Kirby Problem 4.106

Definitions and mathematical statement

For a smooth four-manifold $X$, let $\pi:T^*X\to X$ be its cotangent bundle. Its canonical one-form is defined at a covector $p\in T_x^*X$ by \[ (\lambda_X)_p(v)=p(d\pi_p(v)). \] The two-form $\omega_X=d\lambda_X$ is symplectic. An exact symplectomorphism from $T^*X$ to $T^*Y$ is a diffeomorphism $\Phi$ satisfying \[ \Phi^*\lambda_Y-\lambda_X=dF \] for a globally defined smooth function $F:T^*X\to\mathbb R$. In particular, it preserves the canonical symplectic forms. No compact-support requirement is imposed on $F$ or $\Phi$.

How much of the smooth type of a four-manifold does the exact symplectomorphism type of its cotangent bundle determine? In particular, do there exist homeomorphic but nondiffeomorphic smooth four-manifolds $X,Y$ whose canonical cotangent bundles are not exactly symplectomorphic? Or do homeomorphic bases always give the same exact symplectomorphism type? Ask also whether equality of the simple homotopy types of the bases suffices.

Simple homotopy equivalence means homotopy equivalence generated by elementary expansions and collapses of finite CW models; closed bases provide the principal setting for this version. If open bases are considered, specify the applicable simple-homotopy convention.

The question allows symplectomorphisms moving the zero section; requiring them to preserve it would already impose a diffeomorphism of the bases. The canonical exact form, rather than an arbitrary symplectic structure on the same total space, is the object being compared.

Short English statement

Can canonical exact cotangent geometry distinguish exotic smooth structures on four-manifolds?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2982 (KP-4.106), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.106. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.