# Connecting Homomorphism in LES of fibration

Let $p:E\rightarrow B$ be a Serre fibration of path connected spaces with fiber $F$. Are the connecting homomorphisms $\partial:\pi_{n+1}(B)\rightarrow \pi_{n}(F)$ in the long exact sequence of $p$ induced by a continuous map $\Omega B\rightarrow F$?

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## 1 Answer

Consider the path fibration $PB \to B$, assuming you've chosen a base-point in $B$. This map is null homotopic. A constant map $PB \to B$ lifts to unique base-point preserving map $PB \to E$ (constant the base-point in $F$). So provided your fibration satisfies the homotopy lifting property for the special case of the path fibration over $B$, you get a map $PB \to E$ covering the path fibration $PB \to E$. Restricting this to the loop space $\Omega B \subset PB$ gives a map $\Omega B \to F$.

I presume it's known and pretty easy to check whether or not Serre fibrations have this property, but certainly general fibrations have them (say, taking the fibration definition in Hatcher's book).

Were you interested in this technical detail on the distinction between fibrations and Serre fibrations, or were you more interested in the general question of when the connecting homomorphism is induced by an actual map of spaces?

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 I have seen that this is true for general (Hurewicz) fibrations but was wondering if this also was true for Serre fibrations since Serre is enough to get a LES. – J.K.T. Nov 16 '10 at 21:56 Off the top of my head I don't know the answer but in many reasonable situations the answer seems to be yes. For example, if $B$ is a simplicial complex I believe the answer should be yes. – Ryan Budney Nov 16 '10 at 23:01