# Alexander A. GaifullinRussian Academy of Sciences | RAS · Steklov Mathematical Institute

Alexander A. Gaifullin

PhD (Russian Candidate), DrSci

## About

48

Publications

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226

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Introduction

**Skills and Expertise**

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July 2007 - present

## Publications

Publications (48)

Получена полная классификация конечно порожденных инволютивных коммутативных двузначных групп. Построены три серии таких двузначных групп: основная, унипотентная и специальная - и показано, что любая конечно порожденная инволютивная коммутативная двузначная группа изоморфна двузначной группе, принадлежащей одной из этих серий. Получен ряд классифик...

The Torelli group of a closed oriented surface of genus is the subgroup of the mapping class group consisting of all mapping classes that act trivially on the homology of . One of the most intriguing open problems concerning Torelli groups is the question of whether the group is finitely presented. A possible approach to this problem relies on the...

Группа Торелли замкнутой ориентированной поверхности $S_g$ рода $g$ - это подгруппа $\mathcal{I}_g$ группы классов отображений $\operatorname{Mod}(S_g)$, состоящая из всех классов отображений, которые тривиально действуют на гомологиях поверхности $S_g$. Одна из самых интересных открытых проблем, касающихся групп Торелли, - вопрос, является ли груп...

The Torelli group of a genus $g$ oriented surface $S_g$ is the subgroup $\mathcal{I}_g$ of the mapping class group $\mathrm{Mod}(S_g)$ consisting of all mapping classes that act trivially on the homology of $S_g$. One of the most intriguing open problems concerning Torelli groups is the question of whether the group $\mathcal{I}_3$ is finitely pres...

Professor Mikhail Ivanovich Shtogrin (born September 25, 1938) is widely known due to his contributions to discrete geometry (including regular tilings and Dirichlet-Voronoi partitions) and geometrical crystallography (including cubical complexes). The paper contains a short description of his life, scientific activities, and a photo.

The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $\mathcal{I}_g$ and the Johnson kernel $\mathcal{K}_g$. By a fundamental result...

Professor Mikhail Ivanovich Shtogrin (born September 25, 1938) is widely known due to his contributions to discrete geometry (including regular tilings and Dirichlet-Voronoi partitions) and geometrical crystallography (including cubical complexes). The paper contains a short description of his life, scientific activities, and a photo.

We prove that the Dehn invariant of any flexible polyhedron in n-dimensional Euclidean space, where n ≥ 3, is constant during the flexion. For n = 3 and 4 this implies that any flexible polyhedron remains scissors congruent to itself during the flexion. This proves the Strong Bellows Conjecture posed by R. Connelly in 1979. It was believed that thi...

Let $\mathcal{I}_g$ be the Torelli group of an oriented closed surface $S_g$ of genus $g$, that is, the kernel of the action of the mapping class group on the first integral homology group of $S_g$. We prove that the $k$th integral homology group of $\mathcal{I}_g$ contains a free Abelian subgroup of infinite rank, provided that $g\ge 3$ and $2g-3\...

We prove that the Dehn invariant of any flexible polyhedron in Euclidean space of dimension greater than or equal to 3 is constant during the flexion. In dimensions 3 and 4 this implies that any flexible polyhedron remains scissors congruent to itself during the flexion. This proves the Strong Bellows Conjecture posed by Connelly in 1979. It was be...

We consider a category (Formula presented.) whose morphisms are (Formula presented.)-dimensional pseudomanifolds equipped with certain additional structures (coloring and labeling of some cells), multiplication of morphisms is similar to a concatenation of cobordisms. On the other hand, we consider the product (Formula presented.) of (Formula prese...

An oriented connected closed manifold $M^n$ is called a URC-manifold if for any oriented connected closed manifold $N^n$ of the same dimension there exists a nonzero degree mapping of a finite-fold covering $\widehat{M}^n$ of $M^n$ onto $N^n$. This condition is equivalent to the following: For any $n$-dimensional integral homology class of any topo...

We discuss some recent results on flexible polyhedra and the bellows conjecture, which claims that the volume of any flexible polyhedron is constant during the flexion. Also, we survey main methods and several open problems in this area.

The bellows conjecture claims that the volume of any flexible polyhedron of dimension 3 or higher is constant during the flexion. The bellows conjecture was proved for flexible polyhedra in the Euclidean spaces of dimensions 3 and higher, and for bounded flexible polyhedra in the odd-dimensional Lobachevsky spaces. Counterexamples to the bellows co...

A flexible polyhedron in an -dimensional space of constant curvature is a polyhedron with rigid -dimensional faces and hinges at -dimensional faces. The Bellows conjecture claims that, for , the volume of any flexible polyhedron is constant during the flexion. The Bellows conjecture in Euclidean spaces was proved by Sabitov for (1996) and by the au...

We construct examples of embedded flexible cross-polytopes in the spheres of
all dimensions. These examples are interesting from two points of view. First,
in dimensions 4 and higher, they are the first examples of embedded flexible
polyhedra. Notice that, unlike in the spheres, in the Euclidean spaces and the
Lobachevsky spaces of dimensions 4 and...

We consider a category whose morphisms are bordisms of $n$-dimensional
pseudomanifolds equipped with a certain additional structure (coloring). On the
other hand, we consider the product $G$ of $(n+1)$ copies of infinite symmetric
group. We show that unitary representations of $G$ produce functors from the
category of $(n-1)$-dimensional bordisms t...

We construct self-intersected flexible cross-polytopes in the spaces of
constant curvature, that is, the Euclidean spaces, the spheres, and the
Lobachevsky spaces of all dimensions. In dimensions greater than or equal to 5,
these are the first examples of flexible polyhedra. Moreover, we classify all
flexible cross-polytopes in each of the spaces o...

For n greater than or equal to 4, the square of the volume of an n-simplex
satisfies a polynomial relation with coefficients depending on the squares of
the areas of 2-faces of this simplex. First, we compute the minimal degree of
such polynomial relation. Second, we prove that the volume an n-simplex
satisfies a monic polynomial relation with coef...

In the end of the 19th century Bricard discovered a phenomenon of flexible
polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit
non-trivial flexes. One of the most important results in this field is a
theorem of Sabitov asserting that the volume of a flexible polyhedron is
constant during the flexion. In this paper we study...

In 1996 Sabitov proved that the volume of an arbitrary simplicial polyhedron
P in the 3-dimensional Euclidean space $\R^3$ satisfies a monic (with respect
to V) polynomial relation F(V,l)=0, where l denotes the set of the squares of
edge lengths of P. In 2011 the author proved the same assertion for polyhedra
in $\R^4$. In this paper, we prove that...

In 1940s Steenrod asked if every homology class $z\in H_n(X,\mathbb{Z})$ of
every topological space $X$ can be realised by an image of the fundamental
class of an oriented closed smooth manifold. Thom found a non-realisable
7-dimensional class and proved that for every $n$, there is a positive integer
$k(n)$ such that the class $k(n)z$ is always re...

We study oriented closed manifolds M^n possessing the following Universal
Realisation of Cycles (URC) Property: For each topological space X and each
integral homology class z of it, there exist a finite-sheeted covering \hM^n of
M^n and a continuous mapping f of \hM^n to X such that f takes the fundamental
class [\hM^n] to kz for a non-zero intege...

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a
3-dimensional Euclidean space is a root of certain polynomial with coefficients
depending on the combinatorial type and on edge lengths of the polyhedron only.
Moreover, the coefficients of this polynomial are polynomials in edge lengths
of the polyhedron. This result impl...

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based on the notion of a universal...

We consider the classical N. Steenrod’s problem of realization of cycles by continuous images of manifolds. Our goal is to
find a class $
\mathcal{M}_n
$
\mathcal{M}_n
of oriented n-dimensional closed smooth manifolds such that each integral homology class can be realized with some multiplicity by an image
of a manifold from the class $
\mathcal{...

We construct and study a new 15-vertex triangulation X of the complex projective plane ℂP2. The automorphism group of X is isomorphic to S
4 × S
3. We prove that the triangulation X is the minimal (with respect to the number of vertices) triangulation of ℂP2 admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizat...

To every oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The resulting transformation is the main object of study in this paper. We pose an inversion problem for and show that this problem is closely related to Steenrod's problem on the realization of cycles and to the Rok...

We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a finite-fold covering over the manifold M^n, where M^n is the isospectral manifold of real symmetric tridiagonal (n=1)x(n...

We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinatorial construction that for a...

Let be the characteristic class of a combinatorial manifold given by a polynomial in the rational Pontryagin classes of . We prove that for any polynomial there is a function taking each combinatorial manifold to a cycle in its rational simplicial chains such that: 1) the Poincaré dual of represents the cohomology class ; 2) the coefficient of each...

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We propose an approach to the characterization of triangulations of manifolds by means of the theory of m-valued groups.

This paper is devoted to the well-known problem of computing the Stiefel-Whitney classes and the Pontryagin classes of a manifold from a given triangulation of the manifold. In 1940 Whitney found local combinatorial formulae for the Stiefel-Whitney classes. The first combinatorial formula for the first rational Pontryagin class was found by Gabriel...

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In this paper we consider configurations of straight lines in
general position in a plane with all intersection points marked to show
which of the two lines is above the other. We prove that there exist two
isotopic configurations such that one of them can be obtained as a
projection of a collection of straight lines in 3-space, and the other
no...

A simple complete combinatorial invariant for elements of the braid group is found. It admits some generalizations, e.g. a complete invariant of spherical braids and a complete invariant of cylindrical braids. Values of the invariants are well recognizable, i.e.: they provide the complete algorithmic classification of elements in the named braid gr...