arXiv subjects

Kevin I. Piterman

Publications and source records attributed to Kevin I. Piterman.

4 recordsLinked to original sources

arXiv2024

The common basis complex and the partial decomposition poset

For a finite-dimensional vector space $V$, the common basis complex of $V$ is the simplicial complex whose vertices are the proper non-zero subspaces of $V$, and $\sigma$ is a simplex if and only if there exists a basis $B$ of $V$ that contains a basis of $S$ for all $S\in \sigma$. This complex was introduced by Rognes in 1992 in connection with stable buildings. In this article, we prove that the common basis complex is homotopy equivalent to the proper part of the poset of partial direct sum decompositions of $V$. Moreover, we establish this result in a more general combinatorial context, including the case of free groups, matroids, vector spaces with non-degenerate sesquilinear forms, and free modules over commutative Hermite rings, such as local rings or Dedekind domains.

math.CO↗

arXiv2022

Some results on Quillen's Conjecture via equivalent-poset techniques

We extend the Main Theorem of Aschbacher and Smith on Quillen's Conjecture from $p>5$ to the remaining odd primes $p = 3,5$. In the process, we develop further combinatorial and homotopical methods for studying the poset of nontrivial elementary abelian $p$-subgroups of a finite group. The techniques lead to a number of further results on the Conjecture, often reducing dependence on the CFSG; in particular, we also provide some partial results toward the case of $p=2$.

math.GR↗

arXiv2021

Eliminating components in Quillen's Conjecture

We generalize an earlier result of Segev, which shows that {\em some\/} component in a minimal counterexample to Quillen's conjecture must admit an outer automorphism. We show in fact that {\em every\/} component must admit an outer automorphism. Thus we transform his restriction-result on components to an elimination-result: namely one which excludes any component which does not admit an outer automorphism. Indeed we show that the outer automorphisms admitted must include $p$-outers: that is, outer automorphisms of order divisible by $p$. This gives stronger, concrete eliminations: for example if $p$ is odd, it eliminates sporadic and alternating components -- thus reducing to Lie-type components (and typically forcing $p$-outers of field type). For $p = 2$, we obtain similar but less restrictive results. We also provide some tools to help eliminate suitable components that do admit $p$-outers in a minimal counterexample.

math.GR↗

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↗