Topological lower bounds on the sizes of simplicial complexes and simplicial sets

TL;DR

This paper establishes lower bounds on the number of top-dimensional faces in triangulations and simplicial set representations of certain n-dimensional spaces, like the n-torus, based on topological properties.

Contribution

It provides new topological lower bounds on the size of simplicial complexes representing specific spaces, linking topology to combinatorial complexity.

Findings

Any triangulation of the n-torus has at least 2^n n-dimensional faces.

Simplicial sets with contractible faces representing these spaces also have at least 2^n n-dimensional faces.

The bounds depend on the topological conditions satisfied by the space.

Abstract

We prove that if an $n$-dimensional space $X$ satisfies certain topological conditions then any triangulation of $X$ as well as any its representation as a simplicial set with contractible faces has at least $2^n$ faces of dimension $n$. One example of such $X$ is the $n$-dimensional torus $(S^1)^n$.