On Smooth Closed 3-manifolds with Finite Fundamental Groups

TL;DR

This paper proves that all orientable smooth closed 3-manifolds with finite fundamental groups are topologically equivalent to quotients of the 3-sphere by finite cyclic groups, confirming a geometrization conjecture.

Contribution

It establishes the geometrization conjecture for orientable smooth closed 3-manifolds with finite fundamental groups, showing they are homeomorphic to spherical space forms.

Findings

All such manifolds are homeomorphic to $S^3/G$ for some finite cyclic subgroup G.

The conjecture is confirmed for the class of manifolds with finite fundamental groups.

Provides a classification of these 3-manifolds as spherical space forms.

Abstract

In this paper, we prove a geometrization conjecture, every orientable smooth closed 3-manifold with finite fundamental group is homeomorphic to $S^3/G$ for some finite cyclic subgroup $G\subset {Isom}^+(S^3)$.