Simplicial Complexes and Matroids with Vanishing $T^2

• Published in: The Electronic journal of combinatorics Vol. 32; no. 2
• Main Authors: Constantinescu, Alexandru, Klein, Patricia, Nguyễn, Thái Thành, Singh, Anurag, Venturello, Lorenzo
• Format: Journal Article
• Language: English
• Published: 25.04.2025
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/13236

We investigate quotients by radical monomial ideals for which $T^2$, the second cotangent cohomology module, vanishes. The dimension of the graded components of $T^2$, 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 $T^ 2 = 0$. We characterize the graded components of $T^ 2$ when the simplicial complex is a uniform matroid. Finally, we show that $T^2$ vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing $T^2$ are of corank at most two.