The Rosenberg ${\mathbb{S}}^1$-Stability Conjecture for $\chi (X)=0$

Let $X$ be a closed, oriented manifold with $\dim X⩾5$. In this article, we show that 2006 Rosenberg’s ${\mathbb{S}}^1$-stability holds when $X$ has zero Euler characteristic. The 2006 Rosenberg-Stolz Conjecture for $X\times \mathbb{R}$ also follows under the same assumption, provided that the Riemannian metric $g$ on $X\times \mathbb{R}$ is complete, is of bounded curvature, and whose smallest eigenvalue is uniformly bounded below by some positive constant. We then show a ${\mathbb{T}}^n$-stability theorem with the same hypothesis of $X$.