Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define $$\deg_2f:=\text{#}\{\sigma\in S_n(K)|f_\text{#}(\sigma)=\tau\}\mod 2$$ and $\tau\in S_n(L)$.
Borsuk Theorem Let $f$ be an antipode-preserving map. Then $\deg_2f=1$.
We think about the lower dimensional cases. For computing the degree of antipode-preserving map between $\mathbb S^1$, we can trangulate it as a diamond, denoted by $K$. For there is always a simplicial approximation $g:Sd^m K\to K$ of $f$, which is also antipode-preserving.
My problems are
I can not figure out why $$\text{#}\{\sigma\in S_1(Sd^m K)|g_\text{#}(\sigma)=\tau\}$$ is odd.
How about the higher dimensional cases by using this combinatorial method.
Any advice is helpful. Thank you.