The face numbers of homology spheres
| Autor: | Chong, Kai Fong Ernest, Tay, Tiong Seng |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\Delta$, we use ideas from rigidity theory to construct a graded Artinian $\mathbb{R}$-algebra $\Psi(\Delta,\nu)$ of stresses on a PL realization $\nu$ of $\Delta$ in $\mathbb{R}^d$, where overlapping realized $d$-faces are allowed. In particular, we prove that if $\Delta$ is a simplicial $\mathbb{R}$-homology sphere, then for generic PL realizations $\nu$, the stress algebra $\Psi(\Delta,\nu)$ is Gorenstein and has the weak Lefschetz property. Comment: The multiplication of stresses in Thm. 5.2 is not well-defined. We have a corrected multiplication map, which introduces a coefficient that is no longer always 1 for each summand. However, subsequent proof approaches for Sec. 8-11 require this (incorrect) coefficient 1 for each summand, which we do not know how to fix. Thus, our proof approaches for all main results do not work |
| Databáze: | arXiv |
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