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I came across a theorem which states that a simplicial complex is a geometric realization of the nerve of the stars of its vertices. Here, we consider open stars (set of open simplices of which the vertex is a face). I feel that the collection of stars of all the vertices of the complex will give the complex itself, and the complex is then trivially the geometric realization of the nerve of itself. Could someone help me with visualizing the collection of stars of all the vertices of a simplicial complex? Thanks

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A face of the nerve of the covering of open stars is (by definition) precisely a non-empty finite set of vertices, whose open stars intersect non-trivially.

That is $(v_1v_2\cdots v_r)$ is a face of the nerve of the stars if and only if there is some point in the interior of a simplex, whose vertices include $v_1,v_2,\cdots,v_r$. This happens if and only if $(v_1v_2\cdots v_r)$ is a simplex in the original simplicial complex.

That is you end up with the same simplicial complex that you started with, so of course they have the same geometric realisation.

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  • $\begingroup$ Okay, so that reassures my visualization that the nerve of the vertex stars gives the same (abstract) simplicial complex.. right? $\endgroup$
    – psj
    Nov 11 '20 at 11:44
  • $\begingroup$ Yes absolutely. $\endgroup$
    – tkf
    Nov 11 '20 at 11:52
  • $\begingroup$ Okay, thanks a lot $\endgroup$
    – psj
    Nov 11 '20 at 12:21

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