Atiyah-Bott's fixed point theorem via categorification

Abstract: A famous result of Atiyah and Bott in geometric topology says that a smooth action by a cyclic p-group on a smooth closed orientable manifold cannot have just a single fixed point when p is an odd prime. This result was proved using the Atiyah-Singer index theorem. In this talk, I will explain a different, purely homotopical, proof which in particular exhibits that the theorem is really a consequence of ``global'' homotopical reasons rather than ``local'' geometric ones. To this end, I will introduce a theory of Poincare duality for arbitrary topoi together with a suite of ``basechange'' principles. I will then sketch how this abstract theory reduces the theorem to an elementary Tate cohomology calculation by working with an equivariant topos. This is based on joint work with D. Kirstein and C. Kremer from arXiv:2405.17641.

algebraic geometryrepresentation theory

Audience: researchers in the topic

Algebra and Geometry Seminar @ HKUST

Series comments: Algebra and Geometry seminar at the Hong Kong University of Science and Technology (HKUST).

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