A potential smooth counterexample in dimension 4 to the Poincare conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture

WE DESCRIBE a homotopy 4-sphere X4, built with the usual zero and 4-handle and two lhandles and two 2-handles (Figure 28). Of course X4 is homeomorphic to S4 [Freedman] but considerable effort has not led to a proof that X4 is diffeomorphic to S4. X4 has the following virtues: (A) Although it is easy to construct smooth homotopy 4-spheres (e.g. the Gluck construction on knotted 2-spheres or via non-trivial presentations of the trivial group), this is the only (except S4) example we know without 3-handles and with so few handles altogether. (B) The presentation of the trivial group arising from X4 (see $2) is {x, y lxyx = yxy, x5 = y4}; it is easy to show that this group is trivial, but it seems difficult to do so using Andrews-Curtis moves ([l] or [lo] 5.1). (C) Let Z,, be E without the 4-handle; then we can add two 2-handles and two 3-handles and a 4-handle to get (smoothly) S4 (see $2). Applying the topological Schoenflies theorem to X& in S4, we see directly that X0 is homeomorphic to B4. (D) z&, is an interesting smooth S 3 in S4. The smooth Schoenflies conjecture is unsettled in dimension 4 and z&, is a good test case. So in $4, we give a critical level imbedding of X& in S4 (Fig. Sl-Sll). Scharlemann [12] has used critical level imbeddings to prove the conjecture for genus 2 imbeddings; this one is genus 51. (E) &, is the result of the Gluck construction on a knot K in S4 (Fig. 16). K is constructed from two distinct ribbons for the f&, knot (see [ll]). E was first defined as the double cover of a certain exotic RP4 of Cappell and Shaneson [7]. It was built by decomposing RP4 into a 2-disk bundle over RP’ and the non-trivial 3disk bundle over S ‘, and then replacing the latter by a punctured 3-torus bundle over S1 with 010 monodromy ( 1 0 0 1 . We thought we proved (in [4]) that the double cover E of this -1 10 exotic RP4 was diffeomorphic to S”. However Iain Aitchison and J. H. Rubenstein ([2], [3]) discovered an error (the last sentence on page 77 of [4] states that a framing is zero when it should be odd). We actually proved that X4 is the Gluck construction on a knotted 2-sphere in S4 (this was discussed in Remarks 2 and 3 of [4]). As mentioned above, we are still unable to prove that X4 is diffeomorphic to S4. In the meantime however, Fintushel and Stern [S] have constructed, by different methods, an exotic RP4 whose double cover is S4. Note that both these exotic RP4’s are homotopy RP”s which are s-cobordant to RP’, and then homeomorphic to RP4 by Freedman’s recent proof of the topological s-cobordism theorem for many fundamental groups including Z/2. After some definitions in 0 1, we begin in $2 with a handlebody description of X4 from [4, Fig. 51. We simplify this handlebody presentation by sliding handles over other handles and by handle cancellations and births to get the properties of X4 mentioned above in (A), (B), (C) and (E). It is worth remarking that it is usually hard to see how to add a cancelling pair of handles (a birth) in any useful way; this is done with a (2-3)-pair in Fig. 19 and later in Fig. 35. Some problems are suggested by this work: (1) Does every homotopy 4-ball with boundary S3 smoothly imbed in S4? (2) Do the results of this paper hold for the other fake RP4’s of [3]?