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arXiv · 2305.03220

Combinatorics of higher-dimensional tropical covers

Alejandro Vargas
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Abstract

We develop a combinatorial framework to study certain polyhedral maps which are higher-dimensional analogues of tropical covers between metric graphs. Under a mild combinatorial assumption, we show that a map satisfies the so-called balancing condition if and only if it is an indexed branched cover, i.e.~locally over connected sets the count with multiplicity of points in every fibre is a constant, which in particular gives a well-defined global degree when the target is connected. Given a balanced map $(\Sigma, m_\Sigma) \to \Delta$, we lift several connectivity properties of $\Delta$ to~$\Sigma$. Using these lifting results we determine whether a multiplicity $m_{\mathcal U}$ that is defined only on the interiors of maximal cells of $\Sigma$ can be extended to all $\Sigma$ in a balanced manner. This relies on a strong connectivity assumption; we give a counterexample when this is missing.

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Alejandro Vargas. 2023-05-05. Combinatorics of higher-dimensional tropical covers. https://arxiv.org/abs/2305.03220

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