Affine Hirsch foliations on 3–manifolds

Yu, Bin

doi:10.2140/agt.2017.17.1743

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Affine Hirsch foliations on $3$–manifolds

Bin Yu

Algebraic & Geometric Topology 17 (2017) 1743–1770

DOI: 10.2140/agt.2017.17.1743

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 $n \in ℕ$ , 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.

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Keywords

affine Hirsch foliation, classification, exchangeable braid

Mathematical Subject Classification 2010

Primary: 57M50, 57R32

Secondary: 37E10, 57M25

References

Publication

Received: 16 April 2016

Revised: 12 October 2016

Accepted: 18 December 2016

Published: 17 July 2017

Authors

Bin Yu
School of Mathematical Sciences Tongji University 200092 Shanghai China

Volume 17, issue 3 (2017)