We compare two naturally arising notions of ‘unknotting number’ for 2‐spheres in the 4‐sphere: namely, the minimal number of 1‐handle stabilizations needed to obtain an unknotted surface, and the minimal number of Whitney moves required in a regular homotopy to the unknotted 2‐sphere. We refer to these invariants as the stabilization number and the Casson–Whitney number of the sphere, respectively. Using both algebraic and geometric techniques, we show that the stabilization number is bounded above by one more than the Casson–Whitney number. We also provide explicit families of spheres for which these invariants are equal, as well as families for which they are distinct. Furthermore, we give additional bounds for both invariants, concrete examples of their non‐additivity, and applications to classical unknotting number of 1‐knots.

We show that the homotopy type of a finite oriented Poincaré 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and Bauer, this applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO( 3). An important class of examples are elliptic surfaces with finite fundamental group.

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We consider slice disks for knots in the boundary of a smooth compact 4-manifold X 4 . We call a knot K ⊂ ∂X deep slice in X if there is a smooth properly embedded 2-disk in X with boundary K, but K is not concordant to the unknot in a collar neighborhood ∂X × I of the boundary.We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0-and a nonzero number of 2-handles always has a deep slice knot in the boundary.We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented 4-manifold V with spherical boundary such that every knot K ⊂ S 3 = ∂V is slice in V via a null-homologous disk.