Perelman's solution

[Shadow]

I've been meaning to ask this for a while. From a layman's point of view (me being the layman), Perelman's solution breaks the most basic rules of topology.

 

Topology [...] is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing.

- wiki

 

Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until eventually he is left with a collection of round three-dimensional spheres. Then he rebuilds the original manifold by connecting the spheres together with three-dimensional cylinders, morphs them into a round shape and sees that, despite all the initial confusion, the manifold was in fact homeomorphic to a sphere.

- wiki

 

So...what's the deal?

[Dave]

A disclaimer that I haven't done topology in a good few years now, but whilst it's true that properties which are preserved in a topological sense are important and useful, techniques involving gluing or cutting can be used just as well too. Topological constants aren't everything

[Shadow]

At a second glance, I think I misunderstood the former excerpt. I understood cutting and gluing in topology to be the same as division by zero in algebra (or anywhere else). Thanks dave.

[DrRocket]

At a second glance, I think I misunderstood the former excerpt. I understood cutting and gluing in topology to be the same as division by zero in algebra (or anywhere else). Thanks dave.

 

There are some established techniqies for doing cutting and gluing that are, no kidding, called "surgery" and "plumbing".

 

 

http://en.wikipedia..../Surgery_theory

 

http://mathworld.wol...m/Plumbing.html

 

 

http://www.maths.ed....rs/kirbysch.pdf

 

 

Added in edit. John Milnor, who introduced surgery, wrote this brief piece on 3-manifold classification with commments on Perelman's work shortly after Perelman's announcement and before it had been carefully checked by the community.

 

http://www.math.sunysb.edu/~jack/PREPRINTS/tpc.pdf

Edited June 20, 2011 by DrRocket