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Let $p : E \to B$ be a Serre fibration with $b_0 \in B$ and $F=p^{-1}(b_0)$ and $e_0 \in F$. I have been trying to understand how this induces a long exact sequence $$\dots\xrightarrow{\quad \partial\quad}\pi_n(F,e_0) \xrightarrow{\quad i_*\quad}\pi_n(E,e_0) \xrightarrow{\quad p_*\quad}\pi_{n}(B,b_0)\xrightarrow{\quad \partial\quad}\pi_{n-1}(F,e_0)\xrightarrow{\quad i_*\quad}\dots$$

but I don't understand the maps in question. I'm also aware that this is covered in many books on algebraic topology which I've looked at, but it seems that many authors have drastically different ways to define these and I'm quite confused.

If I understood correctly $i_*:\pi_n(F,e_0) \to \pi_n(E,e_0)$ is induced from the inclusion $i:F \to E$ so that $i_*([\alpha])=[i\circ \alpha]$? Here $\alpha : (I^n,\partial I^n) \to (F,e_0)$ and I guess $i\circ \alpha$ makes sense?

The map $p_*$ is probably defined also like this. The main difficulty is the boundary map. I have no idea how it is defined in this case. I know how its defined in singular homology, but this isn't similar to that.

Jonathan
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  • In what sense do these authors have different definitions? Could you write into your question examples of these and explain what you don't understand? – Shrugs Dec 11 '23 at 17:54
  • May uses something called a loop space which I have not seen before and Tammo merely gives this as a theorem (6.3.2) without a proof. @K02 – Jonathan Dec 11 '23 at 18:01
  • Here's one approach. If you replace $F$ with (homotopy equivalent) homotopy fibre $I_p$ in the above sequence, then the connecting homomorphism can be understood as $\Omega^n i: \Omega^{n+1} B \rightarrow \Omega^n I_p$, where $i: \Omega B \rightarrow I_p$ is the inclusion. – abstractnonsense Dec 11 '23 at 22:48
  • Does this answer your question? It looks like it ought to, in which case your question is a duplicate. – Lee Mosher Dec 12 '23 at 15:01

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