# Non-prime 3-Manifolds with Open Book Genus Two

## TL;DR

This paper characterizes non-prime 3-manifolds with open book genus two, showing they are connected sums of lens spaces with open book genus one, supporting the conjecture that open book genus is additive under connected sum.

## Contribution

It proves that non-prime 3-manifolds with open book genus two are decomposable into prime pieces of open book genus one, confirming additivity in this case.

## Key findings

- Non-prime 3-manifolds with open book genus 2 are connected sums of lens spaces with genus 1.
- Supports Ozbagci's conjecture on the additivity of open book genus.
- No counterexamples exist with connected sum of open book genus 2.

## Abstract

An open book decomposition of a 3-manifold $M$ induces a Heegaard splitting for $M$, and the minimal genus among all Heegaard splittings induced by open book decompositions is called the \emph{open book genus} of $M$. It is conjectured by Ozbagci \cite{O} that the open book genus is additive under the connected sum of 3-manifolds. In this paper, we prove that a non-prime 3-manifold which has open book genus 2 is homeomorphic to $L(p,1)\#L(q,1)$ for some integers $p,q\neq\pm1$, that is, it has non-trivial prime pieces of open book genus 1. In particular, there cannot be a counter-example to additivity of the open book genus such that the connected sum has open book genus 2.