In this lecture we first define what it means for a sequence

$$
\cdots \to X_{n+1} \to X_n \to X_{n-1} \to \cdots
$$

of pointed spaces and pointed maps to be exact in the category $ \mathsf{hTop}_*$.

We then define the mapping fibre $Mf$ associated to a pointed map $f$. We prove the Puppe Sequence, which says that if $f \colon X \to Y$ is continuous pointed map then there is a long exact sequence in $ \mathsf{hTop}_*$ of the form

$$
\cdots \to \Omega^n(Mf) \to \Omega^n X \to \Omega^n Y \to \Omega^{n-1}(Mf) \to \cdots
$$

Finally we use this to define the long exact sequence axiom for homotopy groups, which takes the following form: if $X' \subset X$ then the pointed inclusion $X \hookrightarrow X$ gives rise to an long exact sequence
$$
\cdots \to \pi_n(X') \to \pi_n(X) \to \pi_n(X,X') \to \pi_{n-1}(X') \to \cdots
$$

of (relative) homotopy groups.


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