On the Genesis of the Woods Hole Fixed Point Theorem

The Woods Hole fixed point theorem is a farreaching extension of the classical Lefschetz fixed point theorem to vector bundles. It has as corollaries a holomorphic Lefschetz formula for complex manifolds and the Weyl character formula for the irreducible representations of a compact Lie group. Apart from its importance in its own right, the Woods Hole fixed point theorem is crucial in the history of mathematics as a precursor to the Atiyah–Bott fixed point theorem for elliptic complexes [7], one of the crowning glories of the analysis and topology of manifolds. On the algebraic side it led to Verdier’s Lefschetz fixed point theorem in étale cohomology ([13], [23]). Indeed, Atiyah was awarded the Fields Medal in 1966 and the citation reads in part that Atiyah “proved jointly with Singer the index theorem of elliptic operators on complex manifolds” and “worked in collaboration with Bott to prove a fixed point theorem related to the Lefschetz formula.” The discovery of these fixed point theorems dates back to the AMS Summer Research Institute on Algebraic Geometry at Woods Hole, a small town by the sea in Massachusetts, in 1964. With the passage of time, memories of how the theorems came about have become somewhat murky. In 2001 Raoul Bott unwittingly ignited a controversy through an interview published in the Notices of the AMS, a controversy that to this day has not been settled. There are three protagonists in this drama: Michael Atiyah, Raoul Bott, and Goro Shimura— all three giants of twentieth-century mathematics.