Geometrization

Check your comprehension

Check your comprehension

~ What is the subject of topology?

~ Which shapes are considered the same in topology?

~ What is the ‘simple connectivity test’?

The question raised became known as the Poincaré conjecture. Over the years, many outstanding mathematicians tried to solve it--Poincaré himself, Whitehead, Bing, Papakirioukopolos, Stallings, and others. While their efforts frequently led to the creation of significant new mathematics, each time a flaw was found in the proof. In 1961 came astonishing news. Stephen Smale, then of the University of California at Berkeley (now at the City University of Hong Kong) proved that the analogue of the Poincaré conjecture was true for spheres of five or more dimensions. The higher-dimensional version of the conjecture required a more stringent version of Poincaré's test; it asks whether a so-called homotopy sphere is a true sphere. Smale's theorem was an achievement of extraordinary proportions. It did not, however, answer Poincaré's original question. The search for an answer became all the more alluring.

Smale's theorem suggested that the theory of spheres of dimensions three and four was unlike the theory of spheres in higher dimension. This notion was confirmed a decade later, when Michael Freedman, then at the University of California, San Diego, now of Microsoft Research Station Q, announced a proof of the Poincaré conjecture in dimension four. His work used techniques quite different from those of Smale. Freedman also gave a classification, or kind of species list, of all simply connected four-dimensional manifolds.

Both Smale (in 1966) and Freedman (in 1986) received Fields medals for their work.

There remained the original conjecture of Poincaré in dimension three. It seemed to be the most difficult of all, as the continuing series of failed efforts, both to prove and to disprove it, showed. In the meantime, however, there came three developments that would play crucial roles in Perelman's solution of the conjecture.

~ For spheres of which dimensions the Poincaré conjecture was first proved?

~ What did continuing attempts to prove or disprove the original Poincaré conjecture show?

The first of these developments was William Thurston's geometrization conjecture. It laid out a program for understanding all three-dimensional shapes in a coherent way, much as had been done for two-dimensional shapes in the latter half of the nineteenth century. According to Thurston, three-dimensional shapes could be broken down into pieces governed by one of eight geometries, somewhat as a molecule can be broken into its constituent, much simpler atoms. This is the origin of the name, "geometrization conjecture."

A remarkable feature of the geometrization conjecture was that it implied the Poincaré conjecture as a special case. Such a bold assertion was accordingly thought to be far, far out of reach--perhaps a subject of research for the twenty-second century. Nonetheless, in an imaginative tour de force that drew on many fields of mathematics, Thurston was able to prove the geometrization conjecture for a wide class of shapes (Haken manifolds) that have a sufficient degree of complexity. While these methods did not apply to the three-sphere, Thurston's work shed new light on the central role of Poincaré's conjecture and placed it in a far broader mathematical context.

Дата добавления: 2014-10-31