arXiv subjects

Nikolai Erokhovets

Publications and source records attributed to Nikolai Erokhovets.

8 recordsLinked to original sources

arXiv2026

Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes

We build a large family of hyperbolic links with an explicit decomposition of the complement into right-angled hyperbolic polytopes of finite volume. Namely, in a series of papers A.D.Mednykn and A.Yu.Vesnin introduced a construction that for a given right-angled polytope $P$ in geometry $\mathbb L^3$, $\mathbb R^3$, $\mathbb S^3$, $\mathbb L^2\times \mathbb R$, $\mathbb S^2\times \mathbb R$ and a Hamiltonian cycle, theta-subgraph or $K_4$-subgraph $Γ$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,Γ)$ with an involution $τ$ such that $N(P,Γ)/\langleτ\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,Γ)\to S^3$ is a link $C_Γ\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $Γ$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,Γ)$. We give a criterion when $S^3\setminus C_Γ$ has a complete hyperbolic structure of finite volume and generalize this criterion to similar links in $3$-manifolds different from $S^3$. We prove that hyperbolic links $C_Γ$ are parametrized by nonselfcrossing Eulerian cycles, Eulerian theta-subgraphs and Eulerian $K_4$-subgraphs in hyperbolic right-angled $3$-polytopes of finite volume in $\mathbb L^3$ with $0$, $2$ or $4$ finite vertices. The complement $S^3\setminus C_Γ$ is glued of $4$, $8$ or $16$ copies of the corresponding right-angled polytope. We give a criterion when the link $C_Γ$ consists of mutually unlinked circles and prove that if such a link is nontrivial, then it contains the Borromean rings. The latter problem is motivated by the Efimov effect in quantum mechanics. We consider higher-dimensional analogues of hyperbolic links $C_Γ$.

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arXiv2026

Hyperelliptic four-manifolds defined by vector-colorings of simple polytopes

Toric topology assigns to each simple convex $n$-polytope $P$ with $m$ facets an $n$-dimensional real moment angle manifold $\mathbb RZ_P$ with a canonical action of $\mathbb Z_2^m=(\mathbb Z/2\mathbb Z)^m$. We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on $\mathbb RZ_P$. The orbit space $N(P,H)=\mathbb RZ_P/H$ has an action of $\mathbb Z_2^m/H$. For general $n$ we introduce the notion of a Hamiltonian $C(n,k)$-subcomplex in the boundary of an $n$-polytope $P$ generalizing the notions of a Hamiltonian cycle ($k=2$), Hamiltonian theta-subgraph ($k=3$) and Hamiltonian $K_4$-subgraph ($k=4)$ in the $1$-skeleton of a $3$-polytope. Each $C(n,k)$-subcomplex $C\subset \partial P$ corresponds to a subgroup $H_C\subset\mathbb Z_2^m$ such that $N(P,H_C)\simeq S^n$. We prove that in dimensions $n\leqslant 4$ this correspondence is a bijection. Any subgroup $H\subset \mathbb Z_2^m$ defines a complex $C(P,H)\subset \partial P$. We prove that each Hamiltonian $C(n,k)$-subcomplex $C\subset C(P,H)$ inducing $H$ corresponds to a hyperelliptic involution $τ_C\in\mathbb Z_2^m/H$ on the manifold $N(P,H)$ (that is, an involution with the orbit space homeomorphic to $S^n$) and in dimensions $n\leqslant 4$ this correspondence is a bijection. We prove that for the geometries $\mathbb X= \mathbb S^4$, $\mathbb S^3\times\mathbb R$, $\mathbb S^2\times \mathbb S^2$, $\mathbb S^2\times \mathbb R^2$, $\mathbb S^2\times \mathbb L^2$, and $\mathbb L^2\times \mathbb L^2$ there exists a compact right-angled $4$-polytope $P$ with a free action of $H$ such that the geometric manifold $N(P,H)$ has a hyperelliptic involution in $\mathbb Z_2^m/H$, and for $\mathbb X=\mathbb R^4$, $\mathbb L^4$, $\mathbb L^3\times \mathbb R$ and $\mathbb L^2\times \mathbb R^2$ there are no such polytopes.

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arXiv2025

On uniqueness of the equivariant smooth structure on a real moment-angle manifold

The paper is devoted to the well-known problem of smooth structures on moment-angle manifolds. Each real or complex moment-angle manifold has an equivariant smooth structure given by an intersection of quadrics corresponding to a geometric realisation of a polytope. In 2006 F.Bosio and L.Meersseman proved that complex moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic. Using arguments from calculus we derive from this result that real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic and the polytopes are diffeomorphic as manifolds with corners.

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arXiv2024

Manifolds realized as orbit spaces of non-free $\mathbb Z_2^k$-actions on real moment-angle manifolds

We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on the real moment-angle manifold $\mathbb R\mathcal{Z}_P$ corresponding to a simple convex $n$ polytope $P$ with $m$ facets. The criterion when the orbit space $\mathbb R\mathcal{Z}_P/H$ is a topological manifold (perhaps with a boundary) can be extracted from results by M.A. Mikhailova and C. Lange. For any dimension $n$ we construct series of manifolds $\mathbb R\mathcal{Z}_P/H$ homeomorphic to $S^n$ and series of manifolds $M^n=\mathbb R\mathcal{Z}_P/H$ admitting a hyperelliptic involution $τ\in\mathbb Z_2^m/H$, that is an involution $τ$ such that $M^n/\langleτ\rangle$ is homeomorphic to $S^n$. For any simple $3$-polytope $P$ we classify all subgroups $H\subset\mathbb Z_2^m$ such that $\mathbb R\mathcal{Z}_P/H$ is homeomorphic to $S^3$. For any simple $3$-polytope $P$ and any subgroup $H\subset\mathbb Z_2^m$ we classify all hyperelliptic involutions $τ\in\mathbb Z_2^m/H$ acting on $\mathbb R\mathcal{Z}_P/H$. As a corollary we obtain that a $3$-dimensional small cover has $3$ hyperelliptic involutions in $\mathbb Z_2^3$ if and only if it is a rational homology $3$-sphere and if and only if it correspond to a triple of Hamiltonian cycles such that each edge of the polytope belongs to exactly two of them.

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arXiv2021

Canonical geometrization of orientable $3$-manifolds defined by vector-colourings of $3$-polytopes

In short geometrization conjecture of W.\,Thurston (finally proved by G.~Perelman) says that any oriented $3$-manifold can be canonically partitioned into pieces, which have a geometric structure of one of the eight types. In the seminal paper (1991) M.\,W.\,Davis and T.\,Januszkiewicz introduced a wide class of $n$-dimensional manifolds -- small covers over simple $n$-polytopes. We give a complete answer to the following problem: to build an explicit canonical decomposition for any orientable $3$-manifold defined by a vector-colouring of a simple $3$-polytope, in particular for a small cover. The proof is based on analysis of results in this direction obtained before by different authors.

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arXiv2020

$B$-rigidity of ideal almost Pogorelov polytopes

Toric topology assigns to each $n$-dimensional combinatorial simple convex polytope $P$ with $m$ facets an $(m+n)$-dimensional moment-angle manifold $\mathcal{Z}_P$ with an action of a compact torus $T^m$ such that $\mathcal{Z}_P/T^m$ is a convex polytope of combinatorial type $P$. A simple $n$-polytope is called $B$-rigid, if any isomorphism of graded rings $H^*(\mathcal{Z}_P,\mathbb Z)= H^*(\mathcal{Z}_Q,\mathbb Z)$ for a simple $n$-polytope $Q$ implies that $P$ and $Q$ are combinatorially equivalent. An ideal almost Pogorelov polytope is a combinatorial $3$-polytope obtained by cutting off all the ideal vertices of an ideal right-angled polytope in the Lobachevsky (hyperbolic) space $\mathbb L^3$. These polytopes are exactly the polytopes obtained from any, not necessarily simple, convex $3$-polytopes by cutting off all the vertices followed by cutting off all the "old" edges. The boundary of the dual polytope is the barycentric subdivision of the boundary of the old polytope (and also of its dual polytope). We prove that any ideal almost Pogorelov polytope is $B$-rigid. This produces three cohomologically rigid families of manifolds over ideal almost Pogorelov manifolds: moment-angle manifolds, canonical $6$-dimensional quasitoric manifolds and canonical $3$-dimensional small covers, which are "pullbacks from the linear model".

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arXiv2020

$B$-rigidity of the property to be an almost Pogorelov polytope

Toric topology assigns to each $n$-dimensional combinatorial simple convex polytope $P$ with $m$ facets an $(m+n)$-dimensional moment-angle manifold $\mathcal{Z}_P$ with an action of a compact torus $T^m$ such that $\mathcal{Z}_P/T^m$ is a convex polytope of combinatorial type $P$. We study the notion of $B$-rigidity. A property of a polytope $P$ is called $B$-rigid, if any isomorphism of graded rings $H^*(\mathcal{Z}_P,\mathbb Z)= H^*(\mathcal{Z}_Q,\mathbb Z)$ for a simple $n$-polytope $Q$ implies that it also has this property. We study families of $3$-dimensional polytopes defined by their cyclic $k$-edge-connectivity. These families include flag polytopes and Pogorelov polytopes, that is polytopes realizable as bounded right-angled polytopes in Lobachevsky space $\mathbb L^3$. Pogorelov polytopes include fullerenes -- simple polytopes with only pentagonal and hexagonal faces. It is known that the properties to be flag and Pogorelov polytope are $B$-rigid. We focus on almost Pogorelov polytopes, which are strongly cyclically $4$-edge-connected polytopes. They correspond to right-angled polytopes of finite volume in $\mathbb L^3$. There is a subfamily of ideal almost Pogorelov polytopes corresponding to ideal right-angled polytopes. We prove that the properties to be an almost Pogorelov polytope and an ideal almost Pogorelov polytope are $B$-rigid. As a corollary we obtain that $3$-dimensional associahedron $As^3$ and permutohedron $Pe^3$ are $B$-rigid. We generalize methods known for Pogorelov polytopes. We obtain results on $B$-rigidity of subsets in $H^*(\mathcal{Z}_P,\mathbb Z)$ and prove an analog of the so-called separable circuit condition (SCC). As an example we consider the ring $H^*(\mathcal{Z}_{As^3},\mathbb Z)$.

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arXiv2015

Graph-truncations of $3$-polytopes

In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $Γ$. We give the criterion when the resulting polytope is simple and when it is flag. As a corollary we prove the analog of Eberhard's theorem about the realization of polygon vectors of simple $3$-polytopes for flag polytopes.

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