Strongly isospectral hyperbolic 3manifolds with nonisomorphic rational cohomology rings
Abstract
This paper shows that one cannot "hear" the rational cohomology ring of a hyperbolic 3manifold. More precisely, while it is wellknown that strongly isospectral manifolds have the same cohomology as vector spaces, we give an example of compact hyperbolic 3manifolds 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 DOI:
 10.48550/arXiv.2111.11454
 arXiv:
 arXiv:2111.11454
 Bibcode:
 2021arXiv211111454T
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Differential Geometry
 EPrint:
 11 pages