A Codimension Two Approach to the ${\mathbb{S}}^1$-Stability Conjecture

J. Rosenberg’s ${\mathbb{S}}^1$-stability conjecture states that a closed oriented manifold $X$ admits a positive scalar curvature metric iff $X\times {\mathbb{S}}^1$ admits a positive scalar curvature metric $h$. As pointed out by J. Rosenberg and others, there are known counterexamples in dimension four. We prove this conjecture whenever $h$ satisfies a geometric bound which measures the discrepancy between ${\mathrm{\partial }}_{\theta }\in T{\mathbb{S}}^1$ and the normal vector field to $X\times \{P\}$, for a fixed $P\in {\mathbb{S}}^1\mathrm{.}$