2025/10/23 by Andrea Bianchi, Bianchi, Andrea, Kaif Hilman +5
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2510.20646
We introduce a notion of Poincaré duality for pairs of ∞-categories, extending Poincaré-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a uniform treatment of Wall's Poincaré ads of spaces, iterated Poincaré cobordisms, and in general, diagrams of spaces parametrised by the face poset of a combinatorial manifold. In each of these cases, the theory reduces them to studying a single pair of ∞-categories and the properties of a single functor, the relative cohomology functor. Using this formalism, we prove a very general fibration theorem which, in particular, specialises to a generalisation of Klein-Qin-Su's fibration theorem for Poincaré triads to all ads. This theory also lays the foundation for future work by the authors on Poincaré cobordism categories, isovariant Poincaré spaces and string topology.