Bulletin of the Manifold Atlas (2014) Poincaré’s conjecture*

Abstract. We give an overview of the development on work on Poincaré’s con-jecture in the first half of the twentieth century. 01A55, 01A60 Poincaré’s conjecture states- in modern terms- that every closed 3-manifold with a vanishing fundamental group is homeomorphic to the 3-sphere. There is a generalization of this conjecture for higher-dimensional manifolds, the so-called generalized Poincaré conjecture (formulated for the first time by W. Hurewicz, [5, p.523]). In his series of papers on Analysis situs (1892-1904), H. Poincaré studied the question of how to characterize 3-manifolds by invariants. To that end he introduced the fundamental group and investigated Betti-numbers and torsion coefficients. In a first step he realized that there are closed 3-manifolds with identical Betti-numbers but with different fundamental groups (cf. the Manifold Atlas page on Poincaré’s cube manifolds, [20]). This result was annouced in 1892 and proven in detail in 1895. Motivated by a critique of P. Heegaard, Poincaré introduced the torsio