The complex of non-separating embedded spheres

Abstract

For $n\geq 3$, we shall show that the group $\mbox{Aut\,}(NS(M))$ of simplicial automorphisms of the complex $NS(M)$ of non-separating embedded spheres in the manifold $M=\sharp_{n} S^{2}\times S^{1}$ is isomorphic to the group $\mbox{Out\,}(\mathbf F_n)$ of outer automorphisms of the free group $\mathbf F_n$, where $\mathbf F_n$ is identified with the fundamental group of the manifold $M$ up to conjugacy of the base point in $M$.