Forbidden complexes for the 3-sphere

A simplicial complex is said to be { critical} (or { forbidden}) for the 3-sphere S³ if it cannot be embedded in S³ but after removing any one point, it can be embedded. We show that if a multibranched surface cannot be embedded in S³, it contains a critical complex which is a union of a multibranched surface and a (possibly empty) graph. We exhibit all critical complexes for S³ which are contained in K₅ × S¹ and K3,3 × S¹ families. We also classify all critical complexes for S³ which can be decomposed into G× S¹ and H, where G and H are graphs. In spite of the above property, there exist complexes which cannot be embedded in S³, but they do not contain any critical complexes. From the property of those examples, we define an equivalence relation on all simplicial complexes C and a partially ordered set of complexes (C/~; ), and refine the definition of critical. According to the refined definition of critical, we show that if a complex X cannot be embedded in S³, then there exists [X'] [X] such that $[X']$ is critical for [S³].