#### Volume 25, issue 6 (2021)

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A homology theory for tropical cycles on integral affine manifolds and a perfect pairing

### Helge Ruddat

Geometry & Topology 25 (2021) 3079–3132
##### Abstract

We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. We show the pairing is perfect over $ℚ$ in degree $1$ when the manifold has at worst symple singularities. By joint work with Siebert, the pairing computes period integrals and its perfectness implies the versality of canonical Calabi–Yau degenerations. We also give an intersection-theoretic application for Strominger–Yau–Zaslow fibrations. The treatment of the cap product and Poincaré–Lefschetz by simplicial methods for constructible sheaves might be of independent interest.

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##### Keywords
tropical homology, torus fibration, SYZ, integrable system, toric degeneration, mirror symmetry, versality, affine structure, Picard-Lefschetz
##### Mathematical Subject Classification 2010
Primary: 14J32
Secondary: 05E45, 14D06, 14T05, 32S60, 55U10