Betti Sequences of Closed Smooth Manifolds

Definitions and mathematical statement

For a closed smooth manifold $M$, meaning a compact smooth manifold without boundary, define its rational Betti numbers by \[ b_i(M;{\mathbb{Q}})=\dim_{{\mathbb{Q}}}H_i(M;{\mathbb{Q}}). \] The coefficient field is not specified in the source; the following is an explicit rational-coefficient interpretation requiring confirmation.

For each fixed positive integer $k$, characterize all tuples \[ (b_0,\ldots,b_{4k})\in{\mathbb{Z}}_{\geq 0}^{\,4k+1}, \qquad b_0=b_{4k}=1,\qquad b_{2k}\equiv 1\pmod 2, \] for which there exists a closed smooth manifold $M$ of dimension $4k$ satisfying \[ \dim_{{\mathbb{Q}}}H_i(M;{\mathbb{Q}})=b_i \qquad\text{for every }i\in\{0,\ldots,4k\}. \] The task is to give necessary and sufficient conditions for this existence. Connectedness and orientability need not be imposed separately in this rational interpretation: they follow from the prescribed zeroth and top Betti numbers. The source does not specify the range of $k$; the formulation above treats positive dimensions, and whether $k=0$ is intended remains unresolved.

Short English statement

Determine which finite sequences of nonnegative integers, with first and last entries equal to one and odd middle entry, are realized as the Betti numbers of a closed smooth manifold of dimension $4k$. Rational coefficients and positive $k$ are explicit provisional interpretations because the source leaves these conventions unstated.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003563$, title "Betti Sequences of Closed Smooth Manifolds," Oberwolfach Reports Open Problems collection, source code catalogue entry. The record gives proposed year 2017 and identifies no individual proposer.
• Supplied source citation: Low-dimensional Topology and Number Theory (2018), Oberwolfach Reports, pp. 2363--2426, DOI 10.4171/owr/2017/38. https://doi.org/10.4171/owr/2017/38. This report also appears among the supplied literature sources as "Low dimensional Topology and Number Theory."
• Supplied literature source: Betti numbers of closed manifolds. https://msp.org/agt/2026/26-2/agt-v26-n2-p10-p.pdf.
• Supplied literature source: What are the Betti numbers of closed manifolds. https://www.math.princeton.edu/events/what-are-betti-numbers-closed-manifolds-2023-10-26t170000.
• Literature assessment checked at 2026-08-21: the supplied dated curation labels the problem "partially solved" and states that many Betti number patterns are realized, that the odd middle Betti case is reduced to rational projective plane existence, and that rational projective planes and hence the full realization classification remain unknown in some dimensions. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This is a supplied editorial label, not independent mathematical verification of this draft.