Strongly isospectral manifolds with nonisomorphic cohomology rings

TL;DR

The paper constructs examples of compact flat manifolds that are strongly isospectral yet have nonisomorphic cohomology rings, revealing limitations of spectral data in determining topological invariants.

Contribution

It provides explicit constructions of strongly isospectral manifolds with different cohomology rings, including large families with diverse topological properties.

Findings

Existence of strongly isospectral manifolds with nonisomorphic cohomology rings.

Construction of large Sunada isospectral families with varying topological features.

Identification of manifolds with different primitive form counts.

Abstract

For any $n\geq 7$, $k\geq 3$, we give pairs of compact flat $n$-manifolds $M, M'$ with holonomy groups $\mathbb Z_2^k$, that are strongly isospectral, hence isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology rings. Moreover, if $n$ is even, $M$ is K\"ahler while $M'$ is not. Furthermore, with the help of a computer program we show the existence of large Sunada isospectral families; for instance, for $n=24$ and $k=3$ there is a family of eight compact flat manifolds (four of them K\"ahler) having very different cohomology rings. In particular, the cardinalities of the sets of primitive forms are different for all manifolds.