I have learnt the definitions of the star and link of simplicial complexes (see https://en.wikipedia.org/wiki/Simplicial_complex). However, I have not learnt their usefulness yet, beyond the definition.
May I ask what is the significance of them? Do they play an important role, and are there any theorems involving them?
Thanks.
One use is to tell if a simplicial complex is a manifold. If the open stars are homeomorphic to euclidean space, then it is a manifold. A non-example is that if you do the suspension of a torus, the link of an apex is a torus, so this suspension might not be a manifold. (Though in fact it isn't a manifold. This gives an example for why $3$-cycles in homology are not all just represented by $3$-manifolds.) Unfortunately, this test is not enough to tell whether a simplicial complex isn't a manifold, because there are simplicial complexes with non-spherial links which yet are manifolds.
The link is so named because for an algebraic surface in $\mathbb{C}^2$, the link is a link. As an example, if you take the solutions to $z^2=w^3$ in $\mathbb{C}^2$, which is a two-dimensional surface in four-dimensional space except for a singularity at $(0,0)$, you can examine the link at $(0,0)$. It is equivalent looking at how the solutions intersect a sphere centered at $(0,0)$, and you'll find the link is actually a trefoil knot in that sphere.