Vanishing conjecture for homology below virtual cohomological dimension

Let Γn,s\Gamma_{n,s} be the graph automorphism group with parameters nn and ss, and let 2n+s32n+s-3 be its virtual cohomological dimension. Below-vcd vanishing conjecture. All classes in Hk(Γn,s)H_k(\Gamma_{n,s}) for 0<k<2n+s30<k<2n+s-3 vanish after a single stabilization with respect to nn. This is presented as a consequence of the assembly-generation and stabilization-vanishing conjectures, so it remains open to the extent that those conjectures are unresolved.

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Primary source

James Conant, Allen Hatcher, Martin Kassabov and Karen Vogtmann, “Assembling homology classes in automorphism groups of free groups”, arXiv:1501.02351 (2017).

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