Visibility Complexes and the Baues Problem for Triangulations in the Plane

We give a positive answer for the special case of the Generalized Baues Problem which asks whether the complex of triungulations of a point set A in general position in the plane has the homotopy type of a sphere. In the process, we are led to define the visibility complex for a simplicial complex P whose vertices lie in A, and prove that this visibility complex has the same homotopy type as P. The main technique is a variant of deletion-contraction from matroid theory, along with a new method for proving homotopy equivalence of posets which we call the nerve-flag paradigm.