arXiv subjects

Max Gheorghiu

Publications and source records attributed to Max Gheorghiu.

4 recordsLinked to original sources

arXiv2026

Solid Duality for Profinite Groups

We classify profinite groups that have a Poincar\'e-like duality between their homology and cohomology. Our proofs work over every profinite coefficient ring, and for profinite as well as discrete coefficients. We do not only unify and generalise existing results, but also construct two novel examples of duality groups. We prove our results via the framework of condensed mathematics. This recently developed setting is a natural home for profinite objects, and our work is among the first to leverage this for the study of the (co)homology of profinite groups. Establishing our duality results involves an investigation of homological finiteness properties in condensed mathematics.

math.GR↗

arXiv2024

On a Completion of Cohomological Functors Generalising Tate Cohomology I

Tate cohomology has been generalised by several authors using different constructions that have applications in group theory, ring theory and homotopical algebra. Therefore, there is a need for a uniform account that explains why their underlying approaches all lead to the same conclusions. The key notion in such a uniform theory is a specific completion of cohomological functors that is constructed under mild assumptions. This completion takes Tate cohomology to settings where it has never been introduced such as in condensed mathematics. Through the latter, one can define Tate cohomology for any $T1$ topological group.

math.GR↗

arXiv2024

On a Completion of Cohomological Functors Generalising Tate Cohomology II

Viewing group cohomology as a cohomological functor, G. Mislin has generalised Tate cohomology from finite groups to all discrete groups by defining a completion for cohomological functors in 1994. In a previous paper, we have constructed for a cohomological functor $T^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ its Mislin completion $\widehat{T}^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ under mild assumptions on the abelian categories $\mathcal{C}$ and $\mathcal{D}$, which generalises Tate cohomology to all $T1$ topological groups. In this paper, we investigate the properties of Mislin completions. As their main feature, Mislin completions of Ext-functors detect finite projective dimension of objects in the domain category. We establish a version of dimension shifting, an Eckmann--Shapiro result as well as cohomology products such as external products, cup products and Yoneda products.

math.GR↗

arXiv2023

Loose ear decompositions and their applications to right-angled Artin groups

We characterize planar graphs and graph minors among other graph theoretic notions in terms of right-angled Artin groups (RAAGs). For this, we determine all sets of elements in RAAGs with ears as underlying graphs that are exactly the sets of vertex generators. Generalizing ear decompositions of graphs to loose ear decompositions, we characterize both decompositions in terms of RAAGs. The desired results follow as applications of loose ear decompositions of RAAGs.

math.GR↗