© 1996, Topology Atlas

Browder and others in the 60s used Surgery Theory to show that the isotopy classification of high-dimensional knots with pi_1 = Z is a purely homotopy theoretic problem, e.g. the question of whether a knot is determined by its complement reduces to the existence of a certain surgery doesn't work, but there are also many counterexamples, by Suciu etc. The conjecture is that

knots with pi_1 = Z are determined by their complements.

In my paper [1] I lower Farber's record of slightly, the conjecture is true if the Seifert surface of the knot is approximately n/3-connected. My 1994 paper [1] was the first time anyone actually solved the homotopy theory problem, all other work on knots with pi_1 = Z has been geometric. The technique I used (applied B. Williams's Poincare embedding theorem) was amusing but will not extend any farther. Unpublished work of mine gives a different homotopy theoretic proof that might well be able to lower the record, involving the equivariant homotopy theory of the universal cover. In this language, my proof conks out when the "reduced" universal cover is no longer an equivariant suspension (using Ranicki's theory of CW-pi complexes). But quite likely equivariant analogues of Ganea-type mapping cone constructions ought to help, together with some hard homotopy theory calculations on some examples.

There is a clear bifurcation on how to attack the geometric problem, should we use homotopy theory&the surgery machine (my preference), or use geometric techniques (everybody else)?

If you don't believe the conjecture, and you want a counterexample, a knot with pi_1 = Z which is not determined by it's complement, you just have to find an example, so you can use geometric techniques! I know both Suciu&Cappell have worked on this angle.

But to prove the conjecture, you have to get something of an isotopy classification of these knots, and this is way beyond the reach of any known geometric techniques. So you have to use my homotopy theory approach. For instance, Farber's beautiful proof used Wall's EHP sequence for thickenings, a geometric result that conks out after the metastable range. There's no extension of this to the next range, which is approximately conn(Seifert) < n/4. Now I also use homotopy EHP sequences, so I'm stuck also around n/3 (just a little better than Farber), but there are lots of homotopy theoretic techniques available outside the metastable range, particularly in the next range.

To me the triumph of the Browder-Novikov-* surgery school was that geometric problems were reduced to homotopy theory. But apart from my work there's been very little action, except in cases (spheres, K(pi,n)s) where the homotopy theory was trivial. I think the feeling was that the homotopy theory was intractable, and this lead to the demise of the surgery school, which counted 4.5 Fields medalists (Thom, Smale, Milnor, Novikov&(almost) Sullivan) back in it's heyday. I think my work shows that the homotopy theory is much more tractable than was believed, I'm hoping this will resuscitate the surgery school.

[1] W. Richter, Simple knots are determined by their complements in one more dimension than Farber's range, Proc. Amer. Math. Soc. 120 (1994), 285--294.