Dual Simplicial Complexes – Applied Topology

In the process of designing homework problems for Applied Algebraic Topology (ESE 680-003) last night, I stumbled upon a most beautiful application of the nerve theorem as well as a construction of a dual simplicial complex that is defined for any (locally finite) simplicial complex $K$ . This dual complex has the property that it is always homotopic to the original simplicial complex.

Let $K$ be a simplicial complex with vertex set $V$ . The open star $U_v$ of a vertex $v$ is defined to be the set of simplices $\sigma$ containing $v$ . Visibly, the nerve of the cover $\mathcal{U}=\{U_v | v\in V\}$ is the same as the simplicial complex $K$ .

On the other hand, one can consider a different cover by closed sets, or dually Alexandrov opens when one reverses the partial order. Define a simplex to be maximal if it is not the face of any other simplex. Define the closure $\overline{\sigma}$ of a simplex $\sigma$ to be the set of faces of $\sigma$ , written $\tau\leq\sigma$ , so that $\sigma\leq\sigma$ , i.e. $\sigma$ is a face of itself.

Now define the dual simplicial complex of $K$ to be the nerve of the cover $\mathcal{V}=\{\overline{\sigma} | \sigma \,\mathrm{maximal}\,\}$ .

The nerve theorem works for convex closed sets or open sets with contractible intersections, so we know this dual simplicial complex has to be homotopy equivalent to the simplicial complex $K$ .

If we use Graeme Segal’s construction of the classifying space of the cover category $X_U$ (sometimes called the Mayer-Vietoris blowup complex) in the paper “Classifying Spaces and Spectral Sequences” then we should be able to construct a similar dual space for cell complexes by looking at the open stars of vertices and the closures of maximal cells respectively.