Strongly isospectral hyperbolic 3-manifolds with nonisomorphic rational cohomology rings

TL;DR

This paper demonstrates that strongly isospectral hyperbolic 3-manifolds can have different rational cohomology rings, showing that spectral data does not determine the cohomology ring structure, and introduces a computational method for analyzing cohomology.

Contribution

It provides the first known examples of strongly isospectral hyperbolic 3-manifolds with nonisomorphic rational cohomology rings and develops a computer program to analyze cohomology ring structures.

Findings

Existence of strongly isospectral hyperbolic 3-manifolds with different cohomology rings

Development of a computational tool for cohomology analysis

Spectral data does not determine the rational cohomology ring

Abstract

This paper shows that one cannot "hear" the rational cohomology ring of a hyperbolic 3-manifold. More precisely, while it is well-known that strongly isospectral manifolds have the same cohomology as vector spaces, we give an example of compact hyperbolic 3-manifolds that are strongly isospectral but have nonisomorphic rational cohomology rings. Along the way we implement a computer program which finds the nullity of the cup product map $H^1(M;\mathbb{Q})\wedge H^1(M;\mathbb{Q})\rightarrow H^2(M;\mathbb{Q})$ for any aspherical space in terms of the presentation of the fundamental group.