Canonical definition of quotient bundle

Question

For a vector bundle $\pi:E\to M$ and a subbundle $\pi':E'\to M$ of $E$ where the ranks are $r, r'$, we can define the quotient bundle $E/E'\to M$. The usual construction is to find an open cover where there is a local frame $\sigma_1, \cdots, \sigma_r$ of $E$ such that the first $r'$ sections form a local frame for $E'$, and then the rest would form a frame for $E/E'$. I understand this construction, but it looks noncanonical because the considered open cover is not very canonical.

Is there a way to canonically define the quotient bundle without such choice of open cover? And moreover, would 'canonicality' matter for applications of the quotient bundle?

Answer 1

As always, the ‘best’ way to define such things is via a universal property. Universal properties take care of uniqueness, while existence is usually established by some ad-hoc method. For example, starting at the level of vector spaces first:

Definition/Theorem (Quotient Vector Spaces):

Let $E$ be a vector space over a field $\Bbb{F}$, and $F$ a vector subspace. Then, there is a pair $(\widetilde{E},\pi)$ where $\widetilde{E}$ is a vector space over $\Bbb{F}$ and $\pi:E\to \widetilde{E}$ is a linear map, such that the following condition is satisfied:

• For any vector space $G$ over $\Bbb{F}$, and any linear map $f:E\to G$, if $F\subset \ker f$, then there is a unique linear transformation $\widetilde{f}:\widetilde{E}\to G$ such that $f=\widetilde{f}\circ \pi$ (i.e a certain diagram commutes).

Moreover, the pair $(\widetilde{E},\pi)$ is unique up to unique isomorphism, so we simply denote it as $(E/F,\pi)$ and refer to it as the quotient vector space of $E$ by $F$.

The uniqueness up to unique isomorphism tells us that there is only one such vector space, modulo “implementation details” (although not a perfect analogy, I like to think of it as saying something like every language can express the same information, so an essay in English has just as much information as an Essay in French, as long as you have a dictionary to translate between the two), and also because of the uniqueness, it justifies our notation $E/F$, to express that this vector space really depends only on $E$ and $F$ (though by slight abuse of language we typically suppress mentioning $\pi$).

The existence of a vector space $E/F$ and a linear map $\pi:E\to E/F$ satisfying the condition in the bullet point is of course something which has to be done. It doesn’t matter how you do it, but once you manage to prove the existence, then you have captured all the information it has because the universal property is satisfied.

If you now replace the field $\Bbb{F}$ with $\Bbb{R}$ or $\Bbb{C}$, and “vector space” with “vector bundle”, “vector subspace” with “vector subbundle”, and “linear map” with “vector bundle morphism”, you get the universal definition of quotient bundles.

Such universal property-based definitions are good not only for quotients, but also for tensor products of vector spaces, and by extension, for vector bundles. You can even do it for direct sums.