Thomas McConville, Preparation for “Circuits and Hurwitz action in finite root systems”

Joseph Doolittle, Nerves of Simplicial Complexes and Reconstruction

The standard nerve of a covering $U=\{U_1, \ldots U_j\}$ of some topological space $X$ is the simplicial complex whose faces are subsets $\sigma$ of $U$ such that $\bigcap_{U_i \in \sigma} U_i \neq \emptyset$. In this talk, we introduce a generalization of this definition, and investigate some of its properties when we restrict $X$ to be a simplicial complex, and let $U$ be the set of its facets. We then follow a generalization of a construction given by Grunbaum for the standard nerve to obtain a simplicial complex whose nerve is some desired complex. This generalized construction will give a reconstruction result for simplicial complexes. The generalized nerve is a very powerful tool, and many results have already been discovered. It will have applications in many problems across topological combinatorics.