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arXiv · 1708.09565

Universal simplicial complexes inspired by toric topology

Djordje Baralic·Jelena Grbic·Ales Vavpetic·Aleksandar Vucic
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Abstract

Let $\mathbf{k}$ be the field $\mathbb{F}_p$ or the ring $\mathbb{Z}$. We study combinatorial and topological properties of the universal simplicial complexes $X(\mathbf{k}^n)$ and $K(\mathbf{k}^n)$ whose simplices are certain unimodular subsets of $\mathbf{k}^n$. As a main result we show that $X(\mathbf{k}^n)$, $K(\mathbf{k}^n)$ and the links of their simplicies are homotopy equivalent to a wedge of spheres specifying the exact number of spheres in the corresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz's result that $K(\mathbb{Z}^n)$ and $K(\mathbb{F}_2^n)$ are $(n-2)$-connected simplicial complexes. We discuss applications of these universal simplicial complexes to toric topology and number theory.

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Djordje Baralic, Jelena Grbic, Ales Vavpetic, Aleksandar Vucic. 2020-11-23. Universal simplicial complexes inspired by toric topology. https://arxiv.org/abs/1708.09565

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