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Branched shadows and complex structures of 4-manifolds

Francesco Costantino
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Abstract

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with $Spin^c$-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-handlebody is homotopic to a complex one.

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Francesco Costantino. 2005-02-14. Branched shadows and complex structures of 4-manifolds. https://arxiv.org/abs/math/0502292

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