arXiv research

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

Study 2-complexes' homology properties and torsion growth.

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

The paper shows geometric realisation over specific groups and knots.

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…

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

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

Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.

Rationality proved for curvature invariants of 2-complexes.

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

The paper finds incommensurable lattices in complex models of Baumslag-Solitar groups.

Researchers show a complex structure is not a counterexample to a topological problem.

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…

The main result of this article is that among the family of one-relator presentation 2-complexes that might be expected to be finitely unsplittable (not the union of two proper subpolyhedra with finite first homology groups) almost all have this property. Included among these one-relator presentation 2-complexes are al…

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

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…

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

Fixed point sets of certain group actions are contractible.

Scharlemann constructed a connected simplicial 2-complex $Γ$ with an action by the group ${\mathcal H_{2}}$ of isotopy classes of orientation preserving homeomorphisms of $S^3$ that preserve the isotopy class of an unknotted genus 2 handlebody $V$. In this paper we prove that the 2-complex $Γ$ is contractible. Therefor…

New invariant detects non-homotopy equivalent 4-manifolds.

In this paper, we will use Kahn-Markovic's almost totally geodesic surfaces to construct certain $π_1$-injective 2-complexes in closed hyperbolic 3-manifolds. Such 2-complexes are locally almost totally geodesic except along a 1-dimensional subcomplex. Using Agol and Wise's result that fundamental groups of hyperbolic …

The paper studies a relative version of non-positive immersion for 2-complex pairs and shows conditions under which a transitivity law holds.

Tiny complexes share 3-5 triangles in common coverings.

A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to whic…

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

A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points …

The paper finds a short graph incompressible in a complex with specific properties.

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

If a finite group $G$ is isomorphic to a subgroup of $SO(3)$, then $G$ has the D2-property. Let $X$ be a finite complex satisfying Wall's D2-conditions. If $π_1(X)=G$ is finite, and $χ(X) \geq 1-Def(G)$, then $X \vee S^2$ is simple homotopy equivalent to a finite $2$-complex, whose simple homotopy type depends only on …

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…

We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.

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

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

We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of the fundamental group of Y either is virtually free abelian or …

Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an …

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

Given a finite, connected 2-complex $X$ such that $b_2(X)\le1$ we establish two existence results for representations of the fundamental group of $X$ into compact connected Lie groups $G$, with prescribed values on certain loops. If $b_2(X)=1$ we assume $G=SO(3)$ and that the cup product on the first rational cohomolog…

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…

Minimal genus surfaces solve homology problems in finite complexes.

New rational curvature measures for 2-complexes.

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 study the interplay among Wall's $D(2)$ problem, normal generation conjecture (the Wiegold Conjecture) of perfect groups and Swan's problem on partial Euler characteristic and deficiency of groups. In particular, for a 3-dimensional complex $X$ of cohomological dimension 2 with a finite fundamental group, assuming t…

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2-complex, and an algorithm for co…

A complex disproves a curvature property.

An updated proof of a 1933 theorem of Goeritz, exhibiting a finite set of generators for the group of automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. The group is analyzed via its action on a certain connected 2-complex. (The analogous problem for higher genus Heegaard splittings appears to …

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…