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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 ${\varphi }_i:f^{\ast }{E}_i\to E_i$ commuting with the complex's differential operators. If $\det (1-d{f}_x)\ne 0$ at every fixed point, thenThe lifts, not just the map of the base, determine the action.^[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.

An elliptic complex on a smooth manifold is a finite cochain complex of differential operators between sections of vector bundles,with $D_{i+1}{D}_i=0$, whose principal symbol sequence is exact at every nonzero real cotangent vector. Thus for $\xi \in T_x^{\ast }M\setminus \{0\}$, the kernel of each symbol map equals the image of the preceding one, including injectivity and surjectivity at the ends.^[ref]

The de Rham complex is the basic example. Up to an inessential nonzero scalar convention, the symbol of $d$ is $\alpha \mapsto \xi \wedge \alpha$. Choose a tangent vector $v$ with $\xi (v)=1$. The interior product identitygives a contracting homotopy for that symbol sequence and proves exactness.

The condition belongs to the whole sequence: the individual maps need not be elliptic operators between equal-rank bundles. On a compact manifold without boundary, ellipticity gives finite-dimensional cohomology. Boundary problems require additional conditions. This finiteness is what makes an alternating cohomological trace meaningful in the Atiyah-Bott fixed-point theorem.