Is it possible given a set of Betti numbers to construct a (possibly set of) simplicial complex with the given Betti-described topology? I understand there can be an infinity of simplicial complexes with the same Betti numbers but does a "minimal" simplicial complex construction algorithm exist?
Such an algorithm will greatly help reduce the solution space of constructing simplicial complexes of specified topology.
One way to make things "minimal" (given the lack of any further information) is to construct a simplicial complex whose cup products are all trivial, so the (co)homology generators don't interact with each other in exciting ways. That is to say, just build a simplicial wedge of spheres. All you need to do is have the ability to construct hollow $n$-simplices for all $n>1$ that is smaller than the top non-trivial betti dimension.
More explicitly: if the betti numbers are $(b_0, b_1, \cdots, b_k)$, introduce $b_0$ vertices. Note that $b_0 \geq 1$ if your original simplicial complex is non-empty, so you have at least one vertex, let's call it $a$. Now for each non-zero $b_i$ for $i > 1$, throw in $b_i$ hollow $(i+1)$-simplices which have $a$ as one of their vertices and no other vertices in common.
