Trivial 2-cocycles for invariants of mod p homology spheres and Perron's conjecture

Abstract

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.