arXiv subjects

Antonio Viruel

Publications and source records attributed to Antonio Viruel.

At least 19 recordsLinked to original sources

arXiv2026

Realizing additive monoids as mapping degree sets

We prove that mapping degree sets are stable under multiplication by finite subsets of $\mathbb Z$ containing $0$ and by sets obtained from additive submonoids of $\mathbb Z$ through finitely many sums and products. In particular, every set of the latter type occurs as a mapping degree set. As a consequence, we obtain a broad family of infinite mapping degree sets, including finite unions of arithmetic progressions starting at $0$. These results extend previous work on the realization problem and are related to a question posed by Neofytidis, Wang, and Wang.

math.AT↗

arXiv2026

On the Grothendieck Ring of Finite-Dimensional Non-Degenerate Evolution Algebras

We study the Grothendieck ring of finite-dimensional non-degenerate evolution algebras over a field $\mathbb{K}$, with addition induced by direct sum and multiplication induced by tensor product. Although its underlying abelian group is freely generated by indecomposable isomorphism classes, the ring structure is much smaller: tensor products create systematic non-cancellation phenomena and many zero-divisors. The key invariant is the balance of the directed graph associated with a natural basis. We prove that, for non-degenerate evolution algebras, this balance is independent of the chosen natural basis. We then show that balance controls cancellation after tensoring: indecomposable factors of balance $1$ cancel, whereas factors of larger balance produce zero-divisor relations under mild hypotheses. Finally, we analyze the subring generated by cyclic evolution algebras and prove the predicted zero-divisor criterion for all finite direct sums of cyclic algebras.

math.RA↗

arXiv2025

Every finite group is represented by a finite incidence geometry

We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general framework to represent pairs of finite groups $(G, H)$, where $H \trianglelefteq G$, as pairs of correlation--automorphism groups of suitable incidence geometries. Specifically, we prove that for every such pair $(G, H)$, there exists a finite incidence geometry $Γ$ satisfying that the pair $(\operatorname{Aut}(Γ), \operatorname{Aut}_I(Γ))$ of correlation--automorphism groups of $Γ$ is isomorphic to $(G, H)$. Our construction proceeds in two main steps: first, we realize $(G, H)$ as the correlation and automorphism groups of an incidence system; then, we refine this system into a genuine incidence geometry preserving the same pair of automorphisms groups. We also provide explicit examples, including a family of geometries realizing $(S_n, A_n)$ for all $n \ge 2$.

math.GR↗

arXiv2025

Rigidifying simplicial complexes and realizing group actions

We show that any action of a finite group on a finitely presentable group arises as the action of the group of self-homotopy equivalences of a space on its fundamental group. In doing so, we prove that any finite connected (abstract) simplicial complex $\mathbf{K}$ can be rigidified -- meaning it can be perturbed in a way that reduces the full automorphism group to any subgroup -- while preserving the homotopy type of the geometric realization $| \mathbf{K} |$. We also obtain that every action of a finite group on a finitely generated abelian group is the action of the group of self-homotopy equivalences of a space on one of its higher homotopy groups.

math.AT↗

arXiv2025

Hom-counting functions, combinatorial categories and related problems

Combinatorial categories satisfy a stronger form of Yoneda Lemma, namely, the isomorphism type of an object can be recovered by counting the number of homomorphisms from all other objects into it. In this work, we show that this property holds for sufficiently small categories by studying the algebra of homomorphism-counting functions. We present applications of the results to the isomorphism problem in group, graph and ring theory.

math.CT↗

arXiv2025

Path partial groups

It is well known that not every finite group arises as the full automorphism group of some group. Here we show that the situation is dramatically different when considering the category of partial groups, ${{\mathcal P}art}$, as defined by Chermak: given any group $H$ there exists infinitely many non isomorphic partial groups ${\mathbb M}$ such that $\operatorname{Aut}_{{\mathcal P}art}({\mathbb M})\cong H$. To prove this result, given any simple undirected graph $G$ we construct a partial group ${\mathbb P}(G)$, called the path partial group associated to $G$, such that $\operatorname{Aut}_{{\mathcal P}art}\big({\mathbb P}(G)\big)\cong \operatorname{Aut}_{{\mathcal G}raphs}(G)$.

math.AT↗

arXiv2025

Permutation representations and automorphisms of evolution algebras

We prove that the natural permutation representation of highly transitive finite groups cannot be realized as the full automorphism group of an idempotent, finite-dimensional evolution algebra acting on the set of lines spanned by its natural elements. Specifically, for any sufficiently large integer $n$ and $k \geq 4$, there does not exist an idempotent evolution algebra $X$ of dimension $n$ such that $\operatorname{Aut}(X)$ is isomorphic to a proper $k$-transitive subgroup of $S_n$. Nevertheless, we show that for any finite group $G$, any permutation representation $ξ\colon G \to S_n$, and any field $\Bbbk$, there exists an idempotent, finite-dimensional evolution $\Bbbk$-algebra $X$ such that $\operatorname{Aut}(X) \cong G$, and the induced representation of $\operatorname{Aut}(X)$ on the natural idempotents of $X$ is equivalent to $ξ$.

math.RA↗

arXiv2024

Finite sets containing zero are mapping degree sets

In this paper we solve in the positive the question of whether any finite set of integers, containing the zero, is the mapping degree set between two oriented closed connected manifolds of the same dimension. We extend this question to the rational setting, where an affirmative answer is also given.

math.GT↗

arXiv2024

On the universal and generalized orbifold Euler characteristics

We discuss the universal orbifold Euler characteristic and generalized orbifold Euler characteristics corresponding to finitely generated groups $A$ (the $A$-Euler characteristics). We show that the collection of all $A$-Euler characteristics for $A$ of the form $A'\times Z$ ($Z$ is the group of integers) with finite $A'$ determine the universal orbifold Euler characteristic.

math.AG↗

arXiv2024

On properties of effective topological complexity and effective Lusternik-Schnirelmann category

The notion of effective topological complexity, introduced by Błaszczyk and Kaluba, deals with using group actions in the configuration space in order to reduce the complexity of the motion planning algorithm. In this article we focus on studying several properties of such notion of topological complexity. We introduce a notion of effective LS-category which mimics the behaviour the usual LS-cat has in the non-effective setting. We use it to investigate the relationship between these effective invariants and the orbit map with respect of the group action, and we give numerous examples. Additionally, we investigate non-vanishing criteria based on a cohomological dimension bound of the saturated diagonal.

math.AT↗

arXiv2024

Realization of permutation modules via Alexandroff spaces

We raise the question of the realizability of permutation modules in the context of Kahn's realizability problem for abstract groups and the $G$-Moore space problem. Specifically, given a finite group $G$, we consider a collection $\{M_i\}_{i=1}^n$ of finitely generated $\Z G$-modules that admit a submodule decomposition on which $G$ acts by permuting the summands. Then we prove the existence of connected finite spaces $X$ that realize each $M_i$ as its $i$-th homology, $G$ as its group of self-homotopy equivalences $\E(X)$, and the action of $G$ on each $M_i$ as the action of $\E(X)$ on $H_i(X; \Z)$.

math.AT↗

arXiv2023

The achievement set of generalized multigeometric sequences

We study the topology of all possible subsums of the generalized multigeometric series $k_1f(x)+k_2f(x)+\dots+k_mf(x)+\dots + k_1f(x^n)+\dots+k_mf(x^n)+\dots,$ where $k_1, k_2, \dots, k_m$ are fixed positive real numbers and $f$ runs along a certain class of non-negative functions on the unit interval. We detect particular regions of this interval for which this achievement set is, respectively, a compact interval, a Cantor set and a Cantorval.

math.CA↗

arXiv2023

Automorphism groups of Cayley evolution algebras

In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is isomorphic to $Aut(X)$ where $X$ is a finite-dimensional absolutely simple Cayley evolution $k$-algebra.

math.RA↗

arXiv2022

Groups as automorphisms of dessins d'enfants

It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin d'enfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compact dessins. Moreover, we show that any tame action of a countable group is so realizable.

math.GR↗

arXiv2022

On strongly inflexible manifolds

An oriented closed connected N-manifold M is inflexible if it does not admit self-maps of unbounded degree. In addition, if all the maps from any other oriented closed connected N-manifold have bounded degree, then M is said to be strongly inflexible. The existence of simply-connected inflexible manifolds was established by Arkowitz and Lupton. However, the existence of simply-connected strongly inflexible manifolds is still an open question. We provide an algorithm relying on Sullivan models that allow us to prove that all, but one, of the known examples of simply-connected inflexible manifolds are not strongly inflexible.

math.GT↗

arXiv2021

Regular evolution algebras are universally finite

In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra.

math.RA↗

arXiv2020

Acyclic $2$-dimensional complexes and Quillen's conjecture

Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ for any group $G$ admitting a $G$-invariant acyclic $p$-subgroup complex of dimension $2$. In particular, it follows that Quillen's conjecture holds for groups of $p$-rank $3$. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher--Smith.

math.AT↗

arXiv2020

A K-contact simply connected 5-manifold with no semi-regular Sasakian structure

We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connected symplectic 4-manifold with disjoint symplectic surfaces spanning the homology, all of them but one of genus 1 and the other of genus g>1, (b) to prove a bound on the second Betti number $b_2$ of an algebraic surface with $b_1=0$ and having disjoint complex curves spanning the homology when all of them but one are of genus 1 and the other of genus g>1.

math.DG↗