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. ----- 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.