The Development of Topological 4-manifold Theory

The development of topological 4-manifold theory is described in the form of a flowchart showing the interdependence among many key statements in the theory. In particular, the flowchart demonstrates how the theory crucially relies on the constructions in this book, what goes into the work of Quinn on smoothing, normal bundles, and transversality, and what is needed to deduce the famous consequences, such as the classification of closed, simply connected, topological 4-manifolds, the category preserving Poincaré conjecture, and the existence of exotic smooth structures on 4-dimensional Euclidean space. Precise statements of the results, brief indications of some proofs, and extensive references are provided.

Afterword: PC4 at Age 40

This is not a proof of the Poincaré conjecture, but a discussion of the proof, its context, and some of the people who played a prominent role. It is a personal, anecdotal account. There may be omissions or transpositions, as these recollections are 40 years old and not supported by contemporaneous notes, but memories feel surprisingly fresh....

The s-cobordism Theorem, the Sphere Embedding Theorem, and the Poincaré Conjecture

The s-cobordism theorem, the sphere embedding theorem, and the Poincaré conjecture comprise three key consequences of the disc embedding theorem. The chapter begins by explaining in detail how to use the disc embedding theorem to prove the 5-dimensional s-cobordism theorem and the sphere embedding theorem. The sphere embedding theorem is the output from the disc embedding theorem that one wants in many situations. A version of the Poincaré conjecture is proven, specifically that every smooth homotopy 4-sphere is homeomorphic to the 4-sphere. All the results proved in this chapter are category losing; that is, they require smooth input but only produce homeomorphisms.

Differentiations of Nonlinear Functions Related to States in Magic(n), Cosmology and the Poincare Conjecture

Results for area differentiations and potential differentiations are related to concepts in cosmology, and to magic(n). The shapes of a sphere and other geometries will be discussed for 3 and higher dimensions and the Poincare conjecture is interpreted to fit in this scheme. The novelty of this research note is to relate fundamental concepts and findings of today, in order to calibrate the symbolism and mathematics. Doi: 10.28991/HEF-2020-01-04-05 Full Text: PDF

Poincaré conjecture: A problem solved after a century of new ideas and continued work

The Poincaré conjecture is a topological problem established in 1904 by the French mathematician Henri Poincaré. It characterises three-dimensional spheres in a very simple way. It uses only the first invariant of algebraic topology – the fundamental group – which was also defined and studied by Poincaré. The conjecture implies that if a space does not have essential holes, then it is a sphere. This problem was directly solved between 2002 and 2003 by Grigori Perelman, and as a consequence of his demonstration of the Thurston geometrisation conjecture, which culminated in the path proposed by Richard Hamilton.

Perfect Rigor- A Genius : A Book Review

The solution of Poincare conjecture in topology represents the pinnacle of mathematical logic human brain is capable of. Best mathematicians of the world tried to unlock the answer of this baffling problem. Ultimately Gregory Perelman was able to crack it but in this pursuit transformed himself into an unfathomable riddle.