On Stein spaces with finite homotopy rank-sum
Indranil Biswas, Buddhadev Hajra
TL;DR
This work classifies complex Stein spaces and smooth affine surfaces with finite homotopy rank-sum, connecting complex-analytic geometry to rational homotopy theory. It establishes a dichotomy for Stein surfaces, showing that finite total higher rational homotopy rank forces either a K(π1,1) structure (π2 trivial) or a universal cover homotopy equivalent to S^2 (π2 ≅ Z). For affine non-general-type surfaces with elliptic homotopy type, the results force an Eilenberg–MacLane structure or yield a closely related S^2-type geometry, with precise guidance when π1 is finite (notably Z/2). The paper further leverages ramified coverings, Nori’s lemma, and Friedlander–Halperin bounds to achieve a sharp classification, including a detailed account of C*-fibrations and their fibers. Together, these findings tightly constrain the topology of Stein and affine surfaces under the finite rank-sum hypothesis and reveal when such spaces are EM spaces or S^2-type, with implications for Q- and Z-homology planes.
Abstract
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 consider Stein spaces of arbitrary dimension satisfying the above rational homotopy theoretic property, although most of this article focuses on Stein surfaces only. We characterize all Stein surfaces satisfying the finite homotopy rank-sum property. In particular, if such a Stein surface is affine and every element of its fundamental group is finite, it is either simply connected or has a fundamental group of order $2$. A detailed classification of the smooth complex affine surfaces of the non-general type satisfying the finite homotopy rank-sum property is obtained. It turns out that these affine surfaces are Eilenberg--MacLane spaces whenever the fundamental group is infinite.