A simplicial complex is said to be critical (or forbidden) for the 3-sphere $S^3$ if it cannot be embedded in $S^3$, but becomes embeddable upon removing the open star of any simplex in its second barycentric subdivision.

We classify all critical complexes for $S^3$ that decompose as $(G\times S^1)\cup H$, where $G$ and $H$ are graphs whose intersection $G\cap H$ consists solely of vertices of $H$.

This is a joint work with Mario Eudave-Munoz.

A preprint is available at arXiv:2403.18279.