On the classification of CW complexes with prescribed links
Sylvain Barré, Mikaël Pichot
TL;DR
This work presents a computer-assisted search aimed at classifying 2D CW complexes with prescribed links, motivated by intermediate-rank geometry and the flat closing conjecture. It encodes links as labeled graphs and uses the notion of wired faces to recursively assemble partial complexes, followed by isomorphism checks via graph automorphisms. Key results include the 27 Moebius--Kantor complexes on a single vertex and a concrete $\tilde{B}_2$-type example with 7 vertices and 45 faces, along with a notably large Moebius--Kantor complex (24 vertices, 192 faces). The authors explore implications for constructing intermediate-rank groups and introduce mesoscopic rank considerations, but ultimately abandoned the approach in favor of more abstract methods, while highlighting ideas like random chamber removal from buildings as potential directions.
Abstract
Reporting on a computer--assisted search for nonpositively curved CW complexes of intermediate rank conducted some years ago. Not intended for publication.