Dual Simplicial Complexes

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.


One thought on “Dual Simplicial Complexes

  1. Pingback: Speeding up Homology via Duality? | Applied Algebraic Topology

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