 The Kervaire–Milnor theorem states that if formula_29 is a characteristic sphere in a smooth compact 4-manifold "M", then 
A characteristic sphere is an embedded 2-sphere whose homology class represents the Stiefel–Whitney class formula_1.If formula_1 vanishes, we can take formula_29 to be any small sphere, which has self intersection number 0, so Rokhlin's theorem follows.The Freedman–Kirby theorem states that if formula_29 is a characteristic surface in a smooth compact 4-manifold "M", then 
where formula_36 is the Arf invariant of a certain quadratic form on formula_37.