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Lecture notes on trisections and cohomology

### Peter Lambert-Cole

Vol. 5 (2022), No. 1, 245–264
##### Abstract

We describe several geometric interpretations of ${H}_{2}\left(X\right)$ 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 ${H}_{2}\left(X\right)$ 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 -structures on trisected 4-manifolds.

##### Keywords
4-manifolds, trisections
Primary: 57K40