This will be the 3rd talk on the Hsiang-Kleiner's theorem, which classifies compact positively curved 4-manifolds
with isometric circle actions. We will start with a review of a lemma from the end of the long-forgotten 2nd talk
( Nov 21th, 2016 ), and complete the proof of the theorem in this final talk. Roughly speaking, a contradiction
argument is used to determine a bound on the Betti numbers of these manifolds, then the classification can be
completed using results due to Michael H. Freedman. Several powerful tools from the area of compact Lie group
actions on smooth manifolds will be introduced with adequate details, and then this last part of the theorem can be
proved as an inspiring application of these tools.