The 4-dimensional disc embedding theorem and dual spheres

Abstract

We modify the proof of the disc embedding theorem for $4$-manifolds, which appeared as Theorem 5.1A in the book "Topology of 4-manifolds" by Freedman and Quinn, in order to construct geometrically dual spheres. These were claimed in the statement but not constructed in the proof. We also prove Proposition 1.6 from the Freedman-Quinn book regarding generic homotopies of discs or spheres in a 4-manifolds, which was not proven there.

18 pages, 1 figure. v2: Added citations to Sato, removed previous Sec 9, added Rem 1.5. v3: Substantial rewrite of exposition; proofs unchanged, added details in Lemmas 4.1 and 4.2; added outline of our proof in Sec 1.2 and Rem 1.3 with alternative argument by referee; reordered so that disc embedding theorem proof comes first, then generic homotopies. This version to appear in Selecta Math