On Sat, Dec 8, 2012 at 4:23 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I had a suspicion I knew that space [the intersection of nested tori, each pulled around the previous one so as to link itself], from a previous encounter with it.
Sure enough, it turns out that it's the *complement* of the amazing "Whitehead contractible manifold" W in the 3-sphere.
When you nest 4 linked tori inside each torus at each step, you've proved that the complement of the intersection is not simply connected, hence not contractible. Why doesn't the same proof work in the case where you place a single self-linked torus inside the torus each time? Andy
----- A space is contractible if it can be shrunk to a point within itself, like an n-disk or Euclidean space R^n.
W was discovered in 1935 by J.C. Whitehead, after he erroneously thought he'd proved that every contractible 3-manifold (without boundary) is topologically equivalent to 3-space R^3.
One way to define it is to create the infinite intersection J of tori as below inside the 3-sphere S^3; then the complement S^3 - J is the Whitehead contractible manifold.
The reason it's not topologically R^3 is that W is not "simply-connected at infinity" < http://en.wikipedia.org/wiki/Simply_connected_at_infinity >). -----
--Dan
RWG wrote: <<
I wrote: << (I wonder what happens in the case where there is only one torus in each stage, pulled longitudinally around the previous torus and made to link with itself.)
Ick! I think the nth one winds and unwinds 2^n times around inside the outermost one. But we lose the shrinking to a point effect.
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