Trivial 2-cocycles for invariants of mod p homology spheres and Perron's conjecture
24 Feb 2020
•
Riba
•
Ricard
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.(read more)