What is the Connection Between Fibre Bundles and Their Components?

[matness]

In the definition of fibre bundle we have a structure consist of (E, B,F, G, p, phi)
E:total space
B:base manifold = E/R where R is a relation
p:projection map from E to B
F: fibre
G:lie group acting on F etc.

the relation between E and B is obvious but i don't get connection between F and E also the roles of phi(family of homeomorphisms) or G exactly.

I don't want to just read the defn and pass
I stucked at this defn and really need help.
Can you give any explanation or an example ?

 

[mathwonk]

E is thought of a family of copies of F, parametrized by B.

hopefully by viewing E this way one can combine in formation about B and F into information on E, using G as a way to see how to combine it.basic example is E = tangent bundle to B, with G = linear group of coord changes in the tangent spaces.

the 50 year old book, topology of fibre bundles, by steenrod is still a classic, and just reading the first few pages of examples gives already a good feel for the concept.

 

[matness]

Can we say F consists of p^{-1} (x) where x€B
or does this destroy generality of F?
Do we always construct E starting from fibersc?
..Or are they independently chosen?

(Also thanks for book suggestion, it seems from books.google that what i want is there.Unfortuanetely i have to wait for the library's opening hour)

 

[matness]

thank you again for the book
now everything is clear

 

[andytoh]

the 50 year old book, topology of fibre bundles, by steenrod is still a classic, and just reading the first few pages of examples gives already a good feel for the concept.

I own a copy of the book, and it is good with clear examples. Though the notations are a bit different from today's literature on fiber bundles. It's perhaps a good idea to study covering spaces from Munkres's Topology first.