Normal Tori in $\sharp_n (S^2\times S^1)$

TL;DR

This paper explores the relationship between essential tori and free group splittings in the connected sum of n copies of S^2×S^1, introducing a local minimal intersection concept that extends previous work on spheres.

Contribution

It establishes a correspondence between homotopy classes of embedded essential tori and Z-splittings of the free group, and generalizes minimal intersection notions to tori.

Findings

Established a 1-1 correspondence between tori classes and group splittings.

Defined a local minimal intersection concept for tori.

Extended Hatcher's minimal intersection work from spheres to tori.

Abstract

The fundamental group of $M = \sharp_n (S^2\times S^1)$ is $F_n$, the free group with $n$ generators. There is a 1-1 correspondence between the equivalence classes of $\mathbb{Z}$-- splittings of $F_n$ and homotopy classes of embedded essential tori in $M$. We define and prove a local notion of minimal intersection of a torus with respect to a maximal sphere system in $M$, which generalizes Hatcher's work \cite{H1} on 2-spheres in the same manifold.