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Antonio Díaz Ramos

Publications and source records attributed to Antonio Díaz Ramos.

16 recordsLinked to original sources
arXiv2026

Cohomology of non-finite CL-shellable posets

Shellable complexes are homotopy equivalent to a wedge of spheres of possibly different dimensions, so that the (co)homology of the constant functor over the complex is concentrated in those degrees. In this work, we introduce the concept of a stable functor -a local weakening of fibrancy- over a shellable poset, which ensures the vanishing of the (co)homology of such a functor in specific degrees. The methods are based on a model category structure on the category of functors indexed by a filtered poset and the combinatorial structure of shellable posets. Our techniques work over non-finite and non-pure posets and employ a description of (co)homology via explicit fibrant replacements. Applications include acyclicity criteria for Mackey functors, computation of cohomology of $j$-th exterior powers over arrangement lattices, and homological decompositions for Bianchi groups $Γ_d$ for $d=1,2$.

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arXiv2026

Quillen's conjecture and unitary groups

We prove that the Quillen posets $\mathcal{A}_p(H)$ of $p$-extensions $H$ of simple unitary groups have non-zero homology in the largest possible dimension, with just a few exceptions. This establishes a conjecture raised by Aschbacher-Smith in 1992. In particular, by their work and a more recent article by Piterman-Smith, Quillen's conjecture on the $p$-subgroup posets holds for odd primes.

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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 $n \ge 2$.

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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 $\operatorname{Aut}_{{\mathcal P}art}\big({\mathbb P}(G)\big)\cong \operatorname{Aut}_{{\mathcal G}raphs}(G)$.

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arXiv2024

Mackey functors for posets

We characterize cofibrant objects in the category of functors indexed in a filtered poset and we show that these objects are acyclic. As a consequence, we show that Mackey functors over posets are also acyclic, where we define this type of Mackey functors mimicking the classical notion. As application, we study homotopy colimits over posets and we give a homology decomposition for the classifying space of the Bianchi group $Γ_1$.

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arXiv2021

On the cohomology of pro-fusion systems

We prove the Cartan-Eilenberg stable elements theorem and construct a Lyndon-Hochschild-Serre type spectral sequence for pro-fusion systems. As an application, we determine the continuous mod-$p$ cohomology ring of $\text{GL}_2(\mathbb{Z}_p)$ for any odd prime $p$.

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arXiv2020

A geometric approach to Quillen's conjecture

We introduce {\em admissible collections} for a finite group $G$ and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the {\em Quillen dimension at $p$ property}, a strong version of Quillen's conjecture, at a given odd prime divisor $p$ of $|G|$. Compared to the methods in \cite{AS1993}, our techniques are simpler.

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arXiv2017

Spectral sequences via examples

These are lecture notes for a short course about spectral sequences that was held at Málaga, October 18--20 (2016), during the "Fifth Young Spanish Topologists Meeting". The approach was to illustrate the basic notions via fully computed examples arising from Algebraic Topology and Group Theory.

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arXiv2016

Cohomology of p-groups of nil potency class smaller than p

Let $p$ be a prime number, let $d$ be an integer and let $G$ be a $d$-generated finite $p$-group of nilpotency class smaller than $p$. Then the number of possible isomorphism types for the mod $p$ cohomology algebra $H^*(G;{\mathbb F}_p)$ is bounded in terms of $p$ and $d$.

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arXiv2016

On Quillen's conjecture for p-solvable groups

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without employing it.

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arXiv2015

A local method for posets

We propose some conditions on a poset that produce a small chain complex for its homology. This allows to compare simplicial complexes and Quillen's complexes under the same prism. It turns out they differ in the existence or not of free faces in an acyclic complex.

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arXiv2014

The cohomology of the sporadic group $J_2$ over $\mathbb{F}_3$

We describe the cohomology ring $H^*(J_2;\mathbb{F}_3)$ both as subring of $H^*(3^{1+2}_+;\mathbb{F}_3)$ and with an abstract presentation. We also give its Poincaré series. We use as tool a spectral sequence for the strongly closed $3$-subgroup of $J_2$. This method might be used to compute the cohomology of any finite simple group with a strongly closed $p$-subgroup.

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arXiv2013

A spectral sequence for fusion systems

We build a spectral sequence converging to the cohomology of a fusion system with a strongly closed subgroup. This spectral sequence is related to the Lyndon-Hochschild-Serre spectral sequence and coincides with it for the case of an extension of groups. Nevertheless, the new spectral sequence applies to more general situations like finite simple groups with a strongly closed subgroup and exotic fusion systems with a strongly closed subgroup. We prove an analogue of a result of Stallings in the context of fusion preserving homomorphisms and deduce Tate's p-nilpotency criterion as a corollary.

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