Lemma 3.2.3.2. Let $\vec{\sigma } = (\sigma _0, \sigma _1, \ldots , \sigma _{n+1})$ be an $(n+2)$-tuple of elements of $\Sigma $. The condition that $\vec{\sigma }$ bounds depends only on the sequence of homotopy classes $\{ [ \sigma _ i ] \in \pi _{n}(X,x) \} _{0 \leq i \leq n+1}$. In other words, if $\vec{\sigma }' = (\sigma '_{0}, \sigma '_{1}, \ldots , \sigma '_{n+1})$ is another $(n+2)$-tuple of elements of $\Sigma $ satisfying $[\sigma '_ i] = [\sigma _ i]$ for $0 \leq i \leq n+1$ and $\vec{\sigma }$ bounds, then $\vec{\sigma }'$ also bounds.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let us identify $\vec{\sigma }$ and $\vec{\sigma }'$ with morphisms of simplicial sets $f,f': \operatorname{\partial \Delta }^{n+1} \rightarrow X$ (carrying the $(n-1)$-skeleton of $\operatorname{\partial \Delta }^{n+1}$ to the vertex $x$). For $0 \leq i \leq n+1$, the equality $[\sigma _ i] = [ \sigma '_{i} ]$ allows us choose a homotopy $h_{i}: \Delta ^{1} \times \Delta ^{n} \rightarrow X$ from $\sigma _{i}$ to $\sigma '_{i}$ which carries $\Delta ^{1} \times \operatorname{\partial \Delta }^{n}$ to the vertex $\{ x\} \subseteq X$. These maps can be amalgamated to a homotopy $h$ from $f$ to $f'$: that is, an edge joining $f$ to $f'$ in the simplicial set $\operatorname{Fun}( \operatorname{\partial \Delta }^{n+1}, X)$. If $\vec{\sigma }$ bounds, then $f$ can be extended to an $(n+1)$-simplex $\tau : \Delta ^{n+1} \rightarrow X$. Since $X$ is a Kan complex, the restriction map $\operatorname{Fun}( \Delta ^{n+1}, X) \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^{n+1}, X)$ is a Kan fibration (Corollary 3.1.3.3), so $h$ can be extended to a homotopy $\widetilde{h}$ from $\tau $ to another map $\tau ': \Delta ^{n+1} \rightarrow X$ satisfying $\tau '|_{ \operatorname{\partial \Delta }^{n+1} } = f'$. It follows that the tuple $\vec{\sigma }'$ also bounds. $\square$