Whitehead torsion and the Smith conjecture (Q788333)

M. Cohen has conjectured that if x is a non-trivial element of the Whitehead group of any group G then x has dimension 3. In this paper the author verifies the equivalence of Cohen's conjecture for cyclic groups and the following modification of the Smith conjecture: if \((S^ n,S^{n-2})\) is a knot pair and the k-fold branched covering of \(S^ n\) branched over \(S^{n-2}\) is an n-sphere then either \(\pi_ 1(S^ n- S^{n-2})\) has deficiency less than one or \(\Delta(t)\equiv \pm t^ j mod (t^ k-1)\), \(\Delta\) (t) the Alexander polynomial of the knot. The author draws several consequences of this equivalence which are independent of the truth of the conjectures.