singular homological degree of the identity on the sphere

Given the n-sphere \(S^n\) and the singular homological degree,

\begin{align*} \mathrm{deg}(\mathrm{id}_{S^n}) = 1 \end{align*}

By functorality of the singular homology functor, we get

\begin{align*} \mathrm{H}_n(\mathrm{id}_{S^n}) =& \mathrm{id}_{\mathrm{H}_n(S^n)} \\ \cong& \mathrm{id}_{\mathbb{Z}} \end{align*}

Thus

\begin{align*} \mathrm{id}_{\mathbb{Z}} \cong 1 \end{align*}