Merkulov's homology isomorphism conjecture for the truncated graph complex

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Let Gn\mathrm{G}_n be the Kontsevich graph complex, with differential dd, and let GnMe={Γ∈Gn∣π5Γ=π5dΓ=0}\mathrm{G}^{\mathrm{Me}}_n=\{\Gamma\in\mathrm{G}_n\mid \pi_5\Gamma=\pi_5d\Gamma=0\} be Merkulov's truncated graph complex. Merkulov's conjecture. The inclusion of dg vector spaces

GnMe⊂Gn\mathrm{G}^{\mathrm{Me}}_n\subset\mathrm{G}_n

induces a homology isomorphism

Hk(GnMe)≅Hk(Gn)H_k(\mathrm{G}^{\mathrm{Me}}_n)\cong H_k(\mathrm{G}_n)

for all nn and kk. This asserts that the truncated subcomplex captures the full homology of the graph complex; the supplied text gives no evidence of a resolution.

References

Primary source

Simon Brun and Thomas Willwacher, “Graph homology computations”, arXiv:2307.12668 (2023).

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