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 . This dual complex has the property that it is always homotopic to the original simplicial complex.
Let be a simplicial complex with vertex set . The open star of a vertex is defined to be the set of simplices containing . Visibly, the nerve of the cover is the same as the simplicial complex .
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 of a simplex to be the set of faces of , written , so that , i.e. is a face of itself.
Now define the dual simplicial complex of to be the nerve of the cover .
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 .
If we use Graeme Segal’s construction of the classifying space of the cover category (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.