Isomorphism of cosymplectomorphism groups implies diffeomorphism of manifolds

TL;DR

This paper demonstrates that the group structure of cosymplectomorphisms uniquely determines the underlying manifold up to diffeomorphism, linking group isomorphisms to manifold topology.

Contribution

It establishes a new rigidity result showing that isomorphic cosymplectomorphism groups imply diffeomorphic manifolds, using the Reeb flow and monodromy invariance.

Findings

Group isomorphism preserves Reeb flow as the center of the group.

Isomorphism preserves conjugacy class of monodromy, ensuring bundle equivalence.

Underlying manifolds are diffeomorphic if cosymplectomorphism groups are isomorphic.

Abstract

We prove that if two closed, connected, regular cosymplectic manifolds have isomorphic groups of cosymplectomorphisms (as topological groups), then the underlying manifolds are diffeomorphic. The proof proceeds by characterizing the Reeb flow as the center of the group and descending the isomorphism to the symplectic base manifolds. We show that the isomorphism preserves the conjugacy class of the monodromy of the mapping torus, which ensures that the bundle structures, and thus the total spaces are equivalent.