On the homotopy type of a simplicial complex

The problem of determining the homotopy type of a simplicial complex is very much simplified if the complex happens to be shellable. This means that there exists an enumeration of the facets of a particularly favourable type. But when is a simplicial complex shellable? In general, there is no simple characterization, but we can present a theorem that reduces shellability to some graph-theoretic property of the graph of flats for simple simplicial complexes of dimension 2 which are boolean representable over the superboolean semiring (we remark that all matroids satisfy this property). This is joint work with John Rhodes (Berkeley).