# 45. Fibrations and weak fibrations

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?