论文标题
Mod P同源球和Perron的猜想的琐碎的2个循环
Trivial 2-cocycles for invariants of mod p homology spheres and Perron's conjecture
论文作者
论文摘要
本文的主要目标是解决Perron的猜想。该猜想肯定在mod p torelli组上具有z/p值的某些函数是mod p同源性3键的不变性。为了解决这一猜想，在本论文中，我们首先研究了Mod P同源性3键，理性同源性3 spheres以及可以用粘合图的Heegaard分割的spheres的元素是mod p torelli组的元素。特别是，我们给出一个标准，以确定每当有理性同源性3-Sphere都具有heegaard分裂和粘合图的Heegaard分裂为torelli群体mod p的元素，并且使用此标准，我们证明，并非所有MOD P同源性3个spheres都可以以这种方式实现。接下来，我们将文章“琐碎的共同体和同源性3-Spheres”的不变性扩展到不限制的Abelian群体中，从Torelli组的合适的2个循环系列中获取了一个不变性的构造。特别是，我们解释了Rohlin在这种结构中失去独特性的影响。稍后，使用相同的工具，我们获得了一个不变的理性同源性领域的构造，这些领域的heegaard分裂，粘合图是Mod P Torelli组的一个元素，从Mod P Torelli组的合适的2个循环中的家族中，在Mod P Torelli群体上似乎是Mod P Polyology Spherers不断出现的，这些球员在文献中并没有在文献中表现出相同的作用，而这些角色扮演着rohnlin的作品，这些作品均在Rohlan中扮演的作品。最后，我们证明了Perrron的猜想是错误的，这提供了共同体障碍，这是因为第一个特征性的表面束减少了模量P的特征类别不会消失。
The main target of this thesis is to solve the Perron's conjecture. This conjecture affirms that some function on the mod p Torelli group, with values in Z/p, is an invariant of mod p homology 3-spheres. In order to solve this conjecture, in this thesis we first study the mod p homology 3-spheres, the rational homology 3-spheres and those that can be realized as a Heegaard splitting with gluing map an element of the mod p Torelli group. In particular we give a criterion to determine whenever a rational homology 3-sphere has a Heegaard splitting with gluing map an element of the Torelli group mod p, and using this criterion we prove that not all mod p homology 3-spheres can be realized in such way. Next, we extend the results of the article ''Trivial cocycles and invariants of homology 3-spheres'' obtaining a construction of invariants with values to an abelian group without restrictions, from a suitable family of 2-cocycles on the Torelli group. In particular, we explain the influence of the invariant of Rohlin in the lost of uniqueness in such construction. Later, using the same tools, we obtain a construction of invariants of rational homology spheres that have a Heegaard splitting with gluing map an element of the mod p Torelli group, from a suitable family of 2-cocycles on the mod p Torelli group where appears an invariant of mod p homology spheres which does not appear in the literature, who plays the same role that Rohlin invariant in the lost of uniqueness of our construction. Finally, we prove that Perrron's conjecture is false providing a cohomological obstruction that is given by the fact that the first characteristic class of surface bundles reduced modulo p does not vanish.
