Lefschetz-Hopf fixed-point theorem
= Lefschetz-Hopf fixed-point theorem
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{wiki=Lefschetz_fixed-point_theorem#Lefschetz–Hopf_theorem}
If a smooth self-map of a compact manifold has only isolated nondegenerate fixed points, then
$$
L(f)=\sum_{f(x)=x}\operatorname{sign}\det(I-Df_x).
$$
Every summand has absolute value one, so the number of fixed points is at least $|L(f)|$.