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From Husemoller, page 8 (rephrased):

Theorem: Let $E, B \in \mathbf{Top}_*$, and $F \to E \to B$ be a Serre fibration. Then there is a natural group homomorphism $\partial: \pi_n(B) \to \pi_{n-1}(F)$ such that the sequence $$\ldots \to \pi_n(E) \to \pi_n(B) \ \xrightarrow{\partial}\ \pi_{n-1}(F) \to \pi_{n-1}(E) \to \ldots$$ is exact.

Exercise: apply this theorem to:
1. $\mathbb{Z} \to \mathbb{R} \to S^1$,
2. $\mathbb{Z_2} \to S^n \to \mathbb{R}P^n$,
3. $S^1 \to S^{2n+1} \to \mathbb{C}P^n$.

How to approach this? The definition via homotopy lifting is not very helpful. Husemoller mentions that it is sufficient to check each individual cell in a CW complex, but does it mean that I only have to check for $I^k,\ k = 1,\ldots,n$ [fixed bad typo]? Some hint would be nice, I'm not very familiar with CWs besides what Husemoller himself mentioned earlier (although I did make sure I understood all the proofs he gave).

PS: I asked for a lot of hints recently, it worries me. Were these questions hint-worthy?

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Any fiber bundle is a Serre fibration. –  Grigory M Aug 6 '11 at 9:53
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Alexei, re: PS. If it counts anything for you, I don't think I've seen anything here from you that I'd see as least cause to be worried. Your questions show understanding, you are striving for a deeper grasp of various things and your questions aren't of a silly nature, as far as I can tell. –  t.b. Aug 6 '11 at 16:14

2 Answers 2

up vote 1 down vote accepted

It is equivalent to ask that maps from finite products of unit intervals into the base space lift. That is, you only need to lift maps $f:I^n\rightarrow B$ ($n$ is a non zero integer, $I=[0,1]$). This is no problem for the 2 first examples since they are covering spaces, and the total space is simply connected.

Indeed, under mild local conditions (like Hausdorff, connected and local connectedness I think) for $(X,x_0)$ a pointed space, and $p:E\rightarrow B$ a covering space, $b_0\in B$ a base point, and $e_0\in E_{b_0}\subset E$ a point in the fibre over $b_0$, and $f:X\rightarrow B$, $f$ lifts to $E$ iff $f_*\pi_1(X,x_0)\subset p_*\pi_1(E,e_0)$.

This works for the first 2 cases, because $I^n$ is simply connected, and the two fibrations are covers.

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Where can I find proofs of these, especially the proof that lifting of maps from $I^k$ is sufficient? –  Alexei Averchenko Aug 6 '11 at 9:43

It is also worth noting that any map can be replaced by a fibration. Replaced in the sense that the maps are homotopic, the domain is also changed but its homotopy type is not changed. Since you don't change the homotopy type the homotopy groups and induced maps in homotopy are not changed. You can do this in general when you want to do computations, pretend the maps are fibrations or cofibrations, because up to homotopy everything is.

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(Re: up to homotopy everything is) I'd say, the real question (well, since Serre taught us so) is not whether a map is a (co)fibration, but how to compute its homotopy (co)fiber. And if the map happens to be a (co)fibration already — you're lucky. (But the statement that one can always compute it sounds like... an exaggeration to me.) –  Grigory M Aug 8 '11 at 17:07

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