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In this paper, an edge contraction of a simplicial complex $\Gamma$ is defined as the operation of removing the neighborhood $N_e\Gamma$ of the edge $e=\{0,1\}$ and identifying $N_0\partial N_e\Gamma$ and $N_1\partial N_e\Gamma$. The edge contraction is called valid if the result is again a simplicial complex.

I am very new to this field and have trouble visualising things.

Question:
Is there a simple example of a geometric simplicial complex that is no longer geometric after performing a valid edge contraction?

Why I am asking:
My background is that I want to understand where, in the proof of "Kind-of Fary's theorem" in Adiprasito and Patakova, the interpretation of complexes as geometric ones fails when applied to PL embeddings into $\mathbb{R}^d$. See also this question.

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you can take a triangle, that is, three vertices connected by three edges. If you contract one of the edges, then you would obtain two vertices, connected by two edges, which is not allowed in simplicial complexes :)

Regards, Karim

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    $\begingroup$ Thanks for the answer. But that wouldn't be a valid contraction by definition, would it? I was looking for a geometric simplicial complex such that contracting an edge results in a simplicial complex that is no longer geometric, if something like that exists. $\endgroup$
    – Leo
    Commented 2 days ago

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