The Complex of Non-separating embedded spheres

TL;DR

This paper demonstrates that the automorphism group of the complex of non-separating embedded spheres in a specific 3-manifold is isomorphic to the outer automorphism group of a free group, revealing a deep connection between topology and algebra.

Contribution

It establishes an isomorphism between the automorphism group of a sphere complex in a 3-manifold and the outer automorphism group of a free group, extending understanding of manifold automorphisms.

Findings

Aut(NS(M)) is isomorphic to Out(F_n) for n > 2

The complex NS(M) encodes algebraic automorphism structures

Provides a topological model for Out(F_n)

Abstract

For n >2, we shall show that the group Aut(NS(M)) of simplicial automorphisms of the complex NS(M) of non-separating embedded spheres in the manifold M,connected sum of n copies of S^2 X S^1, isomorphic to the group Out(F_n) of outer automorphisms of the free group F_n, where $F_n$ is identified with the fundamental group of M up to conjugacy of the base point in M.