= Lefschetz-Hopf fixed-point theorem
{c}
Let $f:M\to M$ be a smooth self-map of a \b[<closed manifold>], and suppose every <fixed point> $x$ is nondegenerate, meaning that $I-Df_x$ is an <invertible linear map>. The <fixed points> are isolated and, by <compactness>, finite in number. Their local <fixed point indices> give
$$
L(f)=\sum_{f(x)=x}\operatorname{sign}\det(I-Df_x).
$$
The <determinant> is computed in any local coordinate <basis>; changing that <basis> conjugates the <linear map> and does not change its sign. Near $x$, the displacement map is $v\mapsto (I-Df_x)v$ to first order, explaining its local <mapping degree>. Summing these local degrees gives the global <Lefschetz number>.
Each summand is $+1$ or $-1$, so there are at least $|L(f)|$ <fixed points>. The nondegeneracy condition is essential for this determinant formula; degenerate isolated <fixed points> require their actual local <fixed point indices>. A <manifold with boundary> requires boundary hypotheses and is not included in the displayed statement.
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