We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen’s -conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field . These results provide another self-contained proof of the -conjecture for homology spheres in characteristic .
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Keywords: Simplicial homology spheres, pseudo-manifolds, Stanley-Reisner rings, anisotropy, Hard Lefschetz theorem, $g$-conjecture
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@article{ALCO_2023__6_5_1313_0,
author = {Karu, Kalle and Xiao, Elizabeth},
title = {On the anisotropy theorem of {Papadakis} and {Petrotou}},
journal = {Algebraic Combinatorics},
pages = {1313--1330},
publisher = {The Combinatorics Consortium},
volume = {6},
number = {5},
year = {2023},
doi = {10.5802/alco.298},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.298/}
}
TY - JOUR AU - Karu, Kalle AU - Xiao, Elizabeth TI - On the anisotropy theorem of Papadakis and Petrotou JO - Algebraic Combinatorics PY - 2023 SP - 1313 EP - 1330 VL - 6 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.298/ DO - 10.5802/alco.298 LA - en ID - ALCO_2023__6_5_1313_0 ER -
%0 Journal Article %A Karu, Kalle %A Xiao, Elizabeth %T On the anisotropy theorem of Papadakis and Petrotou %J Algebraic Combinatorics %D 2023 %P 1313-1330 %V 6 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.298/ %R 10.5802/alco.298 %G en %F ALCO_2023__6_5_1313_0
Karu, Kalle; Xiao, Elizabeth. On the anisotropy theorem of Papadakis and Petrotou. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1313-1330. doi : 10.5802/alco.298. https://alco.centre-mersenne.org/articles/10.5802/alco.298/
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