A lower bound on the number of homotopy types of simplicial complexes on $n$ vertices

For $n \in \mathbb{ N}$, let $h(n)$ denote the number of simplicial complexes on $n$ vertices up to homotopy equivalence. Here we prove that $h(n) \geq 2^{ 2^{ 0.02n}}$ when $n$ is large enough. Together with the trivial upper bound of $2^{ 2^n}$ on the number of labeled simplicial complexes on $n$ vertices this proves a conjecture of Kalai that $h(n)$ is doubly exponential in $n$.

Categories: math.AT math.CO math.PR