# Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot

## TL;DR

This paper constructs infinite families of irreducible 3-manifolds with specific properties that cannot be obtained through Dehn surgery on knots in the 3-sphere, answering a longstanding question.

## Contribution

It provides the first known examples of irreducible 3-manifolds with first Betti number one that are not realizable by knot surgery in the 3-sphere.

## Key findings

- Constructed two infinite families of such manifolds.
- Demonstrated these manifolds are not obtained by knot surgery.
- Included examples of Seifert fibered and non-Seifert fibered manifolds.

## Abstract

We give two infinite families of examples of closed, orientable, irreducible 3-manifolds $M$ such that $b_1(M)=1$ and $\pi_1(M)$ has weight 1, but $M$ is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl and Wilton, and provides the first examples of irreducible manifolds with $b_1=1$ that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.