A Kuratowski-Type Classification of Critical Complexes for the 3-Sphere

We give a Kuratowski-type classification of a graph-defined class of minimal piecewise-linear obstructions to embeddability in the 3-sphere. A finite simplicial complex $X$ is called critical for $S^3$ if $\mathrm{\mid }X\mathrm{\mid }$ does not embed in $S^3$, whereas deleting the open star of any simplex in the second barycentric subdivision of $X$ yields a polyhedron embeddable in $S^3$. The main theorem completely classifies critical complexes of the form $(G\times S^1)\cup H$, where $G$ and $H$ are graphs and $H$ is attached along vertices of the branch set of $G\times S^1$. We prove that there are exactly seven such complexes up to homeomorphism: two $K_4$-type complexes, four ${\mathrm{\Theta }}_4$-type complexes, and one $K_{2,3}$-type complex. The proof is combinatorial in nature. By collapsing the $S^1$-factor of $G\times S^1$, we associate to $X$ a reduction graph $\widehat{X}=G\cup H$. Criticality implies that $H$ is a forest, $G$ is planar, and $\widehat{X}$ is inclusion-minimal non-planar. Kuratowski’s theorem therefore reduces the classification to the cases $K_5$ and $K_{3,3}$. A finite analysis of forest attachments, together with a face-incidence criterion for embeddability, leaves precisely the seven models listed above. We also prove that every non-embeddable regular multibranched surface in $S^3$ contains a critical subcomplex of the form $M\cup H$, where $M$ is a regular multibranched surface and $H$ is a graph.