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. |> |> -----------== Posted via Deja News, The Discussion Network ==---------- |> http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own -- 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