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