Is every long exact sequence
$$\cdots\to\pi_{d+1}(B)\to\pi_d(F)\to\pi_d(E)\to\pi_d(B)\to\pi_{d-1}(F)\to\cdots$$
with topological spaces $F,E$ and $B$, where $F$ is a subspace of $E$ with inclusion map $i$, induced by a Serre fibration $p:E\to B$ ? By "induced" I mean that the maps in the sequence are given by
$$p_*:\pi_d(E)\to\pi_d(B)$$
$$i_*:\pi_d(F)\to\pi_d(E)$$
and the boundary map
$$\partial:\pi_{d+1}(B)\to\pi_d(F).$$
If not, what are the conditions under which this is the case?
Thank you!
Best Answer
Ok, so one counterexample to the other interpretation of the question would be to take a fibration $F\to E\to B'$ And let $B$ be a space with the same homotopy groups as $B'$ but not homotopy equivalent to $B$. Note that $B$ must have at least two non-trivial homotopy groups.