Unknotting nonorientable surfaces topologically

Knot invariants are typically used to give a negative answer to the question of when two embeddings are ambiently isotopic, and rarely to give a positive answer. An exception is the celebrated result of Freedman and Quinn that if the complement of a 2-sphere embedded in the 4-sphere has cyclic fundamental group then that 2-sphere is topologically unknotted. We recently showed that the analogous result for closed nonorientable surfaces in the 4-sphere is also true (in most cases). This talk will describe this recent work and highlight some key ideas from the proof. This is joint work with Anthony Conway and Mark Powell.