# The rigidity of the graphs of homology spheres minus one edge

## TL;DR

This paper proves that removing any edge from the graph of a prime homology sphere of dimension at least 2 results in a graph that is generically rigid in the corresponding dimension, confirming a previous conjecture.

## Contribution

It establishes the generic rigidity of graphs derived from prime homology spheres minus one edge, confirming a conjecture by Nevo and Novinsky.

## Key findings

- Graphs of prime homology spheres minus one edge are generically rigid in the corresponding dimension.
- Confirms the conjecture of Nevo and Novinsky for all prime homology spheres of dimension at least 2.
- Provides a new understanding of the rigidity properties of homology sphere graphs.

## Abstract

We prove that for any prime homology $(d-1)$-sphere $\Delta$ of dimension $d-1\geq 3$ and any edge $e\in S$, the graph $G(\Delta)-e$ is generically $d$-rigid. This confirms a conjecture of Nevo and Novinsky.