Kerodon

Example 3.2.3.6. Let $\sigma $ be an element of $\Sigma $, and let $1 \leq i \leq n$. Then the degenerate $(n+1)$-simplex $\tau = s^{n}_{i}(\sigma )$ satisfies $d^{n+1}_{j}(\tau ) = \begin{cases} \sigma & \text{ if } j \in \{ i, i+1\} \\ e & \text{ otherwise.} \end{cases}$ It follows that the multiplication map $m_ i: \pi _{n}(X,x) \times \pi _{n}(X,x) \rightarrow \pi _{n}(X,x)$ of Lemma 3.2.3.5 satisfies the identity $m_ i( [e], [\sigma ] ) = [\sigma ]$. A similar argument shows that $m_ i( [\sigma ], [e] ) = [\sigma ]$.