# Asymptotic topology of random subcomplexes in a finite simplicial complex

## TL;DR

This paper investigates the asymptotic behavior of the topology of random subcomplexes in barycentrically subdivided finite simplicial complexes, providing bounds on Betti numbers, Morse inequalities, and Euler characteristic as subdivisions grow large.

## Contribution

It introduces new asymptotic bounds and formulas for the expected topological invariants of random subcomplexes in iteratively subdivided complexes.

## Key findings

- Derived asymptotic upper and lower bounds for expected Betti numbers.
- Established average Morse inequalities for the random subcomplexes.
- Calculated expected Euler characteristic in the asymptotic regime.

## Abstract

We consider a finite simplicial complex $K$ together with its successive barycentric subdivisions $Sd^d(K), d\geq0,$ and study the expected topology of a random subcomplex in $Sd^d(K), d\gg0$. We get asymptotic upper and lower bounds for the expected Betti numbers of those subcomplexes, together with the average Morse inequalities and expected Euler characteristic.