What Is a Quotient Bundle?

[Bacle]

Hi, everybody:

I have read different sources for quotient bundle: Milnor and Stasheff, Wikipedia
Wolfram, and I still cannot figure out how it's defined. All I know is that it involves
a space X, a subspace Y, and a restriction.

Let Tx be a bundle p:M-->X for X, and Tx|y be the restriction of Tx to y, i.e., the
bundle with top space p^-1 (y):I see a mention of a(n informal) exact
sequence (since there is no actual algebraic map to speak of a kernel). Tx/y is defined
as the "completion of the exact sequence"

0-Ty-Tx|y-Tx/y-0

where it would seem the first two maps are inclusions.

Any hints, ideas, please.?

Thanks.

Since this is not an exact sequence in the algebraic sense, I don't know how to
f

 

[zhentil]

It is an exact sequence in the algebraic sense. Think of the restriction of the tangent bundle to a submanifold. It splits (look at it in a chart) into the tangent bundle of the submanifold and the normal bundle. Once you understand how the normal bundle works, the generalization is straightforward.

 

[lavinia]

you just need to check that the space obtained by modding out the sub-bundle in each fiber is still locally trivial

 

[Bacle]

Thanks , both.

I think I realized the issue (please correct me o.wise): we start with a bundle:

p:E
|
\/
x

And y is a subspace of x.

The sequence is defined pointwise: we have y<x ( y a subspace of x) , so that
we consider T_py ,the tangent space of p , for any p in x , as a subspace of the
tangent space of p in x:

0-T_py-T_px|y-T_px/y-0

then we consider the quotient vector space T_px|y / T_py

at every point in y, right.?. But, how do we define the projection map in T_px/y .?.

I would imagine we compose the original p with a projection onto the first

component of the quotient , otherwise, it becomes a mess to define p on

the cosets.

Is this right.?

Thanks.

 

[lavinia]

the space X is fixed and you have a sub-bundle of the bundle, E, over X. This sub-bundle is also a bundle over X with fiber a subspace of the fiber of E over X. The exact sequence is an exact sequence of fibers. The quotient bundle is another bundle over X.

 

[Bacle]

Thanks, Lavinia, I wanted to know how we defined the projection in the quotient bundle,
tho, since we are dealing with cosets, and p in E-->X is defined on elelments of E. Do we
apply p in Tx/y , to the E component of the quotient.?

 

[zhentil]

Fibrewise (and bundle-wise over a compact manifold) the exact sequence splits, so you can define the projection map to complete the exact triangle.

 

[Bacle]

Thanks again, Zhentil; I forgot that every exact sequence of vector spaces splits,
and if the sequence doesn't split, then you must acquit (remember J.Cochran
on O.J's trial)

 

[lavinia]

Thanks, Lavinia, I wanted to know how we defined the projection in the quotient bundle,
tho, since we are dealing with cosets, and p in E-->X is defined on elelments of E. Do we
apply p in Tx/y , to the E component of the quotient.?

yes. though I don't think that cosets are required. the only thing you need is that the quotient fibers are preserved by the coordinate transformations of E. this automatically gives you a bundle.

Not all bundles are vector bundles yet they still have quotient bundles and these quotients need not be obtained by modding out by a sub-bundle either. For instance, take the unit sphere bundle of a vector bundle with a Riemannian metric. The antipodal map on each fiber commutes with the coordinate transformation so the quotient fibers - which are projective spaces - form a bundle.