# Leray numbers of projections and a topological Helly type theorem

## TL;DR

This paper establishes a bound on the Leray number of projections of simplicial complexes under certain conditions, leading to a topological Helly type theorem extension.

## Contribution

It introduces a new inequality relating Leray numbers of complexes and their projections, extending Helly type theorems in topological combinatorics.

## Key findings

- L((X))  r L(X)+r-1 bound established
- Topological Helly type theorem extended using Leray numbers
- Provides a new tool for analyzing projections of simplicial complexes

## Abstract

Let X be a simplicial complex on the vertex set V. The rational Leray number L(X) of X is the minimal d such that the rational reduced homology of any induced subcomplex of X vanishes in dimensions d and above. Let \pi be a simplicial map from X to a simplex Y, such that the cardinality of the preimage of any point in |Y| is at most r. It is shown that L(\pi(X)) \leq r L(X)+r-1. One consequence is a topological extension of a Helly type result of Amenta.