On Gromov's compactness question regarding positive scalar curvature

In this paper, we give both positive and negative answers to Gromov's compactness question regarding positive scalar curvature metrics on noncompact manifolds. First we construct examples that give a negative answer to Gromov's compactness question. These examples are based on the non-vanishing of certain index theoretic invariants that arise at the infinity of the given underlying manifold. This is a $\sideset{}{^1}\varprojlim$ phenomenon and naturally leads one to conjecture that Gromov's compactness question has a positive answer provided that these $\sideset{}{^1}\varprojlim$ invariants also vanish. We prove this is indeed the case for a class of $1$-tame manifolds.