A Lower Bound on the Number of Homotopy Types of Simplicial Complexes on N Vertices

Abstract

For n ∈ 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.