Geodesic
Simplicial complexes and matroids with vanishing \(T^2\)
Alexandru Constantinescu1 ; Patricia Klein ; Thái Thành Nguyễn2 ; Anurag Singh3 ; Lorenzo Venturello4
1Freie Universität Berlin
2McMaster University
3Indian Institute of Technology Bhilai
4Università di Siena
The electronic journal of combinatorics, Tome 32 (2025) no. 2
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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.
DOI : 10.37236/13236
Classification : 05E45, 05B35, 13D03, 13F55
Mots-clés : uniform matroid, monomial ideals with vanishing \(T^2\)

Alexandru Constantinescu  1   ; Patricia Klein    ; Thái Thành Nguyễn  2   ; Anurag Singh  3   ; Lorenzo Venturello  4

1 Freie Universität Berlin
2 McMaster University
3 Indian Institute of Technology Bhilai
4 Università di Siena 
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     title = {Simplicial complexes and matroids with vanishing {\(T^2\)}},
     journal = {The electronic journal of combinatorics},
     year = {2025},
     volume = {32},
     number = {2},
     doi = {10.37236/13236},
     zbl = {1564.05376},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/13236/}
}
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Alexandru Constantinescu; Patricia Klein; Thái Thành Nguyễn; Anurag Singh; Lorenzo Venturello. Simplicial complexes and matroids with vanishing \(T^2\). The electronic journal of combinatorics, Tome 32 (2025) no. 2. doi: 10.37236/13236

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