A subcomplex of a simplicial $4$-sphere becomes full after subdividing the ambient sphere

Let $L$ be a finite simplicial complex whose geometric realization is homeomorphic to $S^4$, and let a vertex injection $e$ identify a complex $K$ with a subcomplex of $L$: every face of $K$ has image a face of $L$. No fullness is assumed, so $L$ may well have faces all of whose vertices lie in the image of $K$ without those faces belonging to $K$.

The assertion is that $K$ can be made a full subcomplex without changing the ambient homeomorphism type or touching $K$ itself: there are a simplicial complex $L'$ with $|L'| \cong S^4$ and a vertex injection $e'$ of $K$ into $L'$ such that

The mechanism is relative stellar subdivision. Working through the finitely many faces $\tau$ of $L$ whose vertices all lie in the image of $K$ but which are not images of faces of $K$, in order of decreasing dimension, subdivide $L$ stellarly at $\tau$: this introduces one new vertex, the barycentre of $\tau$, and destroys $\tau$ while preserving every simplex that does not contain $\tau$. Since $K$ is closed under passing to subsets and $\tau \notin K$, no face of $K$ contains $\tau$, so all of $K$ survives untouched; and every newly created simplex uses the new vertex, so no new offending face on the old vertices appears. Subdivision does not change the realization up to homeomorphism, so the result is still a triangulation of $S^4$, and after all offending faces have been removed the copy of $K$ is full.

The statement is the combinatorial half of the construction of a full two-dimensional subcomplex of a triangulated four-sphere: it separates the purely piecewise-linear step from the geometry of embedding a three-manifold in $S^4$.

Formalization Note Being a simplicial four-sphere means the barycentric-coordinate realization is homeomorphic to the unit sphere in $\mathbb R^5$; the subcomplex hypothesis and the fullness conclusion are stated through the induced maps on finite vertex sets, and the ambient vertex set is allowed to grow.