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Suppose I have a simplicial complex $K$ constructed by taking two simplicial complexes $K_1$ and $K_2,$ and coning off ever vertex of $K_1$ to all of $K_2$ and vice versa (so, a direct generalization of a complete bipartite graph and also of a suspension). The questions are:

  1. What is this called?
  2. What is the homology of this object?
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    $\begingroup$ Is it the join? $\endgroup$
    – Mark Grant
    Commented Aug 17, 2019 at 17:54
  • $\begingroup$ @MarkGrant I looked in three different places, and got three different definitions :) So, I am not sure. $\endgroup$
    – Igor Rivin
    Commented Aug 17, 2019 at 20:09
  • $\begingroup$ I had in mind the definition given eg here: encyclopediaofmath.org/index.php/Join. But what you have seems to be a subcomplex of it. $\endgroup$
    – Mark Grant
    Commented Aug 17, 2019 at 20:27
  • $\begingroup$ Yes, exactly. Might be homotopic, I guess. $\endgroup$
    – Igor Rivin
    Commented Aug 17, 2019 at 20:32

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