Simplicial complexes and matroids with vanishing T²

We investigate quotients by radical monomial ideals for which T², the second cotangent cohomology module, vanishes. The dimension of the graded components of T², and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete characterization and a full list of one dimensional complexes with T²=0. We characterize the graded components of T² when the simplicial complex is a uniform matroid. Finally, we show that T² vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing T² are of corank at most two.