Buchstaber Invariant of Simple Polytopes

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 …

Hirzebruch genera of manifolds with torus action

A quasitoric manifold is a smooth 2n-manifold M^{2n} with an action of the compact torus T^n such that the action is locally isomorphic to the standard action of T^n on C^n and the orbit space is diffeomorphic, as manifold with corners, to a simple polytope P^n. The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to that of non-singular algebraic toric varieties (or toric manifolds). Unlike toric varieties, quasitoric manifolds may fail to be complex; however, they always admit a stably (or weakly almost) complex structure, and their co…

Bier 球面与环面拓扑

Bier spheres and toric topology

本文计算了任意 Bier 球面的实与复 Buchstaber 数，识别了二维情形下的 13 种组合类型，并应用结果构造 Delzant 多面体的正则正规扇，计算了对应环面簇的上同调环及检验了小覆盖的可定向性。

正复杂度环面作用的矩多面体的室分解

Chamber Decompositions of Moment Polytopes for Torus Actions of Positive Complexity

本文针对环面T^n在复Grassmann流形G_{n,2}上的标准作用，引入容许图概念，给出了容许多面体的完全归纳描述，并揭示了室分解与副扇形的联系。

Combinatorial rigidity of 3-dimensional simplicial polytopes

A simplicial polytope is combinatorially rigid if its combinatorial structure is determined by its graded Betti numbers which are important invariant coming from combinatorial commutative algebra. We find a necessary condition to be combinatorially rigid for 3-dimensional reducible simplicial polytopes and provide some rigid reducible simplicial polytopes.

Moment-angle complexes, monomial ideals, and Massey products

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.

Agrarian and $L^2$-invariants

We develop the theory of agrarian invariants, which are algebraic counterparts to $L^2$-invariants. Specifically, we introduce the notions of agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian polytope for finite free $G$-CW complexes together with a fixed choice of a ring homomorphism from the group ring $\mathbb{Z} G$ to a skew field. For the particular choice of the Linnell skew field $\mathcal{D}(G)$, this approach recovers most of the information encoded in the corresponding $L^2$-invariants. As an application, we prove that for agrarian groups of deficiency $1$, t…

Fullerenes, Polytopes and Toric Topology

The lectures are devoted to a remarkable class of $3$-dimensional polytopes, which are mathematical models of the important object of quantum physics, quantum chemistry and nanotechnology -- fullerenes. The main goal is to show how results of toric topology help to build combinatorial invariants of fullerenes. Main notions are introduced during the lectures. The lecture notes are addressed to a wide audience.

The agrarian polytope of two-generator one-relator groups

Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion-free, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann.

Toric cohomological rigidity of simple convex polytopes

A simple convex polytope $P$ is \emph{cohomologically rigid} if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over $P$. Not every $P$ has this property, but some important polytopes such as simplices or cubes are known to be cohomologically rigid. In this article we investigate the cohomological rigidity of polytopes and establish it for several new classes of polytopes including products of simplices. Cohomological rigidity of $P$ is related to the \emph{bigraded Betti numbers} of its \emph{Stanley--Reisner ring}, another important invariants coming…

关于Weyl群表示不变量与toric簇上同调的不变量

On invariants of representations of Weyl groups associated with the cohomology of toric varieties

本文研究了Weyl群表示不变量与toric簇上同调的关系，构造了显式的代数同构并推广到更一般的情况，回答了两个开放问题。

Torus actions and combinatorics of polytopes

An n-dimensional polytope P^n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope P^n with m codimension-one faces defines an arrangement of even-dimensional planes in R^{2m}. We construct a free action of the group R^{m-n} on the complement of this arrangement. The corresponding quotient is a smooth manifold Z_P invested with a canonical action of the compact torus T^m with the orbit space P^n. For each smooth projective toric variety M^{2n} defined by a simple polytope P^n with the given lattice of faces there exists a subgroup T…