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From configuration spaces to graph complexes via FA-modules
Abstract
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, Φ[n,m], via a cobar construction of 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 Hc∗(F(S1∨ S1,n)). This includes the Payne-Willwacher marked graph complex in genus 2, for which we give a new, explicit decomposition in terms of FA-modules. This allows us to describe gr11Hc∗(ℳ2,n) as the cohomology of a complex of decorated trees and to show, for example, gr11Hcn+1(ℳ2,n)=0.
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BibTeX
@misc{carter2026from,
title = {{From configuration spaces to graph complexes via FA-modules}},
author = {Ayako Carter and Benjamin C. Ward},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.04033},
eprint = {2609.04033},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04033},
}
RIS
TY - UNPB
TI - From configuration spaces to graph complexes via FA-modules
AU - Ayako Carter
AU - Benjamin C. Ward
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.04033
UR - https://arxiv.org/abs/2609.04033
AB - 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, Φ[n,m], via a cobar construction of 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^∗(F(S^1∨ 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 FA-modules. This allows us to describe gr_11H_c^∗(ℳ_2,n) as the cohomology of a complex of decorated trees and to show, for example, gr_11H_c^n+1(ℳ_2,n)=0.
ER -