Source-boundary correction and a twisted-cohomology calculation

Definitions and mathematical statement

The aid's source-boundary correction says numbered Question 1.35 consists only of "Eliashberg made some remarks on the above topics." It contains no unresolved mathematical question. The following paragraphs are Banyaga's unnumbered exposition on locally conformal symplectic geometry; Questions 1.36 and 1.37 are separate records. The registry's question/open metadata is retained without inventing an open conjecture here.

In the source's sign convention, an l.c.s. structure is a nondegenerate two-form $\Omega$ with a closed one-form $\eta$ satisfying \[ {\,\mathrm{d}}\Omega=-\eta\wedge\Omega. \] For closed $\eta$, define ${\,\mathrm{d}}_\eta\beta={\,\mathrm{d}}\beta+\eta\wedge\beta$, so ${\,\mathrm{d}}_\eta^2=0$, ${\,\mathrm{d}}_\eta\Omega=0$, and \[ H_\eta^k(X)=H^k(\Omega^\bullet(X),{\,\mathrm{d}}_\eta) \] is the twisted/Lichnerowicz cohomology in this convention. Renaming the Lee form avoids the source's reuse of $\omega$ for both a one-form and a symplectic form.

For a contact manifold $(N,\alpha)$, on $N\times S^1$ let $\vartheta$ be the pulled-back contact form and $\eta$ a pulled-back nowhere-zero volume one-form on the circle. Then $\Omega={\,\mathrm{d}}\vartheta+\eta\wedge\vartheta={\,\mathrm{d}}_\eta\vartheta$ is the source's l.c.s. example. Another is $\Omega=f\sigma$, where $\sigma$ is symplectic and $f>0$, with $\eta=-{\,\mathrm{d}}\log f$; this is globally conformal symplectic. Conversely, an exact Lee form in this sign convention can be removed by a positive global conformal factor.

The aid develops a strict-contact mapping-torus extension. Let $N$ be a compact contact manifold without boundary, $\phi:N\to N$ a strict contactomorphism $\phi^*\alpha=\alpha$, and $a\in{\mathbb{R}}\setminus\{0\}$. Define \[ X_\phi=(N\times[0,1])/((x,1)\sim(\phi(x),0)), \qquad \eta=a\,{\,\mathrm{d}} t. \] The form $\vartheta$ induced by $\alpha$ descends, and \[ \Omega={\,\mathrm{d}}_\eta\vartheta={\,\mathrm{d}}\vartheta+a{\,\mathrm{d}} t\wedge\vartheta \] is nondegenerate and l.c.s.; its Lee period on the base circle is $a$.

Over ${\mathbb{R}}$, put $A_j=e^a\phi^*-I$ on $H^j(N;{\mathbb{R}})$. The aid reports the Wang short exact sequence \[ 0\longrightarrow\operatorname{coker}A_{k-1} \longrightarrow H_\eta^k(X_\phi) \longrightarrow\ker A_k\longrightarrow0. \] No canonical splitting is asserted. If \[ m_j(e^{-a})=\dim_{\mathbb{R}}\ker(\phi^*-e^{-a}I:H^j(N;{\mathbb{R}})\to H^j(N;{\mathbb{R}})), \] with cohomology outside the degree range zero, then \[ \dim H_\eta^k(X_\phi)=m_k(e^{-a})+m_{k-1}(e^{-a}). \] These are geometric eigenspace multiplicities, not algebraic multiplicities of Jordan blocks. All twisted groups vanish exactly when $e^{-a}$ is absent from the cohomological monodromy spectrum in every degree. In the product case $\phi=\operatorname{id}$ and $a\ne0$, the criterion gives vanishing. This is the aid's developed contextual result, not a solution of a missing numbered question.

Short English statement

The numbered item is only a historical remark; the l.c.s. definitions belong to following exposition. The aid extends the contact-product example to strict-contact mapping tori and computes twisted cohomology from monodromy eigenspaces. Its sign and Lee-period conventions are explicit, and geometric multiplicity is kept distinct from algebraic multiplicity.

Sources

• Local UnsolvedMath registry, permanent ID 20002093, exact source code catalogue entry; canonical aim-geometry-notes.json, notes[430], tag question; AIM Workshop Problem Lists, Geometry, Holomorphic curves in contact geometry. Created and updated 14 August 2026.
• Source URL https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. The supplied verification and aid distinguish the Eliashberg remark numbered 1.35 from Banyaga's following exposition and the separately numbered 1.36--1.37. The official PDF/HTML layout is reported as checked in the supplied evidence; neither was accessed here.
• Lichnerowicz cohomology is named in the source. No separate authoritative citation is supplied for the mapping-torus Wang calculation or its spectral criterion.
• Attempt 1: context_only, problem status at run not_a_problem, rigor label proved, novelty confidence low; classification OPEN-TRIAGE, suggested difficulty L2, registry status open. Candidate: strict-contact mapping-torus spectral criterion. These labels are retained without treating context as an independent conjecture.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.