Invariants of Z/p-homology 3-spheres from the abelianization of the level-p mapping class group.

We study the relation between the set of oriented Z/d-homology 3-spheres and the level-d mapping class groups, the kernels of the canonical maps from the mapping class group of an oriented surface to the symplectic group with coefficients in Z/dZ. We formulate a criterion to decide whenever a Z/d-homology 3-sphere can be constructed from a Heegaard splitting with gluing map an element of the level-d mapping class group. Then, we give a tool to construct invariants of Z/d-homology 3-spheres from families of trivial 2-cocycles on the level-d mapping class groups. We apply this tool to find all the invariants of Z/p-homology 3-spheres constructed from families of 2-cocycles on the abelianization of the level-p mapping class group with p prime and to disprove the conjectured extension of the Casson invariant modulo a prime p to rational homology 3-spheres due to B. Perron. [ABSTRACT FROM AUTHOR]