Let $X$ be a finite simplical complex, and let $g:X\to X$ be a continuous simplicial map such that $g(\sigma)\cap \sigma=\emptyset$ for every simplex $\sigma $ in $X$. Then why is the diagonal of the matrix of the map $g_*:H_n(X^n,X^{n-1})\to (X^n,X^{n-1})$ is zero? (This question is from Hatcher's Algebraic Topology. Hatcher uses this statement in the proof of Lefschetz's fixed point theorem(Theorem 2C.3), but I can't see why this is true.)
First, I know that since $g$ is simplicial, we have $g(X^n)\subset X^n$ for all $n$, so $g$ induces a map $g_*:H_n(X^n,X^{n-1})\to H_n(X^n,X^{n-1})$. Also, I know that $H_n(X^n,X^{n-1})$ is free abelian with basis corresponding to the $n$-cells of $X$. But I can't see how the result follows. Thanks in advance.