Cobordism范畴的桥接和缠结表示
Bridged Links and Tangle Presentations of Cobordism Categories
摘要：
我们开发了一个演算的手术数据，称为桥接链接，其中包括除了链接也对球，描述一个手柄附件。与Kirby等人通常的链接演算相反，这个描述只使用了基本的局部移动（即修改和孤立的取消），并且在非单连通和不连通流形上也有效。特别是，它允许我们通过对具有相同边界的任何其他3-流形进行手术来给出3-流形的表示。本文利用平面带状缠结及其组合规则，对2+1维协边范畴的表示进行了Cerf理论推导。作为一个应用程序，我们给出了一个不同的，更自然的证明的Matveev-Polyak介绍的映射类群，并进一步发现系统的手术介绍一般映射环面。我们讨论了一个自然扩张的ReshetikhinTuraev不变的演算桥接链接。不变性如下-类似于纽结不变量-从简单的标识的基本移动与基本范畴关系的不变性或协积分，分别。因此，我们避免了冗长的计算和不自然的Fenn-Rourke减少原来的证明。此外，我们能够从一个弱得多的“模块性”条件开始，这意味着Turaev的条件。概括的介绍，以配边的表面与边界的概述。
英文摘要：
We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is valid also on non-simply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3-manifold by doing surgery on any other 3-manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerf-theoretical derivation of presentations of 2+1-dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the Matveev-Polyak presentations of the mapping class group, and, furthermore, find systematically surgery presentations of general mapping tori. We discuss a natural extension of the Reshetikhin Turaev invariant to the calculus of bridged links. Invariance follows now - similar as for knot invariants - from simple identifications of the elementary moves with elementary categorial relations for invariances or cointegrals, respectively. Hence, we avoid the lengthy computations and the unnatural Fenn-Rourke reduction of the original proofs. Moreover, we are able to start from a much weaker ``modularity''-condition, which implies the one of Turaev. Generalizations of the presentation to cobordisms of surfaces with boundaries are outlined.
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