Chapter 28: Poincaré's Conjecture

Poincaré’s conjecture stands alone among the millennium problems: it is proved. Grigori Perelman completed the proof in 2002–2003 via Hamilton’s Ricci flow with surgery, earning the Millennium Prize in 2010 (which he declined). The theorem asserts that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere S³. This chapter reviews the established result and explains why simple connectivity matters for the τ-framework: trivial fundamental group guarantees that local Hartogs bulk projections glue globally without obstruction. Unlike every other millennium problem, Poincaré requires no conjectural bridge, no τ-effective approximation. It is simply true. The next chapter reinterprets it categorically without altering its proof status.