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Elias Gabriel Minian

Publications and source records attributed to Elias Gabriel Minian.

At least 19 recordsLinked to original sources
arXiv2026

A topological version of the Cartan-Hadamard Theorem and asphericity complexity

We investigate a topological version of the Cartan-Hadamard theorem that allows one to study asphericity of spaces via distinguished families of paths. A topological space has the distinguished path property (dpp) if there exists a continuous map from the space $Π(X)$ of homotopy classes of paths (relative endpoints) to the path space $X^I$ that is a right inverse to the canonical quotient map. For complete metric spaces endowed with a locally convex metric, the distinguished paths are precisely the local geodesics. We show that if $X$ has the dpp, then its universal cover is contractible and, in particular, $X$ is aspherical. The space $Π(X)$ is a fiber bundle over $X$ whose fiber is the universal cover of $X$, and in the case of Riemannian manifolds of non-positive curvature it is naturally isomorphic to the tangent bundle. We prove that every aspherical CW-complex that is locally finite or countable has the dpp, and this allows us to reinterpret asphericity of CW-complexes in terms of the existence of continuous sections. We also define and study the notion of asphericity complexity of spaces by means of local sections from $Π(X)$ to $X^I$ and relate it to the concept of equivariant topological complexity introduced by Colman and Grant.

math.GT↗
arXiv2025

$δ$-core subsampling, strong collapses and TDA

We introduce a subsampling method for topological data analysis based on strong collapses of simplicial complexes. Given a point cloud and a scale parameter $δ$, we construct a subsampling that preserves both global and local topological features while significantly reducing computational complexity of persistent homology calculations. We illustrate the effectiveness of our approach through experiments on synthetic and real datasets, showing improved persistence approximations compared to other subsampling techniques.

cs.CG↗
arXiv2025

Weakly $G$-slim complexes and the non-positive immersion property for generalized Wirtinger presentations

We investigate weakly $G$-slim complexes, a more flexible variant of Helfer and Wise's slim complexes, which can be defined on any regular $G$-covering. We prove that if a $2$-complex $X$ associated to a group presentation is weakly $H$-slim for some regular $H$-covering, and $π_1X$ is left-orderable, then $X$ is slim and, in particular, it has non-positive immersions. We apply this result to study conditions on generalized Wirtinger presentations that guarantee the non-positive immersion property for their associated $2$-complexes. This class of presentations includes classical Wirtinger presentations of (perhaps high-dimensional) knots in spheres, some classes of Adian presentations and LOTs presentations. On the one hand, our results provide a wide variety of examples of presentation complexes with the non-positive immersion property via a condition that can be checked with a simple algorithm. On the other hand, as an application of these methods, we derive an extension of a result by Wise on Adian presentations and prove a stronger formulation of a result of Howie on LOT presentations.

math.GR↗
arXiv2023

Local indicability of groups with homology circle presentations

We investigate conditions that guarantee local indicability of groups that admit presentations with the homology of a circle, generalizing a result of J. Howie for two-relator presentations. We apply our results to investigate local indicability of LOT groups and some classes of non-cycle-free Adian presentations, extending previous results in that direction by J. Howie and D. Wise.

math.GR↗
arXiv2022

Morse theory of Bestvina-Brady type for posets and matchings

We introduce a Morse theory for posets of Bestvina-Brady type combining matchings and height functions. This theory generalizes Forman's discrete Morse theory for regular CW-complexes and extends previous results on Morse theory for $h$-regular posets to all finite posets. We also develop a relative version of Morse theory which allows us to compare the topology of a poset with that of a given subposet.

math.AT↗
arXiv2020

Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

We present a metric condition ${\LARGEτ}'$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition ${\LARGEτ}'$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, ${\LARGEτ}'$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation $P=\langle X \mid R\rangle$ of group $G$ satisfies conditions ${\LARGEτ}'-C'(\frac{1}{2})$, the length of any nontrivial word in the free group generated by $X$ representing the trivial element in $G$ is at least that of the shortest relator. We also introduce a strict metric condition ${\LARGEτ}'_{<}$, which implies hyperbolicity. Finally we investigate non-metric and dual variants of these conditions, and study a harmonic mean version of small cancellation theory.

math.GR↗
arXiv2019

Strictly systolic angled complexes and hyperbolicity of one-relator groups

We introduce the notion of strictly systolic angled complexes. They generalize Januszkiewickz and Świątkowski's $7$-systolic simplicial complexes and also their metric counterparts, which appear as natural analogues to Huang and Osajda's metrically systolic simplicial complexes in the context of negative curvature. We prove that strictly systolic angled complexes and the groups that act on them geometrically, together with their finitely presented subgroups, are hyperbolic. We use these complexes to study the geometry of one-relator groups without torsion, and prove hyperbolicity of such groups under a metric small cancellation hypothesis, weaker than $C'(1/6)$ and $C'(1/4)-T(4)$.

math.GR↗
arXiv2019

Simplicial complexity of surface groups and systolic area

The simplicial complexity is an invariant for finitely presentable groups that was recently introduced by Babenko, Balacheff and Bulteau to study systolic area. The simplicial complexity $κ(G)$ was proved to be a good approximation of the systolic area $σ(G)$ for large values of $κ(G)$. In this paper we compute the simplicial complexity of all surface groups (both in the orientable and in the non-orientable case). This settles a problem raised by Babenko, Balacheff and Bulteau. We also prove that $κ(G\ast \mathbb{Z})=κ(G)$ for any surface group $G$. This provides the first partial evidence in favor of the conjecture of the stability of the simplicial complexity under free product with free groups. The general stability problem, both for simplicial complexity and for systolic area, remains open.

math.GT↗
arXiv2019

The fundamental group of the $p$-subgroup complex

We study the fundamental group of the $p$-subgroup complex of a finite group $G$. We show first that $π_1(A_3(A_{10}))$ is not a free group (here $A_{10}$ is the alternating group on $10$ letters). This is the first concrete example in the literature of a $p$-subgroup complex with non-free fundamental group. We prove that, modulo a well-known conjecture of M. Aschbacher, $π_1(A_p(G)) = π_1(A_p(S_G)) * F$, where $F$ is a free group and $π_1(A_p(S_G))$ is free if $S_G$ is not almost simple. Here $S_G = Ω_1(G)/O_{p'}(Ω_1(G))$. This result essentially reduces the study of the fundamental group of $p$-subgroup complexes to the almost simple case. We also exhibit various families of almost simple groups whose $p$-subgroup complexes have free fundamental group.

math.GR↗
arXiv2018

The cylinder of a relation and generalized versions of the Nerve Theorem

We introduce the notion of cylinder of a relation in the context of posets, extending the construction of the mapping cylinder. We establish a local-to-global result for relations, generalizing Quillen's Theorem A for order preserving maps, and derive novel formulations of the classical Nerve Theorem for posets and simplicial complexes, suitable for covers with not necessarily contractible intersections.

math.AT↗
arXiv2017

The covering type of closed surfaces and minimal triangulations

The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the minimum possible number of vertices of a simplicial complex homotopy equivalent to X. In this article we compute the covering type of all closed surfaces. Our results completely settle a problem posed by Karoubi and Weibel, and shed more light on the relationship between the topology of surfaces and the number of vertices of minimal triangulations from a homotopy point of view.

math.GT↗
arXiv2016

The homotopy types of the posets of p-subgroups of a finite group

We study the homotopy properties of the posets of p-subgroups Sp(G) and Ap(G) of a finite group G, viewed as finite topological spaces. We answer a question raised by R.E. Stong in 1984 about the relationship between the contractibility of the finite space Ap(G) and that of Sp(G) negatively, and describe the contractibility of Ap(G) in terms of algebraic properties of the group G.

math.GR↗
arXiv2016

A new test for asphericity and diagrammatic reducibility of group presentations

We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of the conjecture.

math.GR↗
arXiv2014

The second homotopy group in terms of colorings of locally finite models and new results on asphericity

We describe the second homotopy group of any CW-complex $K$ by analyzing the universal cover of a locally finite model of $K$ using the notion of $G$-coloring of a partially ordered set. As applications we prove a generalization of the Hurewicz theorem, which relates the homotopy and homology of non-necessarily simply-connected complexes, and derive new results on asphericity for two-dimensional complexes and group presentations.

math.AT↗
arXiv2014

$G$-colorings of posets, coverings and presentations of the fundamental group

We introduce the notion of a coloring of a poset, which consists of a labeling of the edges in its Hasse diagram by elements in a given group $G$. We use $G$-colorings to describe the covering maps of posets and present a new method based on colorings to obtain concrete and simple presentations of the fundamental group of polyhedra.

math.AT↗
arXiv2014

Homotopy colimits of diagrams over posets and variations on a theorem of Thomason

We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason's theorem on homotopy colimits over posets. In particular this allows us to characterize the homotopy colimits of diagrams of simplicial complexes in terms of the Grothendieck construction on the diagrams of their face posets. We also derive analogues of well known results on homotopy colimits in the combinatorial setting, including a cofinality theorem and a generalization of Quillen's Theorem A for posets.

math.AT↗