Please:
a) Explain the meaning of the above question to someone without any university-level maths.
b) Answer it!
Cheers,
Cryptoderk
That’s the Poincare conjecture, which was recently answered in the affirmative by Grigori Perelman. Wikipedia has a pretty good article that explains the gist of the question.
It depends on what you mean by “closed”. The tecnical word is compact, which in this instance (although not in general–I will let ultrafilter give the general definition) means that if you take a sequence of points in the space, some subsequence converges to a point of the space.
Simply connected embodies two concepts. That of being connected for which your intuitive idea will not lead you astray and of simple connectivity. The latter means that if you draw a circle in the space, it can be continuously deformed into a point. Contrast a square with an annulus. The square is simply connected, but a circle drawn on an annulus that surrounds the hole cannot be deformed into a point because the hole prevents it. Actually a square with a single point removed is also not simply connected but that space is not compact.
Finally we come to the word “manifold”, in this case 3-manifold. Let me explain a 2-manifold; it is easier to visualize. A space is a 2-manifold if some neighborhood of each point looks just like 2-dimensional space. It may not be quite obvious, but the disk of points of distance from the origin less than (but not equal to) 1 looks (to a topologist) just like the entire plane. The set of points of distance less than or equal to 1 does not since the points on the circle of radius 1 do not look like 2-dimensional space (we do say that set is a “half-space”, however). Finally the surface of a sphere is a compact simply connected 2-manifold. The surface of a torus is a compact non-simply connected 2-manifold. A compact 2-manifold is a sphere with some number of handles.
Now repeat all but the last replacing 2 by 3 and you have defined 3-manifold. There is no such simple classification of the compact 3-manifold and Thurston’s geometrization conjecture–apparently a wide generalization of the Poincare conjecture (now I am out of my depth)–is the closest thing to a classification. This is apparently what Perleman has shown.
Well, I didn’t realise that this was a famous conjecture, I presumed that it was an everyday maths problem (given the person who has been asking this question for well over a year now).
Thanks!