  On an n-dimensional manifold, positive $k$th-intermediate Ricci curvature is
a condition interpolating between positive sectional curvature ($k = 1$) and
positive Ricci curvature ($k = n - 1$). In this article, we show that closed
manifolds with positive 2nd-intermediate Ricci curvature and large symmetry
rank are either homeomorphic to spheres in odd dimensions or have positive
Euler characteristic in even dimensions. In the process, we establish new tools
for studying closed manifolds with torus symmetries and positive
$k$th-intermediate Ricci for values of $k \geq 2$. These tools include
generalizations of the isotropy rank lemma, symmetry rank bound, and
connectedness principle from the setting of positive sectional curvature.
