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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.

The de Rham complex of a smooth manifold $M$ is the cochain complex of smooth differential forms with exterior derivative:Here $n=\dim M$ and ${\Omega }^0(M)=C^{\infty }(M)$. The identity $d^2=0$ makes every exact differential form closed. The quotient of closed $k$-forms by exact $k$-forms is the de Rham cohomology group $H_{\mathrm{dR}}^k(M)$.^[ref]

In Euclidean space the Poincaré lemma makes this complex exact in positive degrees. On a general manifold it is locally exact in small coordinate balls, but global exactness can fail. For the unit circle, the restriction of $\omega =-ydx+xdy$ has integral $2\pi$ around one positive turn. It is closed, since the circle has no nonzero two-forms. It cannot equal $df$ for a globally defined smooth function on the circle: the integral of a derivative around a closed loop is zero.

The complex therefore separates local differential identities from global obstructions. In degree zero, $df=0$ means that $f$ is locally constant; in degree one, periods around loops can detect a failure to possess a global potential.

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.