Nerve of cover of sphere

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• Triangulations

Problem. Find an open cover of the sphere S² the nerve of which is homeomorphic to S².

Solution. Two solutions.

One can recognize that the sphere is homeomorphic to the tetrahedron, as a 2-dimensional simplicial complex, and use the stars of its vertices to build an open cover on the sphere. Of course, the open sets aren't triangles here but their complements.

Alternatively, the six hemispheres can serve as the open sets. The resulting simplicial complex is an octahedron.

What happens if you have only four hemispheres?