arXiv subjects

Kevin Ivan Piterman

Publications and source records attributed to Kevin Ivan Piterman.

15 recordsLinked to original sources

arXiv2026

Components in characteristic $p$ and Quillen's conjecture

The purpose of this paper is to show that, under mild inductive assumptions, if a group $G$ contains a component that is a simple group of Lie type in characteristic $p$ and $O_p(G) = 1$, then the Quillen poset of $G$ at $p$ has nonzero rational homology. In particular, this shows that such components cannot arise in a minimal counterexample to Quillen's conjecture. This result is of particular interest at the prime $p=2$, where the conjecture is still open.

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arXiv2025

An analogue of Rognes' connectivity conjecture for free groups

We show that the common basis complex of a free group of rank $n$ has the homotopy type of a wedge of spheres of dimension $2n-3$. This establishes an $\mathrm{Aut}(F_n)$-analogue of the connectivity conjecture that Rognes originally stated for $\mathrm{GL}_n(R)$. To prove this, we provide several homotopy-equivalent models of the common basis complex, both in terms of free factors in free groups and in terms of sphere systems in 3-manifolds.

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arXiv2025

Posets of decompositions in spherical buildings

We propose definitions of the common bases complex, the poset of decompositions, and the poset of partial decompositions for arbitrary spherical buildings. We show that the poset of decompositions is Cohen-Macaulay, and that the poset of partial decompositions is spherical and homotopy equivalent to the common bases complex. To prove these results, we rely on the concepts of opposition, Levi spheres, and convexity in buildings. In particular, our results extend the already known constructions for the linear case (vector spaces) to arbitrary buildings. As a byproduct, we see that the poset of ordered partial decompositions carries the square of the Steinberg representation.

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arXiv2025

Notes on the topology of independence structures

Following Welsh, a pre-independence space (pi-space) is a set $M$ together with a non-empty collection $I(M)$ of subsets of $M$, called independent sets, which is closed under taking subsets, and finite independent sets satisfy the exchange property from matroid theory. We show that $I(M)$, viewed as a poset, is contractible if it is infinite-dimensional, and Cohen-Macaulay otherwise. Moreover, the proper part of the associated poset of flats is also contractible in the infinite-dimensional case, and Cohen-Macaulay otherwise. These results generalize those for independence complexes and geometric lattices of (finite) matroids.

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arXiv2025

On reducible Killing forms for groups of Lie type

Killing forms on finite groups arise as examples of braided Killing forms on braided Lie algebras. For a finite group $G$ and a $G$-stable subset $\mathcal{C}$, the Killing form associated with $\mathbb{C}[\mathcal{C}]$ is given by $K_{\mathcal{C}}(a,b) = |C_G(ab) \cap \mathcal{C}|$ for $a,b\in \mathcal{C}$. Motivated by Cartan's criterion for semisimplicity of Lie algebras, and previous work of L\'opez Pe\~na, Majid, and Rietsch, we study the non-degeneracy and irreducibility of $K_{\mathcal{C}}$ when $\mathcal{C}$ is a conjugacy class of involutions or unipotent elements in a finite simple group of Lie type and Lie rank one. Our approach suggests interesting connections with character theory, related counting formulas, and the study of commuting graphs.

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arXiv2024

Connectivity of partial basis complexes of freely decomposable groups

We show that the complex of partial bases of the free group of rank $n$, where vertices are seen up to conjugation, is Cohen--Macaulay of dimension $n-1$. This positively answers a conjecture raised by Day and Putman. We prove our results in the more general context of freely decomposable groups.

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arXiv2024

Posets arising from decompositions of objects in a monoidal category

Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite height, we define decompositions and partial decompositions of $X$ which are coherent with $\sqcup$, and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness. In well-studied scenarios, the poset of $\sqcup$-complemented subobjects specializes to the poset of free factors of a free group, the subspace poset of a vector space, the poset of non-degenerate subspaces of a vector space with a non-degenerate form, and the lattice of flats of a matroid. The decomposition and partial decomposition posets, the complex of frames and partial bases together with the ordered versions, either coincide with well-known structures, generalize them, or yield new interesting objects. In these particular cases, we provide new results along with open questions and conjectures.

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arXiv2023

Homotopy properties of the complex of frames of a unitary space

Let $V$ be a finite dimensional vector space equipped with a non-degenerate Hermitian form over a field $\mathbb{K}$. Let $\mathcal{G}(V)$ be the graph with vertex set the $1$-dimensional non-degenerate subspaces of $V$ and adjacency relation given by orthogonality. We give a complete description of when $\mathcal{G}(V)$ is connected in terms of the dimension of $V$ and the size of the ground field $\mathbb{K}$. Furthermore, we prove that if $\dim(V) > 4$ then the clique complex $\mathcal{F}(V)$ of $\mathcal{G}(V)$ is simply connected. For finite fields $\mathbb{K}$, we also compute the eigenvalues of the adjacency matrix of $\mathcal{G}(V)$. Then by Garland's method, we conclude that $\tilde{H}_m(\mathcal{F}(V);\mathbb{k}) = 0$ for all $0\leq m\leq \dim(V)-3$, where $\mathbb{k}$ is a field of characteristic $0$, provided that $\dim(V)^2 \leq |\mathbb{K}|$. Under these assumptions, we deduce that the barycentric subdivision of $\mathcal{F}(V)$ deformation retracts to the order complex of the certain rank selection of $\mathcal{F}(V)$ which is Cohen-Macaulay over $\mathbb{k}$. Finally, we apply our results to the Quillen poset of elementary abelian $p$-subgroups of a finite group and to the study of geometric properties of the poset of non-degenerate subspaces of $V$ and the poset of orthogonal decompositions of $V$.

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arXiv2023

On the frame complex of symplectic spaces

For a symplectic space $V$ of dimension $2n$ over $\mathbb{F}_{q}$, we compute the eigenvalues of its orthogonality graph. This is the simple graph with vertices the $2$-dimensional non-degenerate subspaces of $V$ and edges between orthogonal vertices. As a consequence of Garland's method, we obtain vanishing results on the homology groups of the frame complex of $V$, which is the clique complex of this graph. We conclude that if $n < q+3$ then the poset of frames of size $\neq 0,n-1$, which is homotopy equivalent to the frame complex, is Cohen-Macaulay over a field of characteristic $0$. However, we also show that this poset is not Cohen-Macaulay if the dimension is big enough.

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arXiv2023

Maximal subgroups of exceptional groups and Quillen's dimension

Given a finite group $G$ and a prime $p$, let $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. The group $G$ satisfies the Quillen dimension property at $p$ if $\mathcal{A}_p(G)$ has non-zero homology in the maximal possible degree, which is the $p$-rank of $G$ minus $1$. For example, D. Quillen showed that solvable groups with trivial $p$-core satisfy this property, and later, M. Aschbacher and S.D. Smith provided a list of all $p$-extensions of simple groups that may fail this property if $p$ is odd. In particular, a group $G$ with this property satisfies Quillen's conjecture: $G$ has trivial $p$-core and the poset $\mathcal{A}_(G)$ is not contractible. In this article, we focus on the prime $p = 2$ and prove that the $2$-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy the Quillen dimension property, with only finitely many exceptions. We achieve these conclusions by studying maximal subgroups and usually reducing the problem to the same question in small linear groups, where we establish this property via counting arguments. As a corollary, we reduce the list of possible components in a minimal counterexample to Quillen's conjecture at $p = 2$.

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arXiv2021

Group actions on contractible $2$-complexes II

In this second part we prove that, if $G$ is one of the groups $\mathrm{PSL}_2(q)$ with $q>5$ and $q\equiv 5\pmod {24}$ or $q\equiv 13 \pmod{24}$, then the fundamental group of every acyclic $2$-dimensional, fixed point free and finite $G$-complex admits a nontrivial representation in a unitary group $\mathrm{U}(m)$. This completes the proof of the following result: every action of a finite group on a finite and contractible $2$-complex has a fixed point.

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arXiv2020

An approach to Quillen's conjecture via centralizers of simple groups

We show that, for any given subgroup $H$ of a finite group $G$, the Quillen poset $\mathcal{A}_p(G)$ of nontrivial elementary abelian $p$-subgroups, is obtained from $\mathcal{A}_p(H)$ by attaching elements via their centralizers in $H$. We use this idea to study Quillen's conjecture, which asserts that if $\mathcal{A}_p(G)$ is contractible then $G$ has a nontrivial normal $p$-subgroup. We prove that the original conjecture is equivalent to the $\mathbb{Z}$-acyclic version of the conjecture (obtained by replacing contractible by $\mathbb{Z}$-acyclic). We also work with the $\mathbb{Q}$-acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of order divisible by $p$ and $p$-rank at least $2$. This allows to extend results of Aschbacher-Smith and to establish the strong conjecture for groups of $p$-rank at most $4$.

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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.

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arXiv2018

A stronger reformulation of Webb's conjecture in terms of finite topological spaces

We investigate a stronger formulation of Webb's conjecture on the contractibilty of the orbit space of the p-subgroup complexes in terms of finite topological spaces. The original conjecture, which was first proved by Symonds and, more recently, by Bux, Libman and Linckelmann, can be restated in terms of the topology of certain finite spaces. We propose a stronger conjecture, and prove various particular cases by combining fusion theory of finite groups and homotopy theory of finite spaces.

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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.

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