On the fundamental groups of subelliptic varieties

TL;DR

This paper proves that smooth subelliptic varieties have finite fundamental groups and characterizes which groups can occur, providing insights into their homotopy types and addressing a problem related to Gromov's conjecture on Oka manifolds.

Contribution

It establishes the finiteness of fundamental groups of smooth subelliptic varieties and characterizes all finite groups realizable as such fundamental groups.

Findings

Fundamental group of smooth subelliptic varieties is finite.

Every finite group can be realized as such a fundamental group.

Existence of a subelliptic variety homotopy equivalent to an n-sphere iff n>1.

Abstract

We show that the fundamental group of any smooth subelliptic variety is finite. Moreover, it is also proved that every finite group can be realized as the fundamental group of a smooth subelliptic variety. As a consequence, it follows that there exists a smooth subelliptic variety homotopy equivalent to the $n$-sphere if and only if $n>1$. This result can be considered as a negative answer to the algebraic version of Gromov's problem on the homotopy types of Oka manifolds.