= Atiyah-Bott fixed-point theorem
{c}
The <Atiyah-Bott fixed-point theorem> computes an alternating <trace> on the <cohomology> of an <elliptic complex> from local data at <fixed points>. In the isolated, nondegenerate case, let $f:M\to M$ be a smooth map of a closed <smooth manifold>, with compatible lifts $\phi_i:f^*E_i\to E_i$ commuting with the complex's differential operators. If $\det(1-df_x)\ne0$ at every fixed point, then
$$
\sum_i(-1)^i\operatorname{tr}(f^*_{\phi_i}|H^i)
=\sum_{f(x)=x}\frac{\sum_i(-1)^i\operatorname{tr}(\phi_{i,x})}{|\det_{\mathbb R}(1-df_x)|}.
$$
The lifts, not just the map of the base, determine the action.<A Lefschetz Fixed Point Formula for Elliptic Complexes: I by M. F. Atiyah and R. Bott>[Introduction, isolated fixed-point formula]{ref}
For the <de Rham complex> and reflection of a <circle>, there are two <fixed points> with derivative $-1$. At each, the alternating fibre trace is $1-(-1)=2$ and the denominator is two, giving local contribution one. Globally the induced actions on $H^0$ and $H^1$ are $1$ and $-1$, so both sides equal two. The <holomorphic Lefschetz fixed-point formula> applies the same idea to holomorphic complexes.
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