Acyclicity conjecture for the Quilt operad

About 6 years old · traced to 2020

Let $\operatorname{\textit{Quilt}}$ be the operad of quilts, and let $H_k(\operatorname{Quilt})$ denote its homology in degree $k$ . Acyclicity conjecture for the Quilt operad. $\operatorname{\textit{Quilt}}$ is acyclic in the sense that

H_k(\operatorname{\textit{Quilt}})=0

for every $k>0$ . This conjecture concerns the homological structure of the operad governing operations on the Hochschild bicomplex of a diagram of algebras. The supplied text gives no evidence that the claim has been resolved.

References

Primary source

Eli Hawkins, “Operations on the Hochschild Bicomplex of a Diagram of Algebras”, arXiv:2002.00886 (2024).

Posted 1 month ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A 2025 preprint claims to prove the conjecture, but the supplied record contains no independent verification.

The conjecture asserts that the Quilt operad has no homology in positive degrees. It is attributed to Hawkins and concerns operations on the Hochschild bicomplex of a diagram of algebras.

March 13, 2025 preprint

Campos and Hermans state that the projection $\operatorname{Quilt}\to\operatorname{Brace}$ is a quasi-isomorphism, explicitly presenting this as a proof of Hawkins’s conjecture. Since $\operatorname{Brace}$ has no positive-degree homology, their result implies $H_k(\operatorname{Quilt})=0$ for every $k>0$ ; the supplied sources report no counterexample, withdrawal, or verification dispute.

Current status (as of September 2026): The conjecture has a claimed proof by Campos and Hermans, but that proof is not independently verified in the supplied record.

Sources

No solutions have been posted yet.