# Universal simplicial complexes inspired by toric topology

# Universal simplicial complexes inspired by toric topology

**Zoom link: https://princeton.zoom.us/j/92116764865**

**Passcode: 114700**

Let k be the field F_p or the ring Z. In this talk I’ll discuss combinatorial and topological properties of the universal complexes X(k^{n}) and K(k^{n}) whose simplices are certain unimodular subsets of k^{n}. I’ll describe their f-vectors, show that they are shellable but not shifted, and mention their applications in toric topology and number theory. As a main result I’ll show that X(k^{n}), K(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(Z^{n}) and K(F_{2}^{n}) are (n - 2)-connected simplicial complexes.

This is joint work with Djordje Baralic, Aleš Vavpetic, and Aleksandar Vucic.