In our algebraic topology lecture we stated the following theorem. Let $F\rightarrow E \rightarrow B$ be a fibration (in the hurcewicz sense). W a pointed space. Then the sequence of pointed sets:
$[W,F]\rightarrow [W,E]\rightarrow[W,B]$ (where "$[W,B]$" denotes pointed homotopy classes) is exact as a sequence of pointed sets.
Our proof did the obvious thing, i.e. to show a map that becomes zero homotopic after pushing to B, comes from F, you lift the homotopy and get a map into the fiber. However, what it failed to show, is that the lifted homotopy is actually a homotopy of $pointed$ maps.
I did some research on the topic and found that different authors, have different outs to this. Whitehead demands $W$ to be a well pointed space and May defines something called "pointed fibrations" which basically do just this.
However I haven't found and failed to construct any explicit counterexamples (which might honestly be kind of hard to construct looking at the setting) where this fails for $W$ not well pointed.
Any ideas would be appreciated.
I feel the bottom end of the usual exact sequence of a fibration is best understood by considering the family of exact sequences of a fibration of groupoids as explained in Topology and Groupoids Chapter 7. A reason for this is that a fibration $p: E \to B$ of spaces gives rise to a fibration of groupoids $\pi_1 p: \pi_1 E \to \pi_1 B$ of fundamental groupoids.