Embedded spheres in S^2\times S^1#...#S^2\times S^1
  We give an algorithm to decide which elements of pi_2(S^2\times S^1#...#S^2\times S^1) can be represented by embedded spheres. Such spheres correspond to splittings of the free group on k generators. Equivalently our algorithm decides whether, for a handlebody N, an element in pi_2(N,\partial N) can be represented by an embedded disc. We also give an algorithm to decide when classes in $\pi_2(S^2\times S^1#...#S^2\times S^1)$ can be represented by disjoint embedded spheres.   We introduce the splitting complex of a free group which is analogous to the complex of curves of a surface. We show that the splitting complex of the free group on k generators embeds in the complex of curves of a surface of genus $k$ as a quasi-convex subset. 
Siddhartha Gadgil