Knots and diffeomorphisms

Summary

In this note I will announce some results concerning the connection between bordism of knots and diffeomorphisms and state some problems.

Consider the bordism group of knots C_k consisting of bordism classes of knots Σ^k → S^k+2, Σ^k a k-dimensional homotopy sphere. Kervaire has shown that this group is zero for k even [2]. Levine has proved that for k ≥ 3 there is an embedding C_2k-1 → W_(-1)k^(Z,Q), the Witt group of (-1)^k-symmetric isometric structures over Q ([6], for definition of W_(-1)k^(Z,Q) compare [7]). Kervaire has shown that W_(-1)k^(Z,Q) is of the form Z^∞ ⊕ (Z/4)^∞ ⊕ (Z/2)^∞. Several people have stated that C_2k-1 is of the form Z^∞ ⊕ (Z/4)^∞ ⊕ (Z/2)^∞, too, but no proof has appeared. It is rather easy to see that this holds for k odd and that for k even C_2k-1 ⊗ Q is Q^∞ and C_2k-1 contains infinitely many torsion elements [7].

Another group which is classified in terms of isometric structures is the bordism group of n-dimensional orientation preserving diffeomorphisms Δ_n. In [3] I have shown that this group n for n odd (n ≠ 3) is classified in terms of the manifold and the mapping torus of the diffeomorphism. For n even (n > 2) we need another invariant given by the isometric structure of a diffeomorphism which lies in W_(-1)k^(Z,Z), the Witt group of (-1)^k-symmetric isometric structures over Z([4], [5]).