What is the/a Quotient Bundle.?

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<sup>-1</sup> (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.

Thanks.

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

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

Thanks , both.

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
Gold Member
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.

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.?

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

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