I am reading Elements of Homotopy Theory by George W. Whitehead. In the section about maps of the $n$-sphere into itself, in the second last paragraph of the text quoted below, he says that "Then an easy argument shows that $d(f)=p(f)-n(f)$".
May I ask how the "easy argument" is established? Really feeling frustrated when "an easy argument" does not seem obvious to me.
The map $\phi$ is simplicial with respect to some subdivision of $L$ that I'll denote $L'$. To describe this subdivision, for any $n$-simplex $\sigma$ of $L$ such that $f | \sigma$ is not constant, the subdivision $L'$ contains an $n$-simplex I'll denote $\sigma'$ which is contained in the interior of $\sigma$ such that $f$ maps $\sigma'$ to $\Delta^n_0$ by a simplicial homeomorphism (when $\sigma$ is modelled by $\Delta^n_0$ then $\sigma'$ corresponds to $\Delta^*$). Take $c_{L'}$ to be the fundamental $n$-cycle for the subdivision $L'$. Take $c_K$ to be the fundamental $n$-cycle for $K$. Then by construction $f_\#(c_{L'})$, which is an $n$-cycle of $K$, assigns coefficient $p(f)-n(f)$ to the simplex $\Delta^n_0$: for each of the $p(f)$ $n$-simplices $\sigma$ of $L$ that are mapped with positive orientation the corresponding $\sigma'$ maps to $\Delta^n_0$ preserving orientation and so generates a $+1$ coefficient; and each of the $n(f)$ that are mapped with negative orientation generates a $-1$ coefficient. So $d(f) = p(f)-n(f)$.