The coNP conjecture for three-sphere recognition

A triangulation $\mathcal{T}$ is a finite triangulation of a closed 3-manifold, and $S^3$ recognition asks whether $\mathcal{T}$ triangulates the 3-sphere. The complexity class $\operatorname{coNP}$ consists of decision problems whose complements lie in NP. The coNP conjecture for three-sphere recognition. $S^3$ recognition lies in $\operatorname{coNP}$. The preceding theorem establishes that $S^3$ recognition lies in NP, while the complementary coNP membership remains open in the source.