I'll go ahead and plug a paper of mine in the hopes it will generate more interest in it, [Boolean formulae, hypergraphs, and combinatorial topology][1] which I wrote with my student Oliver Thistlethwaite many years ago. If you look at the set of boolean formulae in $n$ variables in $k$-conjunctive normal form, denoted $k\operatorname{SAT}n$, this can be turned into a simplicial complex by using implication as an ordering relation. We proved that this simplicial complex is homotopy equivalent to the Alexander dual of the independence complex of $\operatorname{Cube}(n,n-k)$, the hypergraph of $n-k$-faces of an $n$ dimensional cube: 

$$|k\operatorname{SAT}n|\simeq \Theta(\operatorname{Cube}(n,n-k))$$

where $\Theta$ is defined to be the simplicial complex where simplices are spanned by vertices of the hypergraph which are in the complement of at least one hyperedge. (This is equivalent to the Alexander dual of the independence complex, if you know what those are.)

These simplicial complexes seem rather complicated. We discovered that they are not just wedges of same dimensional spheres but in general have homology in multiple dimensions. (Though in the codimension $2$ case, they do seem to be wedges of spheres, see below.) I would love to understand these spaces better. For example, the following is still an open question:

**Conjecture**: $|\Theta(\operatorname{Cube}(n,n-2)|\simeq \vee_{(2n-3)!!}S^{2n-2}$.

  [1]: https://arxiv.org/pdf/0808.0739.pdf