On Stein spaces with finite homotopy rank-sum
-
Indranil Biswas
and
Buddhadev Hajra
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.
Funding statement: The first-named author acknowledges the support of a J. C. Bose Fellowship (JBR/2023/000003).
Acknowledgements
The authors warmly thank the referee for a very careful reading of the manuscript and making many very useful suggestions to improve the exposition. The authors would like to thank R. V. Gurjar for informing us about the content of the reference [13]. The authors are also thankful to A. J. Parameswaran for his useful discussions at the beginning of this project.
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