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Contractibility of cores (relative Artin complexes for almost spherical subsets)

Establish that for any Artin group A_S and any almost spherical subset S'⊂S whose Coxeter diagrams for S' and S are connected and have no ∞-labeled edges, the relative Artin complex Δ_{S,S'} is contractible.

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Background

The paper introduces a program that reduces the K(pi,1)-conjecture for Artin groups to proving contractibility of certain "core" relative Artin complexes Δ_{S,S'}. Here, S' is almost spherical (i.e., removing any single node from S' yields a spherical Artin subgroup) and serves as a terminal stage in a sequence of deformation retractions.

If this conjecture holds, then the K(pi,1)-conjecture follows for all Artin groups (cited as [Huang 2023, Corollary 7.4]). Thus, the contractibility of such cores is a central unresolved step in the proposed approach.

References

We conjecture that the cores are contractible. Conjecture [] Suppose $S'\subset S$, and $S'$ is almost spherical. Suppose the Coxeter diagrams for $S'$ and $S$ are connected and do not have $\infty$-labeled edges. Then $\Delta_{S,S'}$ is contractible.

— Cycles in spherical Deligne complexes and application to $K(π,1)$-conjecture for Artin groups (2405.12068 - Huang, 20 May 2024) in Section 2.3 (Contractibility of core), Conjecture 2.1