TY - JOUR
T1 - Tight combinatorial manifolds and graded Betti numbers
AU - Murai, Satoshi
N1 - Publisher Copyright:
© 2015, Universitat de Barcelona.
PY - 2015/9/18
Y1 - 2015/9/18
N2 - In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres Si×Sj with j ≥ i is tight if and only if it has exactly i + 2j + 4 vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when j > 2i and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.
AB - In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres Si×Sj with j ≥ i is tight if and only if it has exactly i + 2j + 4 vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when j > 2i and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.
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U2 - 10.1007/s13348-015-0137-z
DO - 10.1007/s13348-015-0137-z
M3 - Article
AN - SCOPUS:84939236120
VL - 66
SP - 367
EP - 386
JO - Collectanea Mathematica
JF - Collectanea Mathematica
SN - 0010-0757
IS - 3
ER -