I was reading Prasolov and Sossinsky's book Knots, Links, Braids and Three-Manifolds and came across the following statement in the proof of Theorem 9.2:
We can assume $L^3$ has a triangulation $K$ such that $T$ is a subcomplex of $K$.
The context is as follows: Let $M^3$ be a compact, oriented manifold with connected boundary $\partial M^3$, and $N^3$ be a handlebody with boundary homeomorphic to $\partial M^3$ and spine $T$ (i.e. a 1-dimensional simplicial subcomplex of $N^3$ such that a regular neighborhood of $T$ is homeomorphic to $N^3$). Attach $N^3$ to $M^3$ along an arbitrary homeomorphism of their boundary to yield $L^3$.
It's not clear to me why we can make the assumption above: is it to do with the way $\partial N^3$ meets $\partial M^3$? Can the same assumption be made for any three-manifold $M$? That is, can we choose a graph $T$ in $M$ and find a triangulation of $M$ that contains $T$ as a subcomplex?
Any advice is much appreciated!
This claim is actually nontrivial and fails for general graphs embedded in 3-manifolds (even for topological circles in $S^3$; there are "wild knots" which cannot be realized as subcomplexes of any triangulation).
In order to prove the claim, note that $N^3$ can be triangulated so that $T$ is a subcomplex. Next, $M$ also admits a triangulation (this is quite nontrivial), so that $\partial M$ is a subcomplex. Moreover, if $M_1, M_2$ are homeomorphic triangulated 3-manifolds, then, after a suitable subdivision of triangulations, the underlying simplicial complexes are isomorphic. Lastly, suppose that $S_1, S_2$ are triangulated surfaces and $f: S_1\to S_2$ is a homeomorphism. Then $f$ is isotopic to a homeomorphism which is piecewise-linear with respect to these triangulations. This is also somewhat nontrivial. Lastly, changing the map $f: \partial N\to \partial M$ by isotopy does not change the homeomorphism type of the manifold $N\cup_f M$.
Moise, Edwin E., Geometric topology in dimensions 2 and 3, Graduate Texts in Mathematics. 47. New York - Heidelberg - Berlin: Springer-Verlag. X, 262 p. DM 45.00; $ 19.80 (1977). ZBL0349.57001.