# P10904: Kirby Problem 4.128

- ID: `P10904`
- Reference: `unsolvedmath-3004-kirby-problem-4-128`
- Page: https://theoremdb.org/statements/P10904
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/unsolvedmath-3004-kirby-problem-4-128?representation=markdown
- Record maturity: Reviewed problem

## The problem

What is the structure of 4-manifolds that admit a Riemannian metric of positive scalar curvature? There are variations of this problem for different classes of manifolds.

(a) Is every closed simply connected PSC 4-manifold diffeomorphic to a connected sum of copies of $\mathbb{CP}^{2}, \mathbb{CP}^{2}$, and $S^{2} \times S^{2}$?

(b) What is the structure of non-simply connected closed PSC 4-manifolds?

(c) Which 4-manifolds with boundary have a PSC metric?

(d) Which non-compact 4-manifolds have a complete PSC metric with uniformly positive scalar curvature?

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `unsolvedmath-3004-kirby-problem-4-128`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.