Rigidity and the lower bound theorem for doubly Cohen-Macaulay complexes

Eran Nevo

doi:10.1007/s00454-007-9017-y

Rigidity and the lower bound theorem for doubly Cohen-Macaulay complexes

Eran Nevo

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We prove that for d ≥ 3, the 1-skeleton of any (d-1)-dimensional doubly Cohen-Macaulay (abbreviated 2-CM) complex is generically d-rigid. This implies that Barnette's lower bound inequalities for boundary complexes of simplicial polytopes (Barnette, D. Isr. J. Math. 10:121-125, 1971; Barnette, D. Pac. J. Math. 46:349-354, 1973) hold for every 2-CM complex of dimension ≥ 2 (see Kalai, G. Invent. Math. 88:125-151, 1987). Moreover, the initial part (g ₀,g ₁,g ₂) of the g-vector of a 2-CM complex (of dimension ≥ 3) is an M-sequence. It was conjectured by Björner and Swartz (J. Comb. Theory Ser. A 113:1305-1320, 2006) that the entire g-vector of a 2-CM complex is an M-sequence.

Original language	English
Pages (from-to)	411-418
Number of pages	8
Journal	Discrete and Computational Geometry
Volume	39
Issue number	1-3
DOIs	https://doi.org/10.1007/s00454-007-9017-y
State	Published - 1 Jan 2008
Externally published	Yes

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s00454-007-9017-y

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abstract = "We prove that for d ≥ 3, the 1-skeleton of any (d-1)-dimensional doubly Cohen-Macaulay (abbreviated 2-CM) complex is generically d-rigid. This implies that Barnette's lower bound inequalities for boundary complexes of simplicial polytopes (Barnette, D. Isr. J. Math. 10:121-125, 1971; Barnette, D. Pac. J. Math. 46:349-354, 1973) hold for every 2-CM complex of dimension ≥ 2 (see Kalai, G. Invent. Math. 88:125-151, 1987). Moreover, the initial part (g 0,g 1,g 2) of the g-vector of a 2-CM complex (of dimension ≥ 3) is an M-sequence. It was conjectured by Bj{\"o}rner and Swartz (J. Comb. Theory Ser. A 113:1305-1320, 2006) that the entire g-vector of a 2-CM complex is an M-sequence.",

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AB - We prove that for d ≥ 3, the 1-skeleton of any (d-1)-dimensional doubly Cohen-Macaulay (abbreviated 2-CM) complex is generically d-rigid. This implies that Barnette's lower bound inequalities for boundary complexes of simplicial polytopes (Barnette, D. Isr. J. Math. 10:121-125, 1971; Barnette, D. Pac. J. Math. 46:349-354, 1973) hold for every 2-CM complex of dimension ≥ 2 (see Kalai, G. Invent. Math. 88:125-151, 1987). Moreover, the initial part (g 0,g 1,g 2) of the g-vector of a 2-CM complex (of dimension ≥ 3) is an M-sequence. It was conjectured by Björner and Swartz (J. Comb. Theory Ser. A 113:1305-1320, 2006) that the entire g-vector of a 2-CM complex is an M-sequence.

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Rigidity and the lower bound theorem for doubly Cohen-Macaulay complexes

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