QM Research Seminar: A sheaf theoretic approach to homology manifolds and Poincaré and Atiyah duality

Speaker: Marco Volpe (MPIM Bonn)

Abstract:

Poincaré and Atiyah duality theorems are fundamental results in the study of the (co)homology of smooth manifolds. However, due to their purely cohomological nature, one may naturally wonder whether it is necessary to impose geometric conditions on a topological space to ensure the validity of these theorems. In this presentation, we will review and reinterpret both Poincaré and Atiyah duality theorems using the framework of sheaves of infinity categories. We will employ the formalism of six operations to extend these duality theorems beyond the realm of manifolds. Even more, we will provide a comprehensive characterization of the class of topological spaces that satisfy Poincaré and Atiyah duality. This class essentially consists of what are known as integral homology manifolds. This is a joint work with Markus Land.