Lefschetz number of a doubled map
= Lefschetz number of a doubled map
{c}
{title2=$L(F)=2L(f)-L(f|_{\partial N})$}
Let $F$ be the self-map of the double $D(N)=N\cup_{\partial N}N$ induced by a self-map $f$ preserving the boundary. Naturality of the <Mayer–Vietoris sequence> and alternating-trace additivity give
$$
L(F)=2L(f)-L(f|_{\partial N}).
$$