Talk:Fibration

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This is wrong as it stands. Every fiber bundle satisfies the HLP, but not every fibration is a fiber bundle. Serre's classic example, \Omega X\to P X\to X is not a fiber bundle. Michael Larsen 12:25, 16 Sep 2003 (UTC)

Well, that all bundles satisfy the HLP is wrong too, although morally right, as the article now explains.--John Z 06:03, 9 August 2005 (UTC)

[edit] Technical accesibility

I propose adding the following definition for a Serre fibration, which I am taking from Novikov, "Topology I (General Survey)":

Let B some topological space be the base space, and let A\subset B be some subset. Let I=[0,1] be the unit interval and let \gamma:I\to B be some continuous path. The total space E is given by

E = \{\gamma | \gamma(1)\in A\}

That is, the total space E consists of all continuous paths \gamma:I\to B that connect points in A to points in B. The projection p is given by

p(\gamma) = \gamma(0)= x\in B

The fiber over a point x\in B is thus

Fx = p − 1(x) = {γ | γ(0) = x}

Properties: The total space E contains a copy of A, given by γ(t) = const. Thus A is a deformation retract of E. When A consists of a single point x_0\in B, then E is contractible. When A consists of a single point x_0\in B, the fiber Fx = p − 1(x) over a point x\in B is commonly denoted by Fx = Ω(x,x0). The fiber F_{x_0}=\Omega(x_0,x_0)=\Omega_{x_0} B over x0 is the loop space of B based at x0.

Would this be an acceptable definition of a Serre fibration? I think this is a whole lot easier, and very concrete, as compared to the mumbo-jumbo about CW-complexes; The CW-complex bit is mostly about the categorification of the thing, as best I understand it, and should be held off for later in the article.

I think it also makes the section on the long exact sequence in the article homotopy group less obtuse.

Next: A localy trivial fibration is called a fiber bundle (where we define a locally trivial fibration to the usual definition of being homeomorphic to the direct product of an open set times the fiber. Would that work? I'll try to think up of a really simple illustration of a fibration that is not a fiber bundle. linas 17:56, 16 December 2006 (UTC)

[edit] a few suggestions

I think that the second heading "categorical definition" is misleading. Under a "categorical definition" I'd understand a rephrasing of the previous stuff in categorical terms, however this is not what the section gives. Also, the references seem not quite appropriate since all of them refer to fibrations for categories, and not for spaces.

If noone objects, I might try to improve upon these points. - Saibot2 11:54, 17 March 2007 (UTC)

BTW, what's a "numerable open cover"? - Saibot2 12:08, 17 March 2007 (UTC)

An open cover is "numerable" if it admits a partition of unity. A fiber bundle is a fibration if it becomes trivial when restricted to the open sets of a numerable open cover. As far as I know, more general locally trivial fiber bundles may not be fibrations in the sense defined in this article! However, I don't know a counterexample! If anyone knows one, please tell me. John Baez (talk) 22:36, 27 January 2008 (UTC)

[edit] Connections to the fibred category article

There is a separate article on fibred category. I propose that this article be renamed "fibration (topology)", and that a redirection notice be included at the top of the article, to guide people to the fibred category article if they want to go there. I would then remove the categorical stuff from this page, which is already on that page. OK? Sam Staton 17:44, 24 October 2007 (UTC)

I've just done all this, except for the renaming. Would anyone object to that? Would "fibration (algebraic topology)" be better? Sam Staton (talk) 19:02, 20 November 2007 (UTC)