6
$\begingroup$

What I currently meet are all fiber bundles.

  • 0
    here http://en.wikipedia.org/wiki/Fibred_category2010-10-11

2 Answers 2

7

There's a canonical example of fibrations used in algebraic topology which are not fibre bundles. That is, given a continuous function $f : X \to Y$ there is a homotopy-equivalence $\phi : X' \to X$ and a fibration $f' : X' \to Y$ such that $f \circ \phi$ is homotopic to $f'$. I believe this idea goes back to Serre (or perhaps earlier). The fibre of $f'$ is called the homotopy-fibre of $f$.

A common usage of this construction is with the Postnikov Tower of a space. In his dissertation Jean-Pierre Serre used this (and some "closure" observations of the Serre spectral sequence of a fibration, then it was called Serre $\mathcal C$-theory, nowadays this technology is subsumed in localization) to show that most of the homotopy-groups of the spheres are finite.

  • 0
    Is it easy to see that $f’$ is not a bundle?2018-01-17
  • 0
    @MatthewKvalheim: Your question at its most wide-open could be asking the same question as the OP. Orthogonal projection of a right triangle onto one of its edges is an example of a fibration that is not a bundle.2018-01-17
  • 0
    I’m not exactly sure what you mean — maybe I need to read the details of the construction of $f’$ somewhere. But the triangle makes sense to me! Thanks a lot for the help.2018-01-18
1

I think that most examples of fibrations that you can think of off the top of your head are going to be fiber bundles (although this is probably only because of my limited experience). On the other hand, it's much easier to just prove that things are fibrations, because the definition is much weaker. As far as homotopy theory is concerned, this is going to be the same thing.