# 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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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 $\Omega B \to F$.
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