# Betti numbers of complexes with highly Connected links

## TL;DR

This paper establishes an upper bound on the (k-1)-th Betti number of a simplicial complex with highly connected links, demonstrating the bound's sharpness through examples based on sum complexes.

## Contribution

It provides a new upper bound on Betti numbers for complexes with highly connected links and shows this bound is asymptotically sharp for fixed parameters.

## Key findings

- Upper bound on the (k-1)-th Betti number derived
- Examples show the bound is asymptotically sharp
- Results apply to complexes with vanishing homology in links

## Abstract

Let X be a k-dimensional simplicial complex such that the (k-j-2)-dimensional homology of the links of all j-dimensional simplices in X vanishes. An upper bound is given on the (k-1)-th Betti number of X. Examples based on sum complexes show that this bound is asymptotically sharp for all fixed j<k.