  A simplicial complex is said to be {\em critical} (or {\em forbidden}) for
the 3-sphere $S^3$ if it cannot be embedded in $S^3$ but after removing any one
point, it can be embedded.
  We show that if a multibranched surface cannot be embedded in $S^3$, it
contains a critical complex which is a union of a multibranched surface and a
(possibly empty) graph. We exhibit all critical complexes for $S^3$ which are
contained in $K_5 \times S^1$ and $K_{3,3} \times S^1$ families. We also
classify all critical complexes for $S^3$ which can be decomposed into $G\times
S^1$ and $H$, where $G$ and $H$ are graphs.
  In spite of the above property, there exist complexes which cannot be
embedded in $S^3$, but they do not contain any critical complexes. From the
property of those examples, we define an equivalence relation on all simplicial
complexes $\mathcal{C}$ and a partially ordered set of complexes
$(\mathcal{C}/\mathord\sim; \subseteqq)$, and refine the definition of
critical. According to the refined definition of critical, we show that if a
complex $X$ cannot be embedded in $S^3$, then there exists $[X']\subseteqq [X]$
such that $[X']$ is critical for $[S^3]$.
