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

Let $X$ be a closed, oriented manifold with $\dim X \geqslant 5$. 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 $X$.