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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 fiber is known/given, that we can define a new fiber pointwise, in manipulating each of the fibers, e.g., we may change the fiber from being R (over itself) to being R(+)R, or from R to R(x)R ; RxR*, (dual)etc. What I not too clear on, is on how one put together all the new fibers coherently into a bundle, i.e., how one construct new trivializations and transition functions to turn the space with altered fiber into a new bundle. I am particularly interest in the quotient bundle, if someone knows. I imagine we use initial charts, trivialization to construct the altered bundle, but I don't see fully how, other than I pretty sure we use multilinear algebra and functoriality somehow. Would be great if someone knew about how to do this for general fiber bundles.

Thanks You from Caracas.

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1 Answer 1

Suppose $T$ is a functor from the category of finite dimensional (real, say) vector spaces to itself. We can say that $T$ is continuous if for every finite dimensional vector space $V$ the associated map $T:\hom(V,V)\to\hom(TV,TV)$ is continuous, when we view its domain and codomain as real vector spaces.

For example, the functor $T=\Lambda^2(\mathord-)$ which computes exterior squares is continuous, as many others.

Now suppose $E$ is a locally trivial vector bundle of dimension $n$ over a space $B$, and suppose $\mathcal U$ is an open covering of $B$ over whose open sets $E$ is trivial. Then for each pair $U$, $V\in\mathcal U$ such that $U\cap V\neq\emptyset$ we have a corresponding transition function $g_{U,V}:U\cap V\to\mathrm{GL}(\mathbb R,n)$, as explained for example in the Wikipedia page for vector bundles. Moreover, we can reconstruct $E$ up to isomorphism from the knowledge of $\mathcal U$ and the family $\{g_{U,V}:U,V\in\mathcal U,U\cap V\neq\emptyset\}$ alone.

Now, suppose $T(\mathbb R^n)$ has dimension $m$. Then the vector bundle $T(E)$ can be defined to be the vector bundle which one can construct from the covering $\mathcal U$ and the family $\{\tilde g_{U,V}:U,V\in\mathcal U,U\cap V\neq\emptyset\}$ of transition functions, where for each $U$, $V\in\mathcal U$ such that $U\cap V\neq\emptyset$, the map $$\tilde g:p\in U\cap V\mapsto T(g_{U,V}(p))\in GL(m).$$

This construction gives a meaning to $T(E)$ for all continuous functors $T$, and it can be generalized for continuous functors of many variables, both covariant and contravariant. If I recall correctly, this is discussed in Milnor and Stasheff's book on Characteristic Classes, for example.

NB: there is another way of doing this...

If $E$ is a vector bundle of dimension $n$ over a space $B$, then one can attach to it a principal $\mathrm{GL(n)}$-bundle $P_E$, called the frame bundle. The fiber of $P_E$ over a point $b\in B$ is the set of all ordered bases of the fiber of $E$ over $b$.

Now, given any $\mathrm{GL}(n)$-representation $V$, we can form the bundle $P_E\times_{\mathrm{GL}(n)}V$. If we take $V=V_{\mathrm{taut}}=\mathbb R^n$ with the tautological action of $\mathrm{GL}(n)$, then $P_E\times_{\mathrm{GL}(n)}V$ is isomorphic to $E$; if we take $V=\Lambda^3(V_{\mathrm{taut}})$, then $P_E\times_{\mathrm{GL}(n)}V$ is the bundle $\Lambda^3(E)$, and so on.

 

Both constructions do the same thing: they pick some gadget which precisely captures the way the fibers of the vector bundle you start with are put together (the set of transition functions, the associated principal bundle) and then pick some other fiber and glue copies together using the same prescription. The two approaches are equivalent, of course. The second one is somewhat more «geometric» because the gadget used to record the way the fibers of $E$ are put together is in fact a geometric object and not some vaporous family of transition functions, also known, if you want to scare the kids, as a $1$-cocycle.

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this is a very nice answer. Two comments (i) you did not define $P_E \times_{\operatorname{GL}(n)} V$ (but it is defined in the wikipedia article you linked to). (ii) I call the object you refer to in your last sentence a $1$-cocycle. Are you sure you don't as well? –  Pete L. Clark Feb 10 '11 at 1:54
    
@Pete: Re: (ii) of course you are right---by now I should be able to tell the degree of a Cech cocycle, one'd think, but i am always off by one! –  Mariano Suárez-Alvarez Feb 10 '11 at 3:11

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