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

TL;DR

This paper establishes a doubly exponential lower bound on the number of distinct homotopy types of simplicial complexes with n vertices, confirming a conjecture by Kalai.

Contribution

The authors prove that the number of homotopy types of simplicial complexes on n vertices grows at least doubly exponentially, resolving a longstanding conjecture.

Findings

h(n)  2^{0.02n} for large n

Number of homotopy types is doubly exponential in n

Confirms Kalai's conjecture on the growth rate

Abstract

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 $n$.