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## On the geometric and algebraic rank of graph manifolds (2003)

Citations: | 16 - 1 self |

### Citations

501 |
The geometry of 3-manifolds
- Scott
(Show Context)
Citation Context ...n. We start by discussing the possible pseudohorizontal surfaces in the relevant Seifert manifolds. Some proofs rely on the theory of 2-dimensional orbifolds and their covering theory as discussed in =-=[11]-=-. These lemmata will be used in our discussion of Heegaard splittings and their untelescopings. But many of these results are more general. We do not necessarily require S to be the splitting surface ... |

67 |
Heegaard splittings of Seifert fibered spaces
- Moriah
- 1988
(Show Context)
Citation Context ...njecture. First counterexamples however were found by M. Boileau and H. Zieschang [1]. These examples were Seifert fibered manifolds with g(M) = 3 and r(M) = 2. The work of Y. Moriah and J. Schultens =-=[4]-=- further shows that this class extends to higher genus examples, i.e. Seifert manifolds with g(M) = n + 1 and r(M) = n. In [13] a class of graph manifolds was found for which g(M) = 3 and r(M) = 2. Th... |

48 |
Heegaard genus of closed orientable Seifert 3-manifolds
- Boileau, Zieschang
- 1984
(Show Context)
Citation Context ...er the converse inequality also holds, i.e., whether g(M) = r(M). A positive answer would have implied the Poincaré conjecture. First counterexamples however were found by M. Boileau and H. Zieschang =-=[1]-=-. These examples were Seifert fibered manifolds with g(M) = 3 and r(M) = 2. The work of Y. Moriah and J. Schultens [4] further shows that this class extends to higher genus examples, i.e. Seifert mani... |

42 |
Thin position for 3-manifolds
- Scharlemann, Thompson
- 1992
(Show Context)
Citation Context ...r Si and Fi cobound a submanifold homeomorphic to Si ×I with 2-handles attached to Si ×{1}. In particular, χ(Si) < χ(Fi). Similarly for Fi and Si+1. The following theorem summarizes the discussion in =-=[7]-=-, [8] and [6, Lemma 2]. Theorem 3. Let M be a 3-manifold and M = V ∪S W a Heegaard splitting. Then M = V ∪S W has a strongly irreducible untelescoping S1, F1, S2, F2, . . . , Sn. Furthermore, −χ(S) = ... |

31 |
Structures of the Haken manifolds with Heegaard splittings of genus two
- Kobayashi
- 1984
(Show Context)
Citation Context ...ds constructed in [13] are not of Heegaard genus 2 relies on the classification of 3-manifolds with nonempty characteristic submanifold that have a genus 2 Heegaard splitting as given by T. Kobayashi =-=[3]-=-. Thus we conjecture that the manifolds Mn constructed below are of Heegaard genus 3n, the same argument as above shows that they can be generated by 2n elements. The graph underlying the manifold Mn ... |

29 |
Seifert fibered spaces
- Jaco, Shalen
- 1979
(Show Context)
Citation Context ...tting of genus 2g + 1 if and only if one of the following holds: (1) N2 is the exterior of a s-bridge knot with s ≤ 2g + 1 and the fiber of N1 is identified with the meridian of N2, i.e. ˆ N2 = S 3 . =-=(2)-=- ˆ N1 admits a horizontal Heegaard splitting of genus 2g. We will further see that all manifolds of this type admit a Heegaard splitting of genus 2g + 2. Furthermore, most of these manifolds do not ad... |

27 |
Heegaard splittings of compact 3–manifolds, Handbook of Geometric Topology
- Scharlemann
- 2002
(Show Context)
Citation Context ...and Fi cobound a submanifold homeomorphic to Si ×I with 2-handles attached to Si ×{1}. In particular, χ(Si) < χ(Fi). Similarly for Fi and Si+1. The following theorem summarizes the discussion in [7], =-=[8]-=- and [6, Lemma 2]. Theorem 3. Let M be a 3-manifold and M = V ∪S W a Heegaard splitting. Then M = V ∪S W has a strongly irreducible untelescoping S1, F1, S2, F2, . . . , Sn. Furthermore, −χ(S) = n� (χ... |

9 |
Some problems on 3-manifolds.
- Waldhausen
- 1978
(Show Context)
Citation Context ... g elements. Thus g(M) ≥ r(M) where r(M) denotes the minimal number of generators of π1(M). Sometimes we will refer to g(M) as the geometric rank and to r(M) as the algebraic rank of M. F. Waldhausen =-=[12]-=- asked whether the converse inequality also holds, i.e., whether g(M) = r(M). A positive answer would have implied the Poincaré conjecture. First counterexamples however were found by M. Boileau and H... |

9 |
Some 3-manifolds with 2-generated fundamental group
- Weidmann
- 1912
(Show Context)
Citation Context ...ifolds with g(M) = 3 and r(M) = 2. The work of Y. Moriah and J. Schultens [4] further shows that this class extends to higher genus examples, i.e. Seifert manifolds with g(M) = n + 1 and r(M) = n. In =-=[13]-=- a class of graph manifolds was found for which g(M) = 3 and r(M) = 2. The original Boileau-Zieschang examples can be interpreted as a special case of these graph manifolds. We here show how the pheno... |

7 | Meridional generators and plat presentations of torus links
- Rost, Zieschang
- 1987
(Show Context)
Citation Context ...m van Kampen’s theorem that π1(M) = π1(N1) ∗C π1(N2) with C ∼ = Z 2 . Note that f1 = xyf l 2 for some l ∈ Z as we assume that the intersection number between f1 and f2 is 1. A simple calculation (see =-=[5]-=-) shows that n = min(p, q) conjugates of f1 generate a subgroup of π1(N2) that maps surjectively onto the orbifold group π1(D(p, q)). We do however need something slightly stronger: Claim: We can choo... |

7 | Heegaard splittings of graph manifolds
- Schultens
(Show Context)
Citation Context ...ndamental group that has Heegaard genus 4n. This paper is organized as follows. In Section 3 we review the structure theorem for Heegaard splittings of totally orientable graph manifolds as proven in =-=[9]-=-. Then we study in more detail how Heegaard surfaces can intersect the Seifert pieces that are the building blocks of our examples. In Section 5 and Section 6 we will then give the proofs of Theorem 1... |

4 | The Classification of Heegaard splittings for (closed orientable surface - Schultens - 1993 |

3 | Tunnel number of the sum of n knots is at least n - Scharlemann, Schultens |