Whitehead3 is a ?Slice? link

Abstract

We show that the Whitehead double t Wh(L) of a two component link L is (topologically flat) slice if and only if the linking number of L is zero. When they exist, the slices may be chosen so that the complement (B*-slices) is homotopy equivalent to a wedge of two circles, S 1 v S 1, with certain meridinal loops of Wh(L) freely generating ~(B4-slices). (Compare this with IF1] where it is shown that the Whitehead double of boundary links are slice.) This is consistent with the surgery conjecture but seems not to illuminate the key 3component case (see I-F2]). In particular Wh3=Wh(WhO is slice. We will describe the argument quite explicitly for this case and indicate in closing remarks what modifications are needed for the general result. First we deal with non-existence. Let S be the obvious two component Seifert surface for Wh(L). A component of S is obtained by composing either of the two punctured tori Tz_cSlx D 2 pictured below with an untwisted identification of S I  2 with a neighborhood of a component of L. Add a band to S, S=Su band, so that S becomes the Seifert surface for some knot, K. The Seifert forms for S and S are identical and in a natural basis will be: