Universal simplicial complexes inspired by toric topology
| Autor: | Baralic, Djordje, Grbic, Jelena, Vavpetic, Ales, Vucic, Aleksandar |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: In the previous preprint, there were gaps in the proofs that $K(\mathbb{Z}^n)$ and $X(\mathbb{Z}^n)$ and their links of its simplices have homotopy type of a wedge of countable infinite number of spheres $S^{n-1}$. The fact was pointed to the authors by unanimous referee who read the previous version carefully. The result is proved using direct approach instead of using discrete Morse functions |
| Databáze: | arXiv |
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