## Towards an Algebraic Characterization of 3-dimensional Cobordisms. ArXiv: math.GT/0106253

Citations: | 7 - 0 self |

### BibTeX

@MISC{Kerler_towardsan,

author = {Thomas Kerler},

title = {Towards an Algebraic Characterization of 3-dimensional Cobordisms. ArXiv: math.GT/0106253},

year = {}

}

### OpenURL

### Abstract

(To appear in Contemp. Math.) Abstract: The goal of this paper is to find a close to isomorphic presentation of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3-dimensional topology. The first is the category Cob of connected surfaces with one boundary component and 3-dimensional relative cobordisms, the second is a

### Citations

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Citation Context ...Alg . . . . . . . . . . . . . . . . . .22 1. INTRODUCTION For some time is has been a puzzling question whether the appearance of Hopf algebras in the world of quantum invariants of 3-manifolds as in =-=[18]-=- is a lucky coincidence that makes computations work or if these structures arise in more fundamental ways out of 3-dimensional topology. It was soon understood that such algebraic structures are in f... |

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Citation Context ...n. 2π-twists in the framing along a strand are depicted by full or empty blobs as follows: = = = = The admissible tangles are subject to the following relations generalizing Kirby’s calculus of links =-=[14]-=-: 1. A Hopf link that is isolated from the rest of the diagram with one component 0-framed and the other either 1- or 0-framed can be added or removed from a diagram: (6) 0 (7) 2. Any strand R1 can be... |

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Citation Context ...question. Theorem 2 1. Let Γ∗ 1,1 be the central extension of the mapping class group of the torus with one hole. Let A ∈ Alg be the generating Hopf algebra object. Then there exists a homomorphism W =-=[1]-=- : ̂ Γ1,1 → Aut(A), such that the composite Γ ∗ 1,1 W [1] −−−−→ Aut(A) G −−−−→ ̂ Γ1,1 ⊂ Cob is the identity. Here Aut(A) denotes the invertible morphisms A → A in Alg , and we use that Γ∗ 1,1 is ident... |

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Citation Context ...if we have to introduce additional relations in Alg . Among the set of relations there have to be the relations for the mapping class group generators. They have been worked explicitly for example in =-=[19]-=-. Moreover, we need relations expressing the fact that some mapping class group generators can be extended to the full handles, as well as relations for Smale-cancellations of handles. Unfortunately, ... |

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Citation Context ..., 18D35, 57M27. 1The three-dimensional pictures of the cobordisms representing the products and coproducts have been found by Crane and Yetter and proven to satisfy the relevant axioms, see [20] and =-=[4]-=-. The same picture emerged independently in investigations by the author via the route of tangle presentations, see [8]. An interpretation of braided Hopf algebra structures in linear abelian braided ... |

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Citation Context ...ing through that arc and, finally, the outgoing strands are connected through that annulus. j + − j j + j − Let us record also a few moves that are implied by the above. The first two can be found in =-=[6]-=-. The third follows from the second, the 2-handle slide and the boundary move above. 1. The Fenn-Rourke Move in which a bunch of parallel strands are slid over a 1-framed annulus surrounding them. As ... |

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Citation Context ...pf algebra structures in linear abelian braided tensor categories was found by Lyubashenko in [15]. This combined with the equivalence of cobordisms categories with certain tangle categories given in =-=[12]-=- leads to the three dimensional interpretation. The cobordisms are easily worked out to be the same as the ones in [20] and [4], see [8]. The tangle pictures are also used as examples for integrals in... |

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Citation Context ...vestigations by the author via the route of tangle presentations, see [8]. An interpretation of braided Hopf algebra structures in linear abelian braided tensor categories was found by Lyubashenko in =-=[15]-=-. This combined with the equivalence of cobordisms categories with certain tangle categories given in [12] leads to the three dimensional interpretation. The cobordisms are easily worked out to be the... |

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Citation Context ...ensional interpretation. The cobordisms are easily worked out to be the same as the ones in [20] and [4], see [8]. The tangle pictures are also used as examples for integrals in braided categories in =-=[3]-=-. The purpose of this note is to prove a theorem that was already stated in [10], which not only asserts the existence of cobordisms representing structure morphisms such as products and coproducts of... |

17 | Genealogy of nonperturbative quantum-invariants of 3–manifolds: The surgical family, from: “Geometry and physics
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Citation Context ... the ones in [20] and [4], see [8]. The tangle pictures are also used as examples for integrals in braided categories in [3]. The purpose of this note is to prove a theorem that was already stated in =-=[10]-=-, which not only asserts the existence of cobordisms representing structure morphisms such as products and coproducts of a Hopf algebra but also that these generate the entire cobordism category. In m... |

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A geometrical presentation of the surface mapping class group and surgery
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Citation Context ...ich are mapped to the generators in Cob from Corollary 6. The mapping class group generators are as follows: Aj = = j− j+ j− j+ (13) Sj = j− j+ (14) Dj = j− j+ (j+1) + (j+1) − They are the same as in =-=[17]-=-. The other three generators for handle additions and framing shift are as follows. (15) 1 1− + H + 0 H − = = Z = 0 1 + 1 − (16) 84.THE CATEGORY Alg The category Alg is described entirely in algebrai... |

6 | On the Connectivity of Cobordisms, and Half-Projective TQFT’s - Kerler - 1995 |

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Citation Context ...30, 17B37, 18D35, 57M27. 1The three-dimensional pictures of the cobordisms representing the products and coproducts have been found by Crane and Yetter and proven to satisfy the relevant axioms, see =-=[20]-=- and [4]. The same picture emerged independently in investigations by the author via the route of tangle presentations, see [8]. An interpretation of braided Hopf algebra structures in linear abelian ... |

1 |
A Topological Hopf Algebra. Talk presented at
- Kerler
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(Show Context)
Citation Context ...ound by Crane and Yetter and proven to satisfy the relevant axioms, see [20] and [4]. The same picture emerged independently in investigations by the author via the route of tangle presentations, see =-=[8]-=-. An interpretation of braided Hopf algebra structures in linear abelian braided tensor categories was found by Lyubashenko in [15]. This combined with the equivalence of cobordisms categories with ce... |