  A topological space (not necessarily simply connected) is said to have finite
homotopy rank-sum if the sum of the ranks of all higher homotopy groups (from
the second homotopy group onward) is finite. In this article, we characterize
the smooth compact complex Kaehler surfaces having finite homotopy rank-sum. We
also prove the Steinness of the universal cover of these surfaces assuming
holomorphic convexity of the universal cover.
