From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: a vector bundle question
Date: 30 Jan 1999 06:59:33 -0500
Newsgroups: sci.math

mikegraber@my-dejanews.com writes:

>Hi, I'm reading Milnor and Stasheff,
>Characteristic Classes.  Maybe one of you
>could help me with a problem I'm having.
>They define a canonical line bundle over
>a real projective space Pn as the subset
>of Pn x Rn+1 consisting of all pairs ((-x,+x),v)
>such that v is a multiple of x.  They prove
>that the canonical bundle is not trivial
>(Theorem 2.1).  Then they
>talk about a "trivial line bundle" over Pn.
>I think by a trivial bundle they mean a
>bundle where all the group transition
>functions are the group identity element.
>But how can I get a line bundle over Pn
>where all the transition functions are
>the identity?  

As (for example) "the subset of Pn x Rn+1 
consisting of all pairs ((-x,+x),v) such that 
v is a multiple" of some fixed non-zero vector 
e in Rn+1.

>How is the canonical
>line bundle different from the trivial
>line bundle over Pn?

Cf. Theorem 2.1 (and do be careful in how 
you use the word "the").

Lee Rudolph
==============================================================================

From: hook@nas.nasa.gov (Ed Hook)
Subject: Re: a vector bundle question
Date: 30 Jan 1999 18:37:35 GMT
Newsgroups: sci.math

In article <78trab$oj4$1@nnrp1.dejanews.com>, mikegraber@my-dejanews.com writes:
|> Hi, I'm reading Milnor and Stasheff,
|> Characteristic Classes.  Maybe one of you
|> could help me with a problem I'm having.
|> They define a canonical line bundle over
|> a real projective space Pn as the subset
|> of Pn x Rn+1 consisting of all pairs ((-x,+x),v)
|> such that v is a multiple of x.  They prove
|> that the canonical bundle is not trivial
|> (Theorem 2.1).  Then they
|> talk about a "trivial line bundle" over Pn.
|> I think by a trivial bundle they mean a
|> bundle where all the group transition
|> functions are the group identity element.

  Well, I guess it actually works out like that, but [really] it's
  much simpler to think of the trivial line bundle over P^n as given
  by the projection map P^n x R^1 --> P^n. And there's nothing special
  here about P^n or R^1 -- given any reasonable space X and any positive
  integer k, the "trivial k-plane bundle over X" is the projection onto
  the first factor X x R^k --> X.

|> But how can I get a line bundle over Pn
|> where all the transition functions are
|> the identity?  How is the canonical
|> line bundle different from the trivial
|> line bundle over Pn?

  Well, the canonical (or "tautological") line bundle over P^n isn't
  trivial :-) Beyond that, you'll eventually discover that the canonical
  line bundle (sort of) encodes the structure of line bundles in general,
  in the sense that a line bundle over a nice space X corresponds to a
  map from X into P^n (for some suitably large n), the correspondence
  being that the line bundle is isomorphic to the pullback of the 
  canonical line bundle under the map. *Whew* -- quite a mouthful, huh ?
  Anyhow, what that means is that you know pretty much everything there
  is to know about line bundles, once you get a good mental picture of
  the canonical line bundles over real projective spaces. (That last is
  probably a good example of exuberant overstatement ...)

|> Thanks, Mike.
|> 
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-- 
 Ed Hook                              |       Copula eam, se non posit
 MRJ Technology Solutions, Inc.       |         acceptera jocularum.
 NAS, NASA Ames Research Center       | I can barely speak for myself, much 
 Internet: hook@nas.nasa.gov          |         less for my employer