5
votes
1answer
96 views

Visualizing topology of a Vector Bundle

I've started reading Milnor, Stasheff - Characteristic Classes and at page $18$ they proved that $\mathbb{R}^n$-bundle $\xi$ is trivial if and only if $\xi$ admits $n$ cross sections $s_1, \dots , ...
1
vote
1answer
24 views

The set of all sections of a vector bundle

At http://en.wikipedia.org/wiki/Vector_bundle we have: "Given a vector bundle $\pi : E \rightarrow X$ and an open subset $U$ of $X$, we can consider sections of $\pi$ on $U$, i.e. continuous ...
0
votes
0answers
28 views

The homotopy between two monomorphisms

$X$ is a compact Hausdorff space and $E,F$ are two complex vector bundles on $X$. If $f$ and $g$ are two homotopic monomorphisms from $E$ to $F$, then can we find a homotopy $f_t$ such that $f_0=f, ...
1
vote
1answer
56 views

Doubt on how to prove proposition about bundles

I've started studying bundles and fiber bundles and to get some practice I've tried to prove the following proposition: "Every vector bundle $(E,B,\pi,F,G)$ is associated to a given principal bundle ...
4
votes
1answer
219 views

Classifying vector bundles

Given a manifold $M$, is there a way of classifying up to isomorphism all possible vector bundles over $M$ of a given rank? Some other questions on this site deal with specific cases, which all seem ...
3
votes
1answer
144 views

Pontryagin classes of a product manifold

I'm imagining there's a way to relate the pontryagin classes of $T(M\times N)$ to the pontryagin classes of $M$ and those of $N$, but I haven't been able to find a helpful reference. Could someone ...
0
votes
1answer
145 views

Is a sub-bundle of a vector bundle a vector bundle?

Could anyone please help me with this question? (1) Let (E, p, B) be a vector bundle where E is the total space, B is the base, and p is the structure map, that is, p:E->B. Now suppose E' is a ...
10
votes
1answer
139 views

Is there a characterization of injective $C(X)$-modules analogous to Serre-Swan?

The Serre-Swan theorem in topology says that if $X$ is compact Hausdorff and $C(X)$ the ring of continuous functions on $X$, then the category of finitely generated projective $C(X)$-modules is ...
5
votes
1answer
197 views

Pasting Together Fibers of a Vector Bundle

Everyone: Please forgive that I do not yet know LaTex, bro, and my English ( I am from UCV in Venezuela). I think I understand concept of bundles almost well, and that, once a vector bundle with a ...