On Stein spaces with finite homotopy rank-sum

TL;DR

This paper characterizes Stein surfaces with finite homotopy rank-sum, providing classifications and showing that certain affine surfaces are Eilenberg--MacLane spaces when their fundamental group is infinite.

Contribution

It offers a complete classification of Stein surfaces with finite homotopy rank-sum, especially for affine surfaces with finite fundamental groups.

Findings

Stein surfaces with finite homotopy rank-sum are classified.

Affine Stein surfaces with finite fundamental groups are either simply connected or have order 2.

Infinite fundamental group Stein surfaces are Eilenberg--MacLane spaces.

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.