What is the nerve of the cover of a simplicial complex by the open stars of its vertices?
Solution
Let \(K\) be a simplicial complex and, for each vertex \(v\), let \(\operatorname{st}^{\circ}(v)\) be its open star. The nerve of this cover has one vertex for each vertex of \(K\).
A finite collection of open stars \[ \operatorname{st}^{\circ}(v_0),\ldots, \operatorname{st}^{\circ}(v_q) \] has nonempty intersection exactly when the vertices \(v_0,\ldots,v_q\) span a simplex of \(K\). Indeed, if they span a simplex \(\sigma\), the barycenter of \(\sigma\) lies in every one of these open stars. Conversely, if a point lies in all the stars, it lies in the interior of some simplex \(\tau\) containing every \(v_i\), so the \(v_i\)’s span a face of \(\tau\).
Thus a set of vertices spans a simplex in the nerve if and only if it spans the corresponding simplex in \(K\). Therefore the identification on vertices extends to a canonical simplicial isomorphism \[ \boxed{N(\{\operatorname{st}^{\circ}(v)\})\cong K.} \]