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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.

For a holomorphic map $f:X\to X$ of a compact complex manifold whose fixed points are isolated and satisfy ${\det }_{\mathbb{C}}(1-d{f}_x)\ne 0$, the holomorphic Lefschetz fixed-point formula for the structure sheaf saysThe left side uses sheaf cohomology of holomorphic functions; the right side uses the complex determinant, rather than its absolute value. This is a holomorphic form of the Atiyah-Bott fixed-point theorem.^[ref]

On the complex projective line, $f(z)=az$, with $a\ne 0,1$, fixes zero and infinity. The two derivatives are $a$ and $a^{-1}$, so the local sum is $1/(1-a)+1/(1-a^{-1})=1$. Globally $H^0(\mathcal{O})=\mathbb{C}$ and $H^1(\mathcal{O})=0$, giving the same answer. Individual local terms need not be integers.

Michael Atiyah connected topology, geometry and analysis. With Isadore Singer, he proved the Atiyah-Singer index theorem, relating an analytic index of an elliptic operator to topological data. Oxford's mathematical memorial explains its connection with Riemann-Roch theorem and Gauss-Bonnet theorem formulas.^[ref]

His work with Raoul Bott provides another local-to-global bridge: a holomorphic Lefschetz fixed-point formula expresses a global alternating trace in terms of contributions at fixed points. Such results make local geometric calculations determine invariants of global spaces of solutions.^[ref]

Raoul Bott made fundamental contributions to topology and differential geometry, with applications to Lie groups and differential operators. Harvard's departmental memorial documents this range of work.^[ref]

With Michael Atiyah, he developed a holomorphic Lefschetz fixed-point formula for elliptic complexes. In the holomorphic setting, global traces on cohomology become a sum of local contributions from nondegenerate fixed points. This connects a map's behavior at isolated points with its action on global solution spaces.^[ref]