On the anisotropy theorem of Papadakis and Petrotou
Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1313-1330.

We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen’s g-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 2 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 g-conjecture for homology spheres in characteristic 2.

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DOI: 10.5802/alco.298
Classification: 13F55, 05E40, 05E45, 14M25
Keywords: Simplicial homology spheres, pseudo-manifolds, Stanley-Reisner rings, anisotropy, Hard Lefschetz theorem, $g$-conjecture

Karu, Kalle 1; Xiao, Elizabeth 1

1 Mathematics Department University of British Columbia 1984 Mathematics Road Vancouver, B.C. Canada V6T 1Z2
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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