An Elementary Proof of the Poincare

Shivam Kumar

Abstract: We provide a comprehensive analysis and (simple) proof of Poincare's conjecture, a fundamental problem in topology. The conjecture asserts that any compact, smooth 3-manifold with the same homotopy type as the 3-sphere (S3) is necessarily diffeomorphic to S3 itself. Index Terms-Poincare's conjecture, compact manifolds, smooth manifolds, homotopy equivalence, diffeomorphism, 3-manifold, 3-sphere, topology, differential topology, algebraic topology.

Keywords: Poincare's Conjecture, topology, homotopy, sphere