Positive Mass on Simply Connected Nonspin Manifolds

Definitions and mathematical statement

A closed manifold is compact and has no boundary. A simply connected manifold is path connected and has trivial fundamental group. In the intended smooth setting, a nonspin manifold is one admitting no spin structure; a simply connected manifold is orientable, so this condition is equivalent to $w_2(TM)\ne 0$, where $w_2(TM)\in H^2(M;{\mathbb{Z}}/2{\mathbb{Z}})$ is the second Stiefel--Whitney class of its tangent bundle.

The supplied problem asks whether, for every integer $n\geq 8$, there exists a closed simply connected nonspin $n$-manifold $M$ such that every admissible metric on $M$ has nonnegative mass.

The record does not define either "admissible metric" or "mass." If $\mathcal A(M)$ denotes the intended class of admissible metrics and if the intended mass is a real number $m_M(g)$ assigned to each $g\in\mathcal A(M)$, the quantifier structure is \[ \forall n\in{\mathbb{Z}},\quad n\geq 8\ \Longrightarrow\ \exists M\quad \left\{ \begin{aligned} &\dim M=n,\quad M\text{ is closed and simply connected},\\ &M\text{ admits no spin structure},\\ &\forall g\in\mathcal A(M),\quad m_M(g)\geq 0. \end{aligned} \right. \] Here $\mathcal A(M)$ and $m_M$ are placeholders for missing source definitions, not definitions of admissibility or mass. In particular, the record does not specify whether mass also depends on a point, an operator, or further geometric data, nor how any such data are quantified. It also does not state whether $\mathcal A(M)$ must be nonempty; without that requirement the universal assertion can be vacuous. Thus this display preserves the supplied quantifier structure but is not a fully specified standalone mathematical problem. The smooth Riemannian conventions and the missing admissibility and mass definitions require source confirmation.

Short English statement

For each integer dimension $n\geq 8$, the question asks for a closed simply connected nonspin manifold on which every admissible metric has nonnegative mass. The supplied record leaves admissibility, the definition and possible additional arguments of mass, and nonemptiness of the admissible class unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002630$, title: Positive Mass on Simply Connected Nonspin Manifolds. Collection: Oberwolfach Reports Open Problems. Exact source problem code: catalogue entry. The record gives proposed year 2014 and names no individual proposer.
• Supplied report citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2015), Oberwolfach Reports, pp. 1991--2046, DOI https://doi.org/10.4171/owr/2014/36. This is also the supplied literature source labeled "original OWR report."
• Supplied literature source labeled "mass of elliptic operators": https://arxiv.org/abs/1401.1614.
• Literature assessment dated exactly 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open, states that the proposed manifolds would provide the missing nonspin high-dimensional positive-mass theorem, and reports that candidate families were identified while the general problem remained unresolved. The background explicitly requests independent verification of time-sensitive status and cited claims.