Ayako Carter | Searcharxiv

arXiv2026

From configuration spaces to graph complexes via FA-modules

Work of Gadish and Hainaut (after Petersen) models the compactly supported cohomology of a wedge of circles as a polynomial functor. We identify the coefficients of this functor, $\Phi[n,m]$, via a cobar construction of $\mathbf{FA}$-modules. This identification formally implies that these coefficients will arise in computations of graph homology, and we use this result to give examples of graph complexes whose homology may be embedded in $H_c^\ast(F(S^1\vee S^1,n))$. This includes the Payne-Willwacher marked graph complex in genus 2, for which we give a new, explicit decomposition in terms of simple $\mathbf{FA}$-modules. This allows us to describe $\mathsf{gr}_{11}H_c^{\ast}(\mathcal{M}_{2,n})$ as the cohomology of a complex of decorated trees and to show, for example, $\mathsf{gr}_{11}H_c^{n+1}(\mathcal{M}_{2,n})=0$.

math.AT↗

arXiv2025

An acyclic $d$-partition of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$

In this paper we introduce a $d$-partition $\mathcal{E}_d^{(r)}=(\Omega_1^{(r,d)}, \Omega_2^{(r,d)},\dots, \Omega_d^{(r,d)})$ of the $r$-uniform complete hypergraph $K_{rd}^{(r)}$. We prove that $\mathcal{E}_d^{(r)}$ is homogeneous and that each hypergraph $\Omega_i^{(r,d)}$ is acyclic (i.e. has zero Betti numbers). As an application, we show that the map $det^{S^r}$ is nontrivial for every $r$, which gives a partial answer to a conjecture from [14].

math.CO↗