arXiv subjects

Anton Ayzenberg

Publications and source records attributed to Anton Ayzenberg.

36 records · Page 2Linked to original sources

arXiv2019

Torus actions of complexity one in non-general position

Let the compact torus $T^{n-1}$ act on a smooth compact manifold $X^{2n}$ effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space $X^{2n}/T^{n-1}$ if the action is cohomologically equivariantly formal (which essentially means that $H^{odd}(X^{2n};\mathbb{Z})=0$). It happens that homology of the orbit space can be arbitrary in degrees $3$ and higher. For any finite simplicial complex $L$ we construct an equivariantly formal manifold $X^{2n}$ such that $X^{2n}/T^{n-1}$ is homotopy equivalent to $Σ^3L$. The constructed manifold $X^{2n}$ is the total space of the projective line bundle over the permutohedral variety hence the action on $X^{2n}$ is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of the action in $j$-general position and prove that, for any simplicial complex $M$, there exists an equivariantly formal action of complexity one in $j$-general position such that its orbit space is homotopy equivalent to $Σ^{j+2}M$.

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arXiv2019

Torus action on quaternionic projective plane and related spaces

For an action of a compact torus $T$ on a smooth compact manifold~$X$ with isolated fixed points the number $\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. In this paper we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that $\mathbb{H}P^2/T^3\cong S^5$ and $S^6/T^2\cong S^4$, for the homogeneous spaces $\mathbb{H}P^2=Sp(3)/(Sp(2)\times Sp(1))$ and $S^6=G_2/SU(3)$. Here the maximal tori of the corresponding Lie groups $Sp(3)$ and $G_2$ act on the homogeneous spaces by the left multiplication. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of 8-dimensional manifolds with the action of $T^3$, generalizing $\mathbb{H}P^2$. We prove that their orbit spaces are homeomorphic to $S^5$ as well. We link this result to Kuiper--Massey theorem and some of its generalizations.

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arXiv2018

Manifolds of isospectral matrices and Hessenberg varieties

We study the space $X_h$ 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 $X_h$ is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that $X_h$ is an equivariantly formal manifold. The equivariant and ordinary cohomology of $X_h$ are described using GKM-theory. The main goal of this paper is to show the connection between the manifolds $X_h$ and the semisimple Hessenberg varieties well-known in algebraic geometry. Both the spaces $X_h$ and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families which can be generalized to other submanifolds of the flag variety.

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arXiv2018

Manifolds of isospectral arrow matrices

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space $M_{St_n,λ}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $λ$. We prove that this space is a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe the topology and combinatorics of its orbit space. If $n\geqslant 3$, the orbit space $M_{St_n,λ}/T^n$ is not a polytope, hence this manifold is not quasitoric. However, there is a natural permutation action on $M_{St_n,λ}$ which induces the combined action of a semidirect product $T^n\rtimesΣ_n$. The orbit space of this large action is a simple polytope. The structure of this polytope is described in the paper. In case $n=3$, the space $M_{St_3,λ}/T^3$ is a solid torus with boundary subdivided into hexagons in a regular way. This description allows to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold $M_{St_3,λ}$ using the general theory developed by the first author. This theory is also applied to a certain $6$-dimensional manifold called the twin of $M_{St_3,λ}$. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.

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arXiv2018

Torus actions of complexity one and their local properties

We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n-1 has a specific topology which we axiomatize in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points.

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arXiv2016

Toric manifolds over 3-polytopes

In this note we gather and review some facts about existence of toric spaces over 3-dimensional simple polytopes. First, over every combinatorial 3-polytope there exists a quasitoric manifold. Second, there exist combinatorial 3-polytopes, that do not correspond to any smooth projective toric variety. We restate the proof of the second claim which does not refer to complicated algebro-geometrical technique. If follows from these results that any fullerene supports quasitoric manifolds but does not support smooth projective toric varieties.

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arXiv2016

Dimensions of multi-fan algebras

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. This construction includes many important examples, such as cohomology of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured that the same holds for any simplicial cycle. We disprove this conjecture by showing that the colors of singular points of the cycle may affect the dimensions. However, the colors of smooth points are irrelevant. By using bistellar moves we show that the number of different dimension vectors arising on a given 3-dimensional pseudomanifold with isolated singularities is a topological invariant. This invariant is trivial on manifolds, but nontrivial in general.

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arXiv2015

Cohomology of toric origami manifolds with acyclic proper faces

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 cohomology rings of a symplectic toric manifold can be described in terms of the corresponding polytope. Recently, Holm and Pires described the cohomology of a toric origami manifold $M$ in terms of the orbit space $M/T$ when $M$ is orientable and the orbit space $M/T$ is contractible. But in general the orbit space of a toric origami manifold need not be contractible. In this paper we study the topology of orientable toric origami manifolds for the wider class of examples: we require that every proper face of the orbit space is acyclic, while the orbit space itself may be arbitrary. Furthermore, we give a general description of the equivariant cohomology ring of torus manifolds with locally standard torus actions in the case when proper faces of the orbit space are acyclic and the free part of the action is a trivial torus bundle.

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arXiv2015

Volume polynomials and duality algebras of multi-fans

We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $Δ$ determines a volume polynomial $V_Δ$ whose values are the volumes of multi-polytopes based on $Δ$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(Δ)$. We study the structure and properties of $V_Δ$ and $\mathcal{A}^*(Δ)$ and give applications and connections to other subjects, such as Macaulay duality, Novik--Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.

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arXiv2015

Locally standard torus actions and sheaves over Buchsbaum posets

We consider a sheaf of exterior algebras on a simplicial poset $S$ and introduce a notion of homological characteristic function. Two natural objects are associated with these data: a graded sheaf $\mathcal{I}$ and a graded cosheaf $\widehatΠ$. When $S$ is a homology manifold, we prove the isomorphism $H^{n-1-p}(S;\mathcal{I})\cong H_{p}(S;\widehatΠ)$ which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence $E^2_{p,q}\cong H^{n-1-p}(S;\mathcal{U}_{n-1+q}\otimes \mathcal{I})\Rightarrow H_{p+q}(S;\widehatΠ)$, where $\mathcal{U}_*$ is the local homology stack on $S$. This spectral sequence, in turn, extends Zeeman--McCrory spectral sequence. This sheaf-theoretical result is applied to toric topology. We consider a manifold $X$ with a locally standard action of a compact torus and acyclic proper faces of the orbit space. A principal torus bundle $Y$ is associated with $X$, so that $X\cong Y/\sim$. The orbit type filtration on $X$ is covered by the topological filtration on $Y$. We prove that homological spectral sequences associated with these two filtrations are isomorphic in many nontrivial positions.

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arXiv2015

Topological model for h"-vectors of simplicial manifolds

Any manifold with boundary gives rise to a Poincare duality algebra in a natural way. Given a simplicial poset $S$ whose geometric realization is a closed orientable homology manifold, and a characteristic function, we construct a manifold with boundary such that graded components of its Poincare duality algebra have dimensions $h_k"(S)$. This gives a clear topological evidence for two well-known facts about simplicial manifolds: the nonnegativity of $h"$-numbers (Novik--Swartz theorem) and the symmetry $h"_k=h"_{n-k}$ (generalized Dehn--Sommerville relations).

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arXiv2015

Homology cycles in manifolds with locally standard torus actions

Let $X$ be a $2n$-manifold with a locally standard action of a compact torus $T^n$. If the free part of action is trivial and proper faces of the orbit space $Q$ are acyclic, then there are three types of homology classes in $X$: (1) classes of face submanifolds; (2) $k$-dimensional classes of $Q$ swept by actions of subtori of dimensions $<k$; (3) relative $k$-classes of $Q$ modulo $\partial Q$ swept by actions of subtori of dimensions $\geqslant k$. The submodule of $H_*(X)$ spanned by face classes is an ideal in $H_*(X)$ with respect to the intersection product. It is isomorphic to $(\mathbb{Z}[S_Q]/Θ)/W$, where $\mathbb{Z}[S_Q]$ is the face ring of the Buchsbaum simplicial poset $S_Q$ dual to $Q$; $Θ$ is the linear system of parameters determined by the characteristic function; and $W$ is a certain submodule, lying in the socle of $\mathbb{Z}[S_Q]/Θ$. Intersections of homology classes different from face submanifolds are described in terms of intersections on $Q$ and $T^n$.

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arXiv2015

Locally standard torus actions and h'-vectors of simplicial posets

We consider the orbit type filtration on a manifold $X$ with locally standard action of a compact torus and the corresponding homological spectral sequence $(E_X)^r_{*,*}$. If all proper faces of the orbit space $Q=X/T$ are acyclic, and the free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms are equal to the $h'$-numbers of the Buchsbaum simplicial poset $S_Q$ dual to $Q$. Betti numbers of $X$ depend only on the orbit space $Q$ but not on the characteristic function. If $X$ is a slightly different object, namely the model space $X=(P\times T^n)/\sim$ where $P$ is a cone over Buchsbaum simplicial poset $S$, we prove that $\dim (E_X)^{\infty}_{p,p} = h''_p(S)$. This gives a topological evidence for the fact that $h''$-numbers of Buchsbaum simplicial posets are nonnegative.

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arXiv2014

Toric origami structures on quasitoric manifolds

We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is reformulated in discrete geometrical terms. The problem reduces to existence of planar triangulations with certain coloring and metric properties.

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arXiv2014

Homology of torus spaces with acyclic proper faces of the orbit space

Let $X$ be 2n-dimensional compact manifold with a locally standard action of a compact torus. The orbit space $X/T$ is a manifold with corners. Suppose that all proper faces of $X/T$ are acyclic. In the paper we study the homological spectral sequence $E^*_{*,*}\Rightarrow H_*(X)$ corresponding to the filtration of $X$ by orbit types. When the free part of the action is not twisted, we describe the whole spectral sequence in terms of homology and combinatorial structure of $X/T$. In this case we describe the kernel and the cokernel of the natural map $k[X/T]/(l.s.o.p.) \to H_*(X)$, where $k[X/T]$ is a face ring of $X/T$ and $(l.s.o.p.)$ is the ideal generated by a linear system of parameters (this ideal appears as the image of $H^{>0}(BT)$ in equivariant cohomology. There exists a natural double grading on $H_*(X)$, which satisfies bigraded Poincare duality. This general theory is applied to compute homology groups of origami toric manifolds with acyclic proper faces of the orbit space. A number of natural generalizations is considered. These include Buchsbaum simplicial complexes and posets. h'- and h''-numbers of simplicial posets appear as the ranks of certain terms in the spectral sequence $E^*_{*,*}$. In particular, using topological argument we show that Buchsbaum posets have nonnegative h''-vectors. The proofs of this paper rely on the theory of cellular sheaves. We associate to a torus space certain sheaves and cosheaves on the underlying simplicial poset, and observe an interesting duality between these objects. This duality seems to be a version of Poincare-Verdier duality between cellular sheaves and cosheaves.

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arXiv2014

Buchstaber numbers and classical invariants of simplicial complexes

Buchstaber invariant is a numerical characteristic of a simplicial complex, arising from torus actions on moment-angle complexes. In the paper we study the relation between Buchstaber invariants and classical invariants of simplicial complexes such as bigraded Betti numbers and chromatic invariants. The following two statements are proved. (1) There exists a simplicial complex U with different real and ordinary Buchstaber invariants. (2) There exist two simplicial complexes with equal bigraded Betti numbers and chromatic numbers, but different Buchstaber invariants. To prove the first theorem we define Buchstaber number as a generalized chromatic invariant. This approach allows to guess the required example. The task then reduces to a finite enumeration of possibilities which was done using GAP computational system. To prove the second statement we use properties of Taylor resolutions of face rings.

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arXiv2013

Simplicial complexes Alexander dual to boundaries of polytopes

In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general polytopes. This technique and recent results of N.Yu.Erokhovets are combined to prove the following: Buchstaber invariant $s(P)$ of a convex polytope equals 1 if and only if $P$ is a pyramid. In general, we describe a procedure to construct polytopes with $s_R(P)>k$. The construction has purely combinatorial consequences. We also apply Gale duality to the study of bigraded Betti numbers and f-vectors of polytopes.

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arXiv2010

The problem of Buchstaber number and its combinatorial aspects

For any simplicial complex on m vertices a moment-angle complex Z_K embedded in C^m can be defined. There is a canonical action of a torus T^m on Z_K, but this action fails to be free. The Buchstaber number is the maximal integer s(K) for which there exists a subtorus of rank s(K) acting freely on Z_K. The similar definition can be given for real Buchstaber number. We study these invariants using certain sequences of simplicial complexes called universal complexes. Some general properties of Buchstaber numbers follow from combinatorial properties of universal complexes. In particular, we investigate the additivity of Buchstaber invariant.

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