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Say an abstract simplicial complex $X$ is acyclic if its reduced integral simplicial homology groups $\tilde{\mathrm{H}}^{\Delta}_p(X)$ vanish for all $p\geq 0$. Is it the case that, for any $n>0$, any $n$-dimensional acyclic simplicial complex $X$ on a set $V$ may be extended to an $n$-dimensional acyclic simplicial complex $Y$ on $V$ with a complete $(n-1)$-skeleton, i.e., satisfying $Y^{n-1}=[V]^n$? Here the last expression denotes the collection of size-$n$ subsets of $V$, the vertex-set of $X$.

I suspect this to be true, but I'm finding it tricky to rigorously argue so I'm partly wondering if I'm missing anything and/or if there are any theorems about that might ease the job. Ultimately I'm interested in extensions $Y$ which conserve further properties of $X$, but this question seems to me the place to begin.

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    $\begingroup$ I don't understand what role $X$ plays here? If $Y^{n-1}=[V]^n$ and $m<n$, then $Y$ contains every $m$-dimensional simplicial complex $X$ on $V$. $\endgroup$
    – Jeremy Rickard
    Commented Jul 3, 2020 at 9:17
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    $\begingroup$ @JeremyRickard, fair point, when dim$(X)$<dim$(Y)$ the question's not very interesting. But (as the edit now emphasizes) when dim$(X)$=dim$(Y)$? $\endgroup$
    – bergfalk
    Commented Jul 3, 2020 at 18:29
  • $\begingroup$ Maybe I am getting something wrong, so lets see: suppose we have a simplicial complex consisting of a single solid triangle with an edge attached to some vertex. This 2-dimensional complex has four vertices and is (as far as I understand) acyclic. What would be its acyclic extension? $\endgroup$
    – M. Winter
    Commented Jul 3, 2020 at 21:30
  • $\begingroup$ @M.Winter It seems to me that you would have 3 choices of 2-dimensional acyclic extensions, each taking the form of all of the edges and 3 of the faces of a tetrahedron. $\endgroup$
    – bergfalk
    Commented Jul 3, 2020 at 22:02
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    $\begingroup$ @bergfalk Oh sorry, I missed that. I understand the setting now. Interesting question. $\endgroup$
    – M. Winter
    Commented Jul 3, 2020 at 22:46

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