## Algebraic & Geometric Topology

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

Bin Yu

#### 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 $n ∈ ℕ$, 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
Revised: 12 October 2016
Accepted: 18 December 2016
First available in Project Euclid: 16 November 2017

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

#### 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

#### References

• S Alvarez, P Lessa, The Teichmüller space of the Hirsch foliation, preprint (2015)
• D Bennequin, Entrelacements et équations de Pfaff, from “Third Schnepfenried geometry conference”, Astérisque 107, Soc. Math. France, Paris (1983) 87–161
• J,S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press (1974)
• A Biś, S Hurder, J Shive, Hirsch foliations in codimension greater than one, from “Foliations 2005” (P Walczak, R Langevin, S Hurder, T Tsuboi, editors), World Sci., Hackensack, NJ (2006) 71–108
• A Constantin, B Kolev, The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math. 40 (1994) 193–204
• A Fathi, F Laudenbach, V Poénaru, Travaux de Thurston sur les surfaces, Astérisque 66, Soc. Math. France, Paris (1979)
• E Ghys, Topologie des feuilles génériques, Ann. of Math. 141 (1995) 387–422
• M Hirsch, A stable analytic foliation with only exceptional minimal sets, from “Dynamical systems” (A Manning, editor), Lecture Notes in Math. 468, Springer (1975) 8–9
• W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, Amer. Math. Soc., Providence, RI (1980)
• H,R Morton, Exchangeable braids, from “Low-dimensional topology” (R Fenn, editor), London Math. Soc. Lecture Note Ser. 95, Cambridge Univ. Press (1985) 86–105
• D Rolfsen, Knots and links, corrected reprint of 1st edition, Math. Lect. Ser. 7, Publish or Perish, Houston, TX (1990)
• J,H Shive, Conjugation problems for Hirsch foliations, PhD thesis, University of Illinois at Chicago (2005) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305371555 {\unhbox0
• J,R Stallings, Constructions of fibred knots and links, from “Algebraic and geometric topology, 2” (R,J Milgram, editor), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, RI (1978) 55–60
• W,P Thurston, Hyperbolic structures on $3$–manifolds, II: Surface groups and $3$–manifolds which fiber over the circle, preprint (1998)
• S,C Wang, $3$–manifolds which cover only finitely many $3$–manifolds, Quart. J. Math. Oxford Ser. 42 (1991) 113–124
• B Yu, Smale solenoid attractors and affine Hirsch foliations, Ergodic Theory and Dynamical Systems (online publication May 2017) 1–23