Cohomology Rings of Almost-Complex Manifolds

Definitions and mathematical statement

A closed smooth manifold is a compact smooth manifold without boundary. An almost-complex structure on such a manifold $M$ is a smooth real vector-bundle endomorphism $J:TM\to TM$ satisfying \[ J\circ J=-\operatorname{id}_{TM}. \] Its existence forces every component of $M$ to have even real dimension. Write \[ H^*(M;{\mathbb{Q}})=\bigoplus_{k\geq 0}H^k(M;{\mathbb{Q}}) \] for singular cohomology with rational coefficients, equipped with its cohomological grading, unit, and cup product.

Characterize the finite-dimensional, nonnegatively graded, graded-commutative unital ${\mathbb{Q}}$-algebras $A$ for which there exist a closed smooth manifold $M$, an almost-complex structure $J$ on $M$, and an isomorphism \[ A\cong H^*(M;{\mathbb{Q}}) \] of graded unital ${\mathbb{Q}}$-algebras. Equivalently, give necessary and sufficient conditions for such a realization.

The supplied question specifies neither a fixed dimension nor connectedness. No connectedness restriction is added here; whether the intended classification concerns only connected manifolds requires clarification. The source also does not specify whether dimension zero is to be included. Interpreting "cohomology rings" as graded unital ${\mathbb{Q}}$-algebras is the convention used in this draft.

The supplied background describes a connected-sum construction and an associated rational-cohomology observation as established, but gives neither their statements nor their hypotheses. They therefore cannot be incorporated as precise subsidiary assertions. The supplied label "partially solved" is a dated curatorial assessment, not an independently verified mathematical conclusion.

Short English statement

Determine exactly which graded rational cohomology rings occur for closed smooth manifolds admitting an almost-complex structure. The intended connectedness and dimension conventions require clarification. The supplied partial-solution assessment remains subject to independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004548$, title "Cohomology Rings of Almost-Complex Manifolds," in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. No individual proposer is supplied. The record lists proposed year $2020$.
• Supplied report citation: Mini-Workshop: Almost Complex Geometry (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/33, pp. 1657--1691. https://doi.org/10.4171/owr/2020/33. The report is supplied as documenting workshop discussions and open problems; the specific problem location and attribution have not been independently verified.
• Albanese--Milivojevic, 2019, as labeled in the supplied literature sources: https://arxiv.org/abs/1805.04751.
• Milivojevic, 2022, as labeled in the supplied literature sources: https://doi.org/10.1515/coma-2021-0133.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem "partially solved," describes the connected-sum construction and associated rational-cohomology observation as established, and describes the broader realization problem as only partially classified. The supplied background requests independent verification of time-sensitive status and cited claims.