$. This holds …

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 lea …

Let $V_{KT}$ be the Kodaira-Thurston 4-manifold $\frac{\mathbb{R^{4}}}{G}$ where $G$ is the subgroup of $Diff(\mathbb{R^{4}})$ generated by unit translations along the $x_{1}$, $x …

I have often heard people talk about, say, "the" twisted $S^2$-bundle over $S^2$. My question is, what do they mean by a twisted bundle? I know that in the above example any $S^2$- …

I have read in some lecture notes in the internet (without reference) the following result: Let $E$ be a differentiable sphere bundle whose base $B$ has dimension $n\geq 2$ and …

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 …

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 w …

Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ …

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 in …