arXiv research

Paper confirms Whitehead's conjecture for aspherical 2-complexes.

We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies…

Diagrammatic reducibility DR and its generalization vertex asphericity VA are combinatorial tools developed for detecting asphericity of a 2-complex. Here we present tests for a relative version of VA that apply to pairs of 2-complexes $(L,K)$, where $K$ is a subcomplex of $L$. We show that a relative weight test holds…

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

The paper shows geometric realisation over specific groups and knots.

We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical 2-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds.

A labeled oriented graph (LOG) is an oriented graph with a labeling function from the edge set into the vertex set. The complexity of a LOG is the minimal cardinality of an initial set $S$ of vertices such that every vertex can be reached successively from $S$ only using edges with labels in $S$ or already visited vert…

The paper shows how reducible complexes affect local indicability.

Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.

Study of knots and links in 2-complexes, defining linking numbers and polynomials.

A 2-complex requires at least 12 colours to avoid edge conflicts.

New infinite family of 2-complexes intrinsically linked in 4D.

Proves universal pairing result for 2-complexes, showing lack of positivity.

Study 2-complexes' homology properties and torsion growth.

3D Schoenflies theorem for simply-connected 2-complexes.

We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary t…

New rational curvature measures for 2-complexes.

The paper shows examples of 2-complexes that can't be embedded in R^4, hiding obstructions in higher Milnor invariants.

Aspherical configuration Lie groupoids complement is proven for a class of orbifolds.

We consider computational complexity of problems related to the fundamental group and the first homology group of (embeddable) $2$-complexes. We show, as an extension of an earlier work, that computing first homology of $2$-complexes is equivalent in computational complexity to matrix diagonalization. That is, the usua…

We prove a finiteness result for the systolic area of groups, answering a question of M. Gromov. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of~2-complexes whose systolic area is uniformly bounded. Furthermore, we prove a uniform systolic inequality for all 2-complexe…

Study L^2-Betti numbers in prime characteristic for a conjecture about 2-complex towers.

The study finds stably free modules and distinct 2-complexes for large ranks.

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …

A complex disproves a curvature property.

We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-com…

Study a relative aspherical conjecture and prove 3-manifold obstruction to positive scalar curvature.

Relative notions of combinatorial asphericity have been used to prove that injective labeled oriented trees (which encode spines of ribbon 2-knots) are aspherical. This article presents an overview and comparison of the different notions of relative combinatorial asphericity. It also contains new results concerning cha…

Rationality proved for curvature invariants of 2-complexes.

Let $X$ be a contractible $2$-complex which is a union of two contractible subcomplexes $Y$ and $Z.$ Is the intersection $Y\cap Z$ contractible as well? In this note, we prove that the inclusion-induced map $π_{1}(Y\cap Z)\rightarrow π_{1}(Z)$ is injective if $Y$ is $π_{1}$-injective subcomplex in a locally CAT(0) 2-co…

The paper proves the Singer conjecture for aspherical complex surfaces and refines Gromov's inequality.

Constructs 4-manifolds with positive Euler characteristic proving a conjecture.

Study new bounds on TC of spaces with subgroup inclusions.

We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold $M$ such that the fundamental group $π_1(M)$ coarsely co…

Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov's inequ…

The study of $PD_3$-pairs extends results for aspherical 3-manifolds.

Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.

The paper explores conditions for sections in Lefschetz fibrations and bundles over 2-complexes.

Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.

This is a survey article on symplectically aspherical manifolds. The paper contains a discussion on constructions of symplectically aspherical manifolds, their topological properties and the role of this class in symplectic topology. Research perspectives are discussed.

It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is self-similar, then a typical leaf has exactly one topological end. We also cons…

3D spaces without boundaries are found that aren't manifolds.

The paper extends a vanishing theorem for hypersurfaces in aspherical manifolds.

A {\em word labeled oriented graph} (WLOG) is an oriented graph $\cal G$ on vertices $X=\{ x_1,\ldots ,x_k\}$, where each oriented edge is labeled by a word in $X^{\pm1}$. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of …

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior f…

We prove a systolic inequality for the phi-relative 1-systole of a phi-essential 2-complex, where phi is a homomorphism from the fundamental group of the complex, to a finitely presented group G. Indeed we show that universally for any phi-essential Riemannian 2-complex, and any G, the area of X is bounded below by 1/8…

The "polyhedral product functor" produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with …

Study shows some complex shapes don't fit a certain property.