Lecture notes on trisections and cohomology

Abstract

We describe several geometric interpretations of H 2 (X ) when X is a trisected 4-manifold. The main insight is that, by analogy with Hodge theory and sheaf cohomology in algebraic geometry, classes in H 2 (X ) can be usefully interpreted as "(1,1)"-classes. First, we reinterpret work of Feller, Klug, Schirmer and Zemke and of Florens and Moussard on the (co)homology of trisected 4-manifolds in terms of the Čech cohomology of presheaves on X , in both the case of singular and de Rham cohomology. We then discuss complex line bundles, almost-complex structures, spin structures and Spin ‫ރ‬ -structures on trisected 4-manifolds. MSC2020: 57K40.