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On the Genesis of the Woods Hole Fixed Point Theorem (Commentary on [118], [119], [123], [124])

  • Loring W. Tu
Chapter
Part of the Contemporary Mathematicians book series (CM)

The Woods Hole fixed point theorem is a farreaching extension of the classical Lefschetz fixed point theoremto 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...

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Authors and Affiliations

  • Loring W. Tu
    • 1
  1. 1.Department of MathematicsTufts UniversityMedfordUSA

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