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 Ck consisting of bordism classes of knots Σk → Sk+2, Σk 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 C2k-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 C2k-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 C2k-1 ⊗ Q is Q∞ and C2k-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 Δn. 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]).