Embedded spheres in

We give an algorithm to decide which elements of π 2 ( # k S 2 × S 1 ) can be represented by embedded spheres up to conjugacy. Such spheres correspond to splittings of the free group F k on k generators. We also give an algorithm to decide when classes in π 2 ( # k S 2 × S 1 ) can be represented by disjoint embedded spheres. thods may be useful in studying the splitting complex of a free group, and hence the group of outer automorphisms.