Universal simplicial complexes inspired by toric topology

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.

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

References in corpus (1)

• The problem of Buchstaber number and its combinatorial aspects