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 ${\text{ Spin}}^{ℂ}$-structures on trisected 4-manifolds.

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