利用广义正则亏格和G-度分类紧致4-流形
Classifying compact 4-manifolds via generalized regular genus and G-degree
摘要：
（d+1）-有色图，即（d+1）-正则的边有色图，已经被证明是紧致PL d-流形的一个有用的表示工具，从而扩展了最初为闭情形发展的理论（称为结晶理论）.在这种情况下，组合定义的PL不变量发挥了相关的作用。本论文特别关注广义正则亏格和G度：第一个扩展到更高的维度的Heegaard亏格的3-流形的经典概念，第二个出现，在理论物理学中，从随机张量理论作为一种方法，量子引力在大于2维。我们建立了几个一般性的结果有关的两个不变量，与不变量的边界和与秩的基本群，以及他们的行为就连接和。我们还计算了一些有趣的紧致d-流形类的广义正则亏格和G-度，如S ~ 2上的D^2-丛、闭体、闭流形与区间的乘积。本文的主要结果涉及到4维，我们得到了广义正则亏格至多为1的紧致PL流形和G度至多为18的紧致PL流形的分类;此外，在空边界或连通边界的情况下，我们将分类推广到广义正则亏格2和G度24。
英文摘要：
(d+1)-colored graphs, i.e. edge-colored graphs that are (d+1)-regular, have already been proved to be a useful representation tool for compact PL d-manifolds, thus extending the theory (known as crystallization theory) originally developed for the closed case. In this context, combinatorially defined PL invariants play a relevant role. The present paper focuses in particular on generalized regular genus and G-degree: the first one extending to higher dimension the classical notion of Heegaard genus for 3-manifolds, the second one arising, within theoretical physics, from the theory of random tensors as an approach to quantum gravity in dimension greater than two. We establish several general results concerning the two invariants, in relation with invariants of the boundary and with the rank of the fundamental group, as well as their behaviour with respect to connected sums. We also compute both generalized regular genus and G-degree for interesting classes of compact d-manifolds, such as handlebodies, products of closed manifolds by the interval and D^2-bundles over S^2. The main results of the paper concern dimension 4, where we obtain the classification of all compact PL manifolds with generalized regular genus at most one, and of all compact PL manifolds with G-degree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to G-degree 24.
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