A Generalization of the Geroch Conjecture with Arbitrary Ends

Using $\mu$-bubbles, we prove that for $3\le n\le 7$, the connected sum of a Schoen-Yau-Schick $n$-manifold with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When either $3\le n\le 5$, $1\le m\le n-1$ or $6\le n\le 7$, $m\in \{1,n-2,n-1\}$, we also show the connected sum $(M^{n-m}\times {\mathbb{T}}^m)\mathrm{\#}{X}^n$ where $X$ is an arbitrary manifold does not admit a metric of positive $m$-intermediate curvature. Here $m$-intermediate curvature is a new notion of curvature introduced by Brendle, Hirsch and Johne interpolating between Ricci and scalar curvature.