The Rosenberg $ \mathbb{S}^{1} $-Stability Conjecture for $ \chi(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 $.