Splittings of a free group correspond to embedded spheres in the 3-manifold $M=\sharp_k S^2\times S^1$. These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in $\pi_2(M)$ can be represented by an embedded sphere.

In this note, we shall give another proof of a theorem of Aramayona and Souto, namely the group of simplicial automorphisms of the sphere complex $\mathbb{S}(M)$ associated to the manifold $M=\sharp_nS^2\times S^1$ is isomorphic to the group Out $(F_n)$ of outer automorphisms of the free group $F_n$ of rank $n\geq 3$.