NORMAL TORI IN #_n(S2xS1) AND THE DEHN TWIST AUTOMORPHISMS OF THE FREE GROUP

Abstract

The 3-dimensional space which renders a Out(F_n) action is M = #_n(S2x S1). The relation between M and Out(F_n) is that the latter is isomorphic to the mapping class group of M up to rotations about 2-spheres in M.

Associated to M is a rich algebraic structure coming from the essential 2-spheres that M contains. Inspired by this and combining the work of Whitehead with that of Laudenbach, Hatcher defined the notion of normal form with respect to a fixed sphere system and proved the existence of normal representatives of spheres in a given isotopy class of spheres in M. This is a local notion of minimal intersection of a sphere system with respect to a maximal sphere system in M.

In this work, a notion of being normal for tori in #_n(S2 xS1) is defined. This notion is crucial to determine minimality of intersections between tori and between spheres and tori. We prove two theorems regarding existence and uniqueness of normal representatives in a given homotopy class of tori. Then we define criteria for minimal intersection in a local sense and prove that a normal representative from a given homotopy class of tori satisfies it.

Just as there is a 1-1 correspondence between the equivalence classes of free splittings of the free group and the isotopy classes of embedded essential spheres in M, we prove that there is a 1-1 correspondence between the equivalence classes of Z- splittings of F_n and homotopy classes of embedded essential tori in M. This gives us the opportunity to understand Dehn twist automorphisms of the free group, since they are

defined with respect to Z- splittings. To this end, we define Dehn twist along a torus in M using the mapping classes of M and describe these twists with respect to their actions on the universal cover of M.

In addition, we give the motivation behind this work by stating possible applications and reasons for the importance of studying tori in this manifold.