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Pontryagin Algebras and the LS-Category of Moment–Angle Complexes in the Flag Case
Proceedings of the Steklov Institute of Mathematics. 2022. Vol. 317. No. 1. P. 55 – 77.
Keywords: moment-angle complex
Vylegzhanin F., Algebraic and Geometric Topology 2025 Vol. 25 No. 9 P. 5619–5663
We develop a general homological approach to presentations of connected graded associative algebras, and apply it to the loop homology of moment-angle complexes Z_K that correspond to flag simplicial complexes K. For an arbitrary coefficient ring, we describe generators of the Pontryagin algebra H_∗(ΩZ_K) and defining relations between them. We prove that such moment-angle complexes ...
Added: December 22, 2025
Bahri A., Limonchenko I., Panov T. et al., Discrete and Computational Geometry 2026 Vol. 76 P. 313–338
We define bigraded persistent homology modules and bigraded barcodes of a finite pseudo-metric space X using the ordinary and double homology of the moment-angle complex associated with the Vietoris–Rips filtration of X. We prove a stability theorem for the bigraded persistent double homology modules and barcodes. ...
Added: October 26, 2025
Vylegzhanin Fedor, Stanton L., / Series arXiv "math". 2025.
Added: June 21, 2025
Fedor Vylegzhanin, Yakov Veryovkin, / Series math "arxiv.org". 2025.
Added: April 15, 2025
Ivan Limonchenko, Taras Panov, Song J. et al., Advances in Mathematics 2023 Vol. 432 Article 109274
We put a cochain complex structure CH*(Z_K) on the cohomology of a moment-angle complex Z_K and call the resulting cohomology the double cohomology, HH*(Z_K). We give three equivalent definitions for the differential, and compute HH*(Z_K) for a family of simplicial complexes containing clique complexes of chordal graphs. ...
Added: October 25, 2023
Khovanskii A., Limonchenko I., Monin L., Filomat 2022 Vol. 36 No. 19 P. 6513–6537
The classical BKK theorem computes the intersection number of divisors on toric variety in terms of volumes of corresponding polytopes. It was observed by Pukhlikov and the first author that the BKK theorem leads to a presentation of the cohomology ring of toric variety as a quotient of the ring of differential operators with constant ...
Added: June 1, 2023
Vylegzhanin F., Proceedings of the Steklov Institute of Mathematics (USA) 2022 Т. 317 С. 64–88
For any flag simplicial complex K, we describe the multigraded Poincaré series, the minimal number of relations, and the degrees of these relations in the Pontryagin algebra of the corresponding moment–angle complex ZK. We compute the LS-category of ZK for flag complexes and give a lower bound in the general case. The key observation is that the Milnor–Moore spectral ...
Added: November 15, 2022
Panov T., Зейникешева И. К., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 317 С. 157–167
We compute the equivariant cohomology $H^*_{T_I}(Z_K)$ of moment-angle complexes $Z_K$ with respect to the action of coordinate subtori $T_I \subset T^m$. We give a criterion for the equivariant formality of $Z_K$ and obtain specifications for the cases of flag complexes and graphs. ...
Added: November 11, 2022
Limonchenko I., Solomadin G., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 317 С. 132–156
In this paper we prove that the quotient of any real or complex moment-angle complex by any closed subgroup in the naturally acting compact torus on it is equivariantly homotopy equivalent to the homotopy colimit of a certain toric diagram. For any quotient we prove an equivariant homeomorphism generalizing the well-known Davis-Januszkiewicz construction for quasitoric ...
Added: October 15, 2022
Limonchenko I., Panov T., Успехи математических наук 2022 Т. 77 № 4 С. 203–204
We give a correct statement and a complete proof of the criterion obtained in a paper of Grbic-Panov-Theriault-Wu for the face rin k[K] of a simplicial complex K to be Golod over a field k. (The original argument depended on the main result of Berglund and Joellenbeck, which was shown to be false by Katthaen.) We also construct ...
Added: September 8, 2022
Massey Products in the Cohomology of the Moment-Angle Manifolds Corresponding to Pogorelov Polytopes
E. G. Zhuravleva, Mathematical notes 2019 Vol. 105 No. 4 P. 519–527
Nontrivial Massey products in the cohomology of the moment-angle manifolds corresponding to polytopes in the Pogorelov class are constructed. This class includes the dodecahedron and all fullerenes, i.e., simple 3-polytopes with only 5- and 6-gonal faces. The existence of nontrivial Massey products implies that the spaces under consideration are not formal in the sense of ...
Added: October 31, 2021
Panov T., Theriault S., Compositio Mathematica 2019 Vol. 155 No. 1 P. 206–228
If K is a simplicial complex on m vertices the flagification of K is the minimal flag complex Kf on the same vertex set that contains K. Letting L be the set of vertices, there is a sequence of simplicial inclusions L→K→Kf. This induces a sequence of maps of polyhedral products (X,A)^L⟶g(X,A)^K⟶f(X,A)^{Kf}. We show that Ωf and Ωf∘Ωg have right homotopy inverses and draw consequences. For a flag complex K the polyhedral product of ...
Added: October 29, 2021
Grbić J., Simmons G., Ilyasova M. et al., Proceedings of the Royal Society of Edinburgh: Section A 2022 Vol. 152 No. 1 P. 128–147
We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $R_K$, to be a one-relator group; and ...
Added: October 29, 2021
Abramyan S., Siberian Mathematical Journal 2019 Vol. 60 No. 2 P. 185–196
We give an example of a simplicial complex whose corresponding moment-angle complex is homotopy equivalent to a wedge of spheres, but there is a sphere that cannot be realized by any linear combination of iterated higher Whitehead products. Using two explicitly defined operations on simplicial complexes, we prove that there exists a simplicial complex that ...
Added: May 10, 2020
Abramyan S., Panov T., Proceedings of the Steklov Institute of Mathematics 2019 Vol. 1 No. 305 P. 1–21
We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment–angle complex $\mathcal{Z_K}$. Namely, we say that a simplicial complex $\mathcal{K}$ realises an iterated higher Whitehead product w if wis a nontrivial element of $\pi_*(\mathcal{Z_K})$. The combinatorial approach to the question of realisability uses the ...
Added: October 28, 2019