We return to the study of fibrations from Lecture 33. We show that that the homotopy type of the fibre of a fibration is well defined, and use the Puppe Sequence from the last lecture to prove the homotopy long exact sequence for fibrations.

We then sketch the proof of Homotopy Theorem for Fibrations, which states that a fibration  $ p \colon E \to X$ induces a functor $ \Pi(Z,X) \to \mathsf{hFib}_Z$ from the homotopy groupoid $ \Pi(Z,X)$ to the homotopy category of fibrations over $Z$ (for any space $Z$).

We then define weak fibrations, and prove that a weak fibration is equivalent to asking that the homotopy lifting property holds for any cube $I^n$. We prove that any fibre bundle is a weak fibration.

Finally, we prove Serre's Theorem: that a weak fibration $ p \colon E \to X$ with fibre $F$ has $ \pi_n(E,F) \cong \pi_n(X)$ for all $ n \ge 1$, and use this to give another proof of the homotopy long exact sequence that also works for weak fibrations (and hence also, fibre bundles).

Remark

Next Wednesday, Berit will go over the harder problems from Problem Sheet Q and R. During the final lecture of the semester (Friday 1st June), I will present an “Epilogue” where I discuss (and sketch the proofs of) all the results I didn't have time to cover this year.



Comments and questions?