Spin torus-action conjecture for simply connected manifolds

About 4 years old · traced to 2022

Let $n\geq 1$ , and let $M$ be a closed, spin, simply connected $(n+2)$ -manifold with an effective $T^n$ -action.

Spin torus-action conjecture. The manifold $M$ is homeomorphic to one of $S^3$ , $S^4$ , $S^5$ , or

\#^{n}_{i=2}m_i\cdot S^i\times S^{n+2-i}.

This is a purely topological classification statement underlying the proposed refinement of the domain-of-outer-communication conjecture. The supplied context says that the refined conjecture is verified in the relevant dimensions $n=2,3,4$ under the stated spin condition, but does not establish the claim for all $n\geq 1$ .

References

Primary source

Vishnu Kakkat, Marcus Khuri, Jordan Rainone and Gilbert Weinstein, “The Geometry and Topology of Stationary Multi-Axisymmetric Vacuum Black Holes in Higher Dimensions”, arXiv:2203.08325 (2022).

Posted 1 month ago

Progress summary

Refreshed 4 weeks ago

Open

No public source reports a proof or counterexample, so the classification remains open beyond the dimensions already covered.

The conjecture asserts that every closed, spin, simply connected manifold of dimension $n+2$ with an effective $T^n$ -action has one of the listed connected-sum types. The catalogued source records the all- $n$ statement as open as of March 2022.

Known results

The corresponding refinement is reported as verified under the spin hypothesis for $n=2,3,4$ .
Orlik and Raymond (1970) classified effective $T^2$ -actions on closed, orientable, simply connected $4$ -manifolds; the source does not formulate this as a proof of the stated conjecture.

Current status (as of September 2026): The cases $n=2,3,4$ are reported as verified under the spin condition, while no public proof, counterexample, or verification of the full claim for all $n\geq 1$ was found.

Sources

No solutions have been posted yet.