arXiv subjects

Fedor Vylegzhanin

Publications and source records attributed to Fedor Vylegzhanin.

7 recordsLinked to original sources

arXiv2026

Iterated Whitehead products in the homotopy groups of polyhedral products

We study structure within the homotopy groups of the Davis-Januszkiewicz space DJ(K) associated with a simplicial complex K. The inclusion of each vertex in K induces a map from the two-sphere into DJ(K). These maps generate a quasi-Lie subalgebra QL(K) via the Whitehead product and a Pi-subalgebra S(K) via the Whitehead product and composition. We describe the quasi-Lie subalgebra QL(K), and show that the Pi-subalgebra S(K) coincides with the whole of the homotopy groups of DJ(K) if and only if K is a flag complex. Extensions to more general polyhedral products are also considered.

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arXiv2026

The commutator subalgebra of the Lie algebra associated with a right-angled Coxeter group

We study the graded Lie algebra $L(RC_K)$ associated with the lower central series of a right-angled Coxeter group. We construct a surjective homomorphism from the polynomial ring over an explicit Lie algebra $N_K$ to the commutator subalgebra of $L(RC_K)$, and conjecture that it is an isomorphism. The homomorphism is defined in terms of a new operation in Lie algebras associated with groups generated by involutions, which corresponds to the squaring and has an analogue in homotopy theory. We show that the universal enveloping algebra $U(N_K)$ is isomorphic to the mod 2 loop homology algebra of the corresponding moment-angle complex $ZK$. This allows us to give a presentation of the Lie algebra $N_K$ by generators and relations.

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arXiv2026

A colimit decomposition for the loop homology of polyhedral products

We show that the loop homology algebras of polyhedral products of the form $(\underline{X},\underline{*})^{\mathcal{K}}$ can be written as a colimit over the flagification of $\mathcal{K}$, and obtain a similar result for the Poincaré series. This effectively reduces the study of the algebras $H_*(Ω(\underline{X},\underline{*})^{\mathcal{K}})$ to the case of 1-neighbourly simplicial complexes. We give presentations of the loop homology of Davis--Januszkiewicz spaces (i.e. Yoneda algebras of Stanley--Reisner rings) and calculate the Poincaré series of looped polyhedral products associated to various families of simplicial complexes, including HMF-presented complexes and skeleta of flag complexes.

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arXiv2026

Anick's conjecture for polyhedral products

We develop a method for studying the pointed loop space of general polyhedral products, showing that many properties are determined by the moment-angle complex. To apply the method, we show that localised away from a finite set of primes, the loop space of a moment-angle complex is homotopy equivalent to a product of loops on spheres. As a consequence, we give p-local loop space decompositions of quasitoric manifolds, certain toric orbifolds and a wide family of polyhedral products. This verifies a conjecture of Anick for such spaces. We also describe the additive structure of loop homology of simply connected polyhedral products in terms of polynomials studied by Backelin and Berglund.

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arXiv2025

Loop homology of moment-angle complexes in the flag case

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 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 are coformal over $\mathbb{Q},$ give a necessary condition for rational formality, and compute their homotopy groups in terms of homotopy groups of spheres.

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arXiv2025

Cartesian subgroups in graph products of groups

The kernel of the natural projection of a graph product of groups onto their direct product is called the Cartesian subgroup of the graph product. This construction generalises commutator subgroups of right-angled Coxeter and Artin groups. Using theory of polyhedral products, we give a lower and an upper bound on the number of relations in presentations of Cartesian groups and on their deficiency. The bounds are related to the fundamental groups of full subcomplexes in the clique complex, and the lower bound coincide with the upper bound if these fundamental groups are free or free abelian. Following Li Cai's approach, we also describe an algorithm that computes "small" presentations of Cartesian subgroups.

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arXiv2022

Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

For any flag simplicial complex $K$, we describe the multigraded Poincare series, the minimal number of relations and the degrees of these relations in the Pontryagin algebra of the corresponding moment-angle complex $Z_K$. We compute the LS-category of $Z_K$ for flag complexes and give a lower bound in the general case. The key observation is that the Milnor-Moore spectral sequence collapses at the second sheet for flag $K$. We also show that the results of Panov and Ray about the Pontryagin algebras of Davis-Januszkiewicz spaces are valid for arbitrary coefficient rings, and introduce the $(\mathbb{Z}\times\mathbb{Z}^m)$-grading on the Pontryagin algebras which is similar to the multigrading on the cohomology of $Z_K$.

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