# Simplicial complex construction from given Betti numbers?

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.

• A complete characterization of pairs $(f,\beta)$, where $f$ is the $f$-vector and $\beta$ the Betti sequence of a simplicial complex, is due to Björner and Kalai, projecteuclid.org/download/pdf_1/euclid.acta/1485890544. You can see from this which ones with fixed $\beta$ have a minimal $f$-vector. – Richard Stanley Dec 11 '17 at 13:40

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.