Algebraic & Geometric Topology

$3$–manifolds built from injective handlebodies

James Coffey and Hyam Rubinstein

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Abstract

This paper studies a class of closed orientable 3–manifolds constructed from a gluing of three handlebodies, such that the inclusion of each handlebody is π1–injective. This construction is the generalisation to handlebodies of the construction for gluing three solid tori to produce non-Haken Seifert fibred 3–manifolds with infinite fundamental group. It is shown that there is an efficient algorithm to decide if a gluing of handlebodies satisfies the disk-condition. Also, an outline for the construction of the characteristic variety (JSJ decomposition) in such manifolds is given. Some non-Haken and atoroidal examples are given.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 6 (2017), 3213-3257.

Dates
Received: 21 February 2006
Revised: 19 January 2017
Accepted: 1 March 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841506

Digital Object Identifier
doi:10.2140/agt.2017.17.3213

Mathematical Reviews number (MathSciNet)
MR3709646

Zentralblatt MATH identifier
1380.57023

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M10: Covering spaces 57M50: Geometric structures on low-dimensional manifolds

Keywords
3–manifolds handlebodies infinite fundamental group non-Haken

Citation

Coffey, James; Rubinstein, Hyam. $3$–manifolds built from injective handlebodies. Algebr. Geom. Topol. 17 (2017), no. 6, 3213--3257. doi:10.2140/agt.2017.17.3213. https://projecteuclid.org/euclid.agt/1510841506


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References

  • I R Aitchison, J H Rubinstein, Localising Dehn's lemma and the loop theorem in $3$–manifolds, Math. Proc. Cambridge Philos. Soc. 137 (2004) 281–292
  • M Freedman, J Hass, P Scott, Closed geodesics on surfaces, Bull. London Math. Soc. 14 (1982) 385–391
  • M Freedman, J Hass, P Scott, Least area incompressible surfaces in $3$–manifolds, Invent. Math. 71 (1983) 609–642
  • M Hall, Jr, Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949) 421–432
  • J Hempel, $3$–Manifolds, Ann. of Math. Studies 86, Princeton Univ. Press (1976)
  • W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, Amer. Math. Soc., Providence, RI (1980)
  • W Jaco, P B Shalen, A new decomposition theorem for irreducible sufficiently-large $3$–manifolds, from “Algebraic and geometric topology, part 2” (R J Milgram, editor), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, RI (1978) 71–84
  • K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Mathematics 761, Springer (1979)
  • S Matveev, Algorithmic topology and classification of $3$–manifolds, Algorithms and Computation in Mathematics 9, Springer (2003)
  • U Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984) 209–230
  • D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Berkeley, CA (1976)
  • C P Rourke, B J Sanderson, Introduction to piecewise-linear topology, Ergeb. Math. Grenzgeb. 69, Springer (1972)
  • P Scott, There are no fake Seifert fibre spaces with infinite $\pi \sb{1}$, Ann. of Math. 117 (1983) 35–70