Kapranov–Voevodsky conjecture on higher Bruhat-to-Stasheff–Tamari maps

Kapranov and Voevodsky introduced the map in 1991 and conjectured that it is induced by an order congruence. The related claim that the map is surjective in all dimensions remains separately unresolved in the later literature.

Known results

• Reading and Speyer (2002) proved surjectivity for $d\le 2$ and found fibers of the original map without maximum elements.
• Williams (2023) proved that a related map $g\colon\mathcal{B}(n,\delta+1)\to\mathcal{S}(n,\delta)$ is surjective, full, and a quotient of posets.
• The 2021 equality of the two higher Stasheff–Tamari orders does not resolve this conjecture.

2020 obstruction and 2023 status

A 2020 paper argues that fibers of the original $f\colon\mathcal{B}(n,d)\to\mathcal{S}(n+2,d+1)$ are not always intervals; under the stated order-congruence notion, this would disprove the conjecture. The same work distinguishes this from a weaker quotient notion, while Williams (2023) still describes the associated all-dimensional surjectivity problem as open.

Current status (as of September 2026): The quotient-by-congruence conjecture is claimed false because some fibers are not intervals, but that obstruction is unverified; all-dimensional surjectivity remains open.