The face numbers of homology spheres
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866911936685277184 |
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| author | Chong, Kai Fong Ernest Tay, Tiong Seng |
| author_facet | Chong, Kai Fong Ernest Tay, Tiong Seng |
| contents | The $g$-theorem is a momentous result in combinatorics that gives a complete numerical characterization of the face numbers of simplicial convex polytopes. The $g$-conjecture asserts that the same numerical conditions given in the $g$-theorem also characterizes the face numbers of all simplicial spheres, or even more generally, all simplicial homology spheres.
In this paper, we prove the $g$-conjecture for simplicial $\mathbb{R}$-homology spheres. A key idea in our proof is a new algebra structure for polytopal complexes. Given a polytopal $d$-complex $Δ$, we use ideas from rigidity theory to construct a graded Artinian $\mathbb{R}$-algebra $Ψ(Δ,ν)$ of stresses on a PL realization $ν$ of $Δ$ in $\mathbb{R}^d$, where overlapping realized $d$-faces are allowed. In particular, we prove that if $Δ$ is a simplicial $\mathbb{R}$-homology sphere, then for generic PL realizations $ν$, the stress algebra $Ψ(Δ,ν)$ is Gorenstein and has the weak Lefschetz property. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1706_03322 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | The face numbers of homology spheres Chong, Kai Fong Ernest Tay, Tiong Seng Combinatorics Commutative Algebra Rings and Algebras Primary: 05E45, Secondary: 05E40, 13A02, 13E10, 13J30, 52B70, 52C25 The $g$-theorem is a momentous result in combinatorics that gives a complete numerical characterization of the face numbers of simplicial convex polytopes. The $g$-conjecture asserts that the same numerical conditions given in the $g$-theorem also characterizes the face numbers of all simplicial spheres, or even more generally, all simplicial homology spheres. In this paper, we prove the $g$-conjecture for simplicial $\mathbb{R}$-homology spheres. A key idea in our proof is a new algebra structure for polytopal complexes. Given a polytopal $d$-complex $Δ$, we use ideas from rigidity theory to construct a graded Artinian $\mathbb{R}$-algebra $Ψ(Δ,ν)$ of stresses on a PL realization $ν$ of $Δ$ in $\mathbb{R}^d$, where overlapping realized $d$-faces are allowed. In particular, we prove that if $Δ$ is a simplicial $\mathbb{R}$-homology sphere, then for generic PL realizations $ν$, the stress algebra $Ψ(Δ,ν)$ is Gorenstein and has the weak Lefschetz property. |
| title | The face numbers of homology spheres |
| topic | Combinatorics Commutative Algebra Rings and Algebras Primary: 05E45, Secondary: 05E40, 13A02, 13E10, 13J30, 52B70, 52C25 |
| url | https://arxiv.org/abs/1706.03322 |