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## Эквивариантные когомологии момент–угол-комплексов относительно координатных подторов

Труды Математического института им. В.А. Стеклова РАН. 2022. Т. 317. С. 157-167.

Panov T., Зейникешева И. К.

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.

Ayzenberg A., Masuda M., / Cornell University. Series arXiv "math". 2019.

Let a compact torus T=T^{n−1} act on a smooth compact manifold X=X^{2n} effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If H^{odd}(X)=0 and the weights of tangent representation at each fixed point are in general position, we prove that the orbit space Q=X/T is a homology (n+1)-sphere. If, in addition, π_1(X)=0, then Q is homeomorphic to S^{n+1}. ...

Added: January 14, 2020

Finkelberg M., Braverman A., Shiraishi J., Providence : American Mathematical Society, 2014

Let G be an almost simple simply connected complex Lie group, and let G/U be its base affine space. In this paper we formulate a conjecture which provides a new geometric interpretation of the Macdonald polynomials associated to G via perverse coherent sheaves on the scheme of formal arcs in the affinizationof G/U. We prove ...

Added: March 5, 2015

Ayzenberg A., Algebraic and Geometric Topology 2020 Vol. 20 No. 6 P. 2957-2994

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $\lambda$. Using the discretized S\edt{c}hr\"{o}dinger operator we describe all spectra $\lambda$ for which $X_{n,\lambda}$ is a topological manifold. The space $X_{n,\lambda}$ carries a natural effective action of a compact $(n-1)$-torus. ...

Added: January 14, 2020

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

Ayzenberg A., Cherepanov V., / Cornell University. Series arXiv "math". 2019. No. 1905.04761.

Let the compact torus Tn−1 act on a smooth compact manifold X2n effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X2n/Tn−1 if the action is cohomologically equivariantly formal (which essentially means that Hodd(X2n;Z)=0). It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any finite ...

Added: October 23, 2019

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

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

Panov T., / Cornell University. Series arXiv "math". 2019.

We describe the basic Dolbealut cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a dga model for the ordinary Dolbeault cohomology algebra. ...

Added: November 1, 2019

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

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

Ayzenberg A., / Arxiv Cornell University Library. Series 1803.11433 "1803.11433 ". 2018. No. 11433.

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space of Hermitian periodic tridiagonal n×n-matrices with a fixed simple spectrum. Using discrete Shroedinger operator we give a condition on the spectrum which guarantees that this space is a manifold. The space carries a natural effective action of ...

Added: October 15, 2018

Buchstaber V., Limonchenko I., / Cornell University. Series math "arxiv.org". 2018. No. 1808.08851.

We introduce the notions of algebraic and geometric direct families of polytopes and develop a theory of such families. The theory is then applied to the problem of existence of nontrivial higher Massey products in cohomology of moment-angle-complexes. ...

Added: September 29, 2019

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

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

Baralic D., Grbic J., Limonchenko I. et al., Filomat 2020 Vol. 34 No. 7 P. 2329-2356

In this paper we illustrate a tight interplay between homotopy theory and combinatorics
within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects
associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and
real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related
small cover. ...

Added: October 12, 2021

Panov T., Ishida H., / Cornell University. Series arXiv "math". 2018.

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a ...

Added: November 1, 2019

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

Ayzenberg A., Masuda M., Park S. et al., Journal of Symplectic Geometry 2017 Vol. 15 No. 3 P. 645-685

A toric origami manifold is a generalization of a symplectic toric manifold (or a toric symplectic manifold). The origami symplectic form is allowed to degenerate in a good controllable way in contrast to the usual symplectic form. It is widely known that symplectic toric manifolds are encoded by Delzant polytopes, and the cohomology and equivariant ...

Added: September 19, 2017

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

Ayzenberg Anton, Buchstaber V.M., International Mathematical Research Notices 2021 Vol. 2021 No. 21 P. 16671-16692

We study the space Xh of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that Xh is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of ...

Added: June 16, 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

Sirotin V., Arkhipova M., Dubrova T. A. et al., Bielsko-Biala : University of Bielsko-Biala Press, 2016

The main attributes of modern enterprises should be the flexibility and the ability of forecasting the future. Constant adaptation to the changing environment and the rapidity of undertaking certain actions which are conditioned by specific situations determine the rules for the future position of market competition. Effective and efficient adjustment of the company in line ...

Added: November 2, 2016

Pahomov F., Известия РАН. Серия математическая 2016 Т. 80 № 6 С. 173-216

Полимодальная логика доказуемости
GLP была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории свободной GLP-алгебры, порожденной константами 0, 1 [1]. В этой статье для любого натурального n решается аналогичный вопрос для логик GLPn, являющихся ...

Added: December 4, 2017

Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016