The Poincaré Homology 3-Sphere

In 1900, Henri Poincaré initially conjectured that any 3-manifold that has the homology groups of a sphere must be the 3-sphere, but later in 1904 discovered a counterexample, named the Poincaré homology 3-sphere. In the process, Poincaré also invented homology, the fundamental group and conjectured problems that have continued to interest mathematicians up to the current day. In this video, we study the Poincaré homology sphere by applying foundational results of algebraic topology such as the Seifert-van Kampen theorem and Poincaré duality. Note: This video is converted from the slides of a talk given to fellow graduate students in a student seminar. I assumed knowledge of definitions and key properties of the fundamental group, homology and cohomology groups that are encountered in a first course on algebraic topology. Even if you have not taken such a course, I hope that the video may communicate some of the underlying concepts and ideas behind algebraic topology :) Animated with Manim Community Edition. v0.20.1. https://www.manim.community/ Music: Purple Planet Music Song: Fresh Thinking https://www.purple-planet.com/