If X×S¹ admits a PSC metric whose circle factor is at angle < 45° to the X-slice, then X itself admits a PSC metric, for any closed oriented X of dimension at least two.
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A Codimension Two Approach to the $\mathbb{S}^1$-Stability Conjecture
If X×S¹ admits a PSC metric whose circle factor is at angle < 45° to the X-slice, then X itself admits a PSC metric, for any closed oriented X of dimension at least two.