On flag spheres with few equators

In this note we construct a flag simplicial $3$-sphere $Δ$ with the following properties: - $Δ$ is not a suspension; - $Δ$ has no edge that can be contracted to obtain another flag sphere; - The only equators (induced subcomplexes which are spheres of codimension $1$) of $Δ$ are vertex links. Our construction has $12$ vertices, the minimum number of vertices such a simplicial complex can have. This answers a question posed by Chudnovsky and Nevo.

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