## Algebraic & Geometric Topology

- Algebr. Geom. Topol.
- Volume 17, Number 3 (2017), 1743-1770.

### Affine Hirsch foliations on $3$–manifolds

#### Abstract

This paper is devoted to discussing affine Hirsch foliations on $3$–manifolds. First, we prove that up to isotopic leaf-conjugacy, every closed orientable $3$–manifold $M$ admits zero, one or two affine Hirsch foliations. Furthermore, every case is possible.

Then we analyze the $3$–manifolds admitting two affine Hirsch foliations (we call these *Hirsch manifolds*). On the one hand, we construct Hirsch manifolds by using exchangeable braided links (we call such Hirsch manifolds *DEBL Hirsch manifolds*); on the other hand, we show that every Hirsch manifold virtually is a DEBL Hirsch manifold.

Finally, we show that for every $n\in \mathbb{N}$, there are only finitely many Hirsch manifolds with strand number $n$. Here the strand number of a Hirsch manifold $M$ is a positive integer defined by using strand numbers of braids.

#### Article information

**Source**

Algebr. Geom. Topol., Volume 17, Number 3 (2017), 1743-1770.

**Dates**

Received: 16 April 2016

Revised: 12 October 2016

Accepted: 18 December 2016

First available in Project Euclid: 16 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.agt/1510841408

**Digital Object Identifier**

doi:10.2140/agt.2017.17.1743

**Mathematical Reviews number (MathSciNet)**

MR3677938

**Zentralblatt MATH identifier**

1376.57018

**Subjects**

Primary: 57M50: Geometric structures on low-dimensional manifolds 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology [See also 58H10]

Secondary: 37E10: Maps of the circle 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

**Keywords**

affine Hirsch foliation classification exchangeable braid

#### Citation

Yu, Bin. Affine Hirsch foliations on $3$–manifolds. Algebr. Geom. Topol. 17 (2017), no. 3, 1743--1770. doi:10.2140/agt.2017.17.1743. https://projecteuclid.org/euclid.agt/1510841408