On manifolds with almost non-negative Ricci curvature and integrally-positive $k^{th}$-scalar curvature

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor. If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds for $k=2$, then we show that $M$ is contained in a neighbourhood of controlled width of an isometrically embedded $1$-dimensional sub-manifold. From this, we deduce several metric and topological consequences: $M$ has at most linear volume growth and at most two ends, the first Betti number of $M$ is bounded above by $1$, and there is precise information on elements of infinite order in $\pi_1(M)$. If $(M^n,g)$ is a Riemannian manifold satisfying such bounds for $k\geq 2$ and additionally the Ricci curvature is asymptotically non-negative, then we show that $M$ has at most $(k-1)$-dimensional behavior at large scales. If $k=n={\rm dim}(M)$, so that the integral lower bound is on the scalar curvature, assuming in addition that the $n-2$-Ricci curvature is asymptotically non-negative, then we prove that the dimension drop at large scales improves to $n-2$. From the above results, we deduce topological restrictions, such as upper bounds on the first Betti number.