The disc embedding theorem for simply connected 4-manifolds was proved by Freedman in 1982 and forms the basis for his proofs of the h-cobordism theorem, the Poincare conjecture, the exactness of the surgery sequence, and the classification of simply connected manifolds, all in the topological category and dimension four. The disc embedding theorem for more general 4-manifolds is proved in the book of Freedman and Quinn. However, the geometrically transverse spheres claimed in the outcome of the theorem are not constructed. We close this gap by constructing the desired transverse spheres. We also outline where and why such transverse spheres are necessary. This is a joint project with Mark Powell and Peter Teichner.