I am wondering if there is an analogue of the nine lemma for fibration sequences of spaces. The situation I am in is that I have the following diagram:
All the columns and the first 2 rows are fibration sequences. Is the 3rd row necessarily a fibration sequence? (possibly assuming that the composition is already nullhomotopic). I'm pretty sure this is true if we work in the stable homotopy category since we can use some version of the nine lemma for triangulated categories, but is it true just for topological spaces? (or possibly even just CW complexes?)