$\textbf{Problem:}$ Given a finite topological space $X$ with $|X|=n$ and $k\in \mathbb{N}_0$, decide if $H_k(X,\mathbb{F})=0$.
Where $\mathbb{F}$ denotes some field and one can also assume that $X$ is $T_0$ and reduced (a minimal finite $T_0$ topological space). In this case, the input finite topological space can be represented by a poset. I am interested if this decision problem is NP-hard. The problem can be viewed as a special case of to the homology problem for clique complexes of graphs which is known to be NP-hard in general. I am thinking about this problem for a while now but i got stuck. I would be very thankful for any suggestions.
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