arXiv subjects

Alexander A. Gaifullin

Publications and source records attributed to Alexander A. Gaifullin.

At least 19 recordsLinked to original sources

arXiv2026

Finite generation, algebraicity, and representation stability for homology of Torelli groups

We solve a long-standing problem of whether the homology groups of the Torelli subgroups $\mathcal{I}_g\le\mathrm{Mod}_g$ are finitely generated in stable range. Namely, we prove that the group $H_k(\mathcal{I}_g;\mathbb{Z})$ is finitely generated, provided that $k\le g-2$. Two main ingredients of our approach are as follows. First, we show that the action of any symplectic transvection $t_x\in\mathrm{Sp}_{2g}(\mathbb{Z})$ on the homology of $\mathcal{I}_g$ satisfies the following unipotency condition: $(t_x-1)^{k+1}H_k( \mathcal{I}_g;\mathbb{Z})=0$. The proof of this fact relies on the study of the spectral sequence for the action of $\mathcal{I}_g$ on the complex of homologous curves on $Σ_g$. The second key ingredient is Tavgen's theorem asserting that the group $\mathrm{Sp}_{2g}(\mathbb{Z})$ is boundedly elementarily generated. For homology with coefficients in $\mathbb{Q}$, we further prove that $H_k(\mathcal{I}_g;\mathbb{Q})$ is an algebraic $\mathrm{Sp}_{2g}(\mathbb{Z})$-representation in the same stable range $k\le g-2$. Kupers and Randal-Williams have obtained a conditional result: they computed the algebraic part of the rational cohomology of Torelli groups in stable range under the assumpition that the rational cohomology groups are finite-dimensional in this stable range. Our results turn this conditional computation into a precise theorem that describes the whole rational cohomology ring of Torelli groups in stable range. As further applications, we, firstly, prove Morita's conjecture asserting that the $\mathrm{Sp}_{2g}(\mathbb{Z})$-invariant part of the rational cohomology of $\mathcal{I}_g$ stabilizes to the polynomial ring $\mathbb{Q}[e_2,e_4,\ldots]$ in the even Miller-Morita-Mumford classes; secondly, we prove the uniform representation stability for the series of groups $\left\{ H_k\bigl(\mathcal{I}_g^1;\mathbb{Q})\right\}_{g=1}^{\infty}$.

math.GT↗

arXiv2026

Exotic spherical flexible octahedra and counterexamples to the Modified Bellows Conjecture

In 2014 the author showed that in the three-dimensional spherical space, alongside with three classical types of flexible octahedra constructed by Bricard, there exists a new type of flexible octahedra, which was called exotic. In the present paper we give a geometric construction for exotic flexible octahedra, describe their configuration spaces, and calculate their volumes. We show that the volume of an exotic flexible octahedron is nonconstant during the flexion, and moreover the volume remains nonconstant if we replace any set of vertices of the octahedron with their antipodes. So exotic flexible octahedra are counterexamples to the Modified Bellows Conjecture proposed by the author in 2015.

math.MG↗

arXiv2026

Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$

Let $Σ_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in\{0,1\}$. The Johnson kernel $\mathcal{K}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}(Σ_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup $\mathrm{IO}_n\subset\mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization of $F_n$. Long standing questions are whether the groups $\mathcal{K}_g^b$ and $[\mathrm{IO}_n,\mathrm{IO}_n]$ or their abelianizations $(\mathcal{K}_g^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_n,\mathrm{IO}_n]^{\mathrm{ab}}$ are finitely generated for $g\ge3$ (respectively, $n\ge3$). During the last 15 years, these questions were answered positively for $g\ge4$ and $n\ge4$, respectively. Nevertheless, the cases of $g=3$ and $n=3$ remained completely unsettled. In this paper, we prove that the abelianizations $(\mathcal{K}_3^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_3,\mathrm{IO}_3]^{\mathrm{ab}}$ are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group.

math.GR↗

arXiv2024

New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane

Brehm and Kühnel (1992) constructed three 15-vertex combinatorial 8-manifolds `like the quaternionic projective plane' with symmetry groups $\mathrm{A}_5$, $\mathrm{A}_4$, and $\mathrm{S}_3$, respectively. Gorodkov (2016) proved that these three manifolds are in fact PL homeomorphic to $\mathbb{HP}^2$. Note that 15 is the minimal number of vertices of a combinatorial 8-manifold that is not PL homeomorphic to $S^8$. In the present paper we construct a lot of new 15-vertex triangulations of $\mathbb{HP}^2$. A surprising fact is that such examples are found for very different symmetry groups, including those not in any way related to the group $\mathrm{A}_5$. Namely, we find 19 triangulations with symmetry group $\mathrm{C}_7$, one triangulation with symmetry group $\mathrm{C}_6\times\mathrm{C}_2$, 14 triangulations with symmetry group $\mathrm{C}_6$, 26 triangulations with symmetry group $\mathrm{C}_5$, one new triangulation with symmetry group $\mathrm{A}_4$, and 11 new triangulations with symmetry group $\mathrm{S}_3$. Further, we obtain the following classification result. We prove that, up to isomorphism, there are exactly 75 triangulations of $\mathbb{HP}^2$ with 15 vertices and symmetry group of order at least 4: the three Brehm-Kühnel triangulations and the 72 new triangulations listed above. On the other hand, we show that there are plenty of triangulations with symmetry groups $\mathrm{C}_3$ and $\mathrm{C}_2$, as well as the trivial symmetry group.

math.CO↗

arXiv2024

On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

In 1987 Brehm and Kühnel showed that any triangulation of a $d$-manifold (without boundary) that is not homeomorphic to the sphere has at least $3d/2+3$ vertices. Moreover, triangulations with exactly $3d/2+3$ vertices may exist only for `manifolds like projective planes', which can have dimensions $2$, $4$, $8$, and $16$ only. There is a $6$-vertex triangulation of the real projective plane $\mathbb{RP}^2$, a $9$-vertex triangulation of the complex projective plane $\mathbb{CP}^2$, and $15$-vertex triangulations of the quaternionic projective plane $\mathbb{HP}^2$. Recently, the author has constructed first examples of $27$-vertex triangulations of manifolds like the octonionic projective plane $\mathbb{OP}^2$. The four most symmetrical have symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$. These triangulations were constructed using a computer program after the symmetry group was guessed. However, it remained unclear why exactly this group is realized as the symmetry group and whether $27$-vertex triangulations of manifolds like $\mathbb{OP}^2$ exist with other (possibly larger) symmetry groups. In this paper we find strong restrictions on symmetry groups of such $27$-vertex triangulations. Namely, we present a list of $26$ subgroups of $\mathrm{S}_{27}$ containing all possible symmetry groups of $27$-vertex triangulations of manifolds like the octonionic projective plane. (We do not know whether all these subgroups can be realized as symmetry groups.) The group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ is the largest group in this list, and the orders of all other groups do not exceed $52$. A key role in our approach is played by the use of Smith and Bredon's results on the topology of fixed point sets of finite transformation groups.

math.CO↗

arXiv2024

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

In 1987 Brehm and Kühnel showed that any combinatorial $d$-manifold with less than $3d/2+3$ vertices is PL homeomorphic to the sphere and any combinatorial $d$-manifold with exactly $3d/2+3$ vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper. The latter possibility may occur for $d\in\{2,4,8,16\}$ only. There exist a unique $6$-vertex triangulation of $\mathbb{RP}^2$, a unique $9$-vertex triangulation of $\mathbb{CP}^2$, and at least three $15$-vertex triangulations of $\mathbb{HP}^2$. However, until now, the question of whether there exists a $27$-vertex triangulation of a manifold like the octonionic projective plane has remained open. We solve this problem by constructing a lot of examples of such triangulations. Namely, we construct $634$ vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Four of them have symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$, and the other $630$ have symmetry group $\mathrm{C}_3^3$ of order $27$. Further, we construct more than $10^{103}$ non-vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Most of them have trivial symmetry group, but there are also symmetry groups $\mathrm{C}_3$, $\mathrm{C}_3^2$, and $\mathrm{C}_{13}$. We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane $\mathbb{OP}^2$. Nevertheless, we have no proof of this fact so far.

math.CO↗

arXiv2024

Commutativity of involutive two-valued groups

In 2022, V. M. Buchstaber, A. P. Veselov, and the author proved a number of classification results for involutive commutative two-valued groups, including a complete classification in the finitely generated case. In the present note, we prove that any involutive two-valued group is commutative, so all these results are valid without the assumption of commutativity.

math.GR↗

arXiv2023

On infinitely generated homology of Torelli groups

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 $2g-3\le k\le 3g-6$. Earlier the same property was known only for $k=3g-5$ (Bestvina, Bux, Margalit, 2007) and in the special case $g=k=3$ (Johnson, Millson, 1992). We also show that the hyperelliptic involution acts on the constructed infinite system of linearly independent homology classes in $\mathrm{H}_k(\mathcal{I}_g;\mathbb{Z})$ as multiplication by $-1$, provided that $k+g$ is even, thus solving negatively a problem by Hain. For $k=2g-3$, we show that the group $\mathrm{H}_{2g-3}(\mathcal{I}_g;\mathbb{Z})$ contains a free abelian subgroup of infinite rank generated by abelian cycles and we construct explicitly an infinite system of abelian cycles generating such subgroup. As a consequence of our results, we obtain that an Eilenberg--MacLane CW complex of type $K(\mathcal{I}_g,1)$ cannot have a finite $(2g-3)$-skeleton. The proofs are based on the study of the spectral sequence for the action of $\mathcal{I}_g$ on the complex of cycles constructed by Bestvina, Bux, and Margalit.

math.GR↗

arXiv2023

Classification of involutive commutative two-valued groups

A complete classification of finitely generated involutive commutative two-valued groups is obtained. Three series of such two-valued groups are constructed: principal, unipotent and special, and it is shown that any finitely generated involutive commutative two-valued group is isomorphic to a two-valued group belonging to one of these series. A number of classification results are obtained for topological involutive commutative two-valued groups in the Hausdorff and locally compact cases. The classification of algebraic involutive two-valued groups in the one-dimensional case is also discussed.

math.GR↗

arXiv2021

On spectral sequence for the action of genus 3 Torelli group on the complex of cycles

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 presented or not. A possible approach to this problem relies upon the study of the second homology group of $\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the action of $\mathcal{I}_3$ on the complex of cycles. In this paper we obtain a partial result towards the conjecture that $H_2(\mathcal{I}_3;\mathbb{Z})$ is not finitely generated and hence $\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ of the spectral sequence is infinitely generated, that is, the group $E^1_{0,2}$ remains infinitely generated after taking quotients by images of the differentials $d^1$ and $d^2$. If one proceeded with the proof that it also remains infinitely generated after taking quotient by the image of $d^3$, he would complete the proof of the fact that $\mathcal{I}_3$ is not finitely presented.

math.GT↗

arXiv2019

On the top homology group of Johnson kernel

The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $Σ_g$ on the lower central series of $π_1(Σ_g)$ 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 of Johnson (1985), $\mathcal{K}_g$ is the subgroup of $\mathrm{Mod}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group $\mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}(\mathcal{K}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is not finitely generated as a module over the group ring $\mathbb{Q}[\mathcal{I}_g]$.

math.GT↗

arXiv2017

Dehn invariant of flexible polyhedra

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 believed that this conjecture was disproved by Alexandrov and Connelly in 2009. However, we find an error in their counterexample. Further, we show that the Dehn invariant of a flexible polyhedron in either sphere or Lobachevsky space of dimension greater than or equal to 3 is constant during the flexion if and only if this polyhedron satisfies the usual Bellows Conjecture, i.e., its volume is constant during every flexion of it. Using previous results due to the first listed author, we deduce that the Dehn invariant is constant during the flexion for every bounded flexible polyhedron in odd-dimensional Lobachevsky space and for every flexible polyhedron with sufficiently small edge lengths in any space of constant curvature of dimension greater than or equal to 3.

math.MG↗

arXiv2016

Small covers of graph-associahedra and realization of cycles

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 topological space $X$, a multiple of it can be realized as the image of the fundamental class of a finite-fold covering $\widehat{M}^n$ of $M^n$ under a continuous mapping $f\colon \widehat{M}^n\to X$. In 2007 the author gave a constructive proof of the classical result by Thom that a multiple of any integral homology class can be realized as an image of the fundamental class of an oriented smooth manifold. This construction yields the existence of URC-manifolds of all dimensions. For an important class of manifolds, the so-called small covers of graph-associahedra corresponding to connected graphs, we prove that either they or their two-fold orientation coverings are URC-manifolds. In particular, we obtain that the two-fold covering of the small cover of the usual Stasheff associahedron is a URC-manifold. In dimensions 4 and higher, this manifold is simpler than all previously known URC-manifolds.

math.AT↗

arXiv2016

Flexible Polyhedra and Their Volumes

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.

math.MG↗

arXiv2016

The bellows conjecture for small flexible polyhedra in non-Euclidean spaces

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 conjecture are known in all open hemispheres of dimensions 3 and higher. The aim of this paper is to prove that, nonetheless, the bellows conjecture is true for all flexible polyhedra in either spheres or Lobachevsky spaces of dimensions greater than or equal to 3 with sufficiently small edge lengths.

math.MG↗

arXiv2015

The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces

A flexible polyhedron in an n-dimensional space of constant curvature, namely, in the Euclidean space, or in the Lobachevsky space, or in the sphere, is a polyhedron with rigid (n-1)-dimensional faces and hinges at (n-2)-dimensional faces. The Bellows conjecture claims that, for n greater than or equal to 3, the volume of any flexible polyhedron is constant during the flexion. The Bellows conjecture in Euclidean spaces was proved by Sabitov in dimension 3 (1996) and by the author in dimensions 4 and higher (2012). Counterexamples to the Bellows conjecture in open hemispheres were constructed by Alexandrov in dimension 3 (1997) and by the author in dimensions 4 and higher (2015). In this paper we prove the Bellows conjecture for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces. The proof is based on the study of the analytic continuation of the volume of a simplex in the Lobachevsky space considered as a function of the hyperbolic cosines of its edge lengths.

math.MG↗

arXiv2015

Embedded flexible spherical cross-polytopes with non-constant volumes

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 higher, still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are non-constant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was built only in dimension 3 (Alexandrov, 1997), and was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type among which there are our counterexamples to the usual Bellows Conjecture. By the way, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write relations on the volumes of their faces of codimensions 1 and 2.

math.MG↗

arXiv2015

Infinite symmetric group and bordisms of pseudomanifolds

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 to the category of Hilbert spaces and bounded linear operators.

math.RT↗