Short answers, pulled from the story.
Henri Poincaré published his paper Analysis Situs in 1895, announcing a new way to study shapes through topological invariants. He introduced the fundamental group as a tool to distinguish between different three-dimensional spaces.
Grigori Perelman posted three papers on the arXiv repository starting the 11th of November 2002 and continuing through 2003. These documents outlined a complete proof using modified Ricci flow with surgery techniques to handle singularities.
Richard S. Hamilton published his foundational paper on Ricci flow in 1982, introducing a method to deform Riemannian metrics on manifolds. The technique mimics how heat diffuses through an object by expanding negative curvature regions while contracting positive ones.
The International Congress of Mathematicians awarded Grigori Perelman the Fields Medal on the 22nd of August 2006 during a ceremony in Madrid. He refused to accept the medal worth fifteen thousand Canadian dollars despite being honored for his work on Ricci flow.
J. H. C. Whitehead claimed to prove the conjecture in the 1930s before retracting his argument entirely. During this process, he discovered examples of simply-connected non-compact three-manifolds not homeomorphic to Euclidean space known as Whitehead manifolds.