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We study the relation between the set of oriented $\mathbb{Z}/d$-homology$3$-spheres and the level-$d$ mapping class groups, the kernels of thecanonical maps from the mapping class group of an oriented surface to thesymplectic group with coefficients in $\mathbb{Z}/d\mathbb{Z}$. We formulate acriterion to decide whenever a $\mathbb{Z}/d$-homology $3$-sphere can beconstructed from a Heegaard splitting with gluing map an element of thelevel-$d$ mapping class group. Then we give a tool to construct invariants of$\mathbb{Z}/d$-homology $3$-spheres from families of trivial $2$-cocycles onthe level-$d$ mapping class groups. We apply this tool to find all theinvariants of $\mathbb{Z}/p$-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 invariantmodulo a prime $p$ to rational homology $3$-spheres due B. Perron.

Invariants of $\mathbb{Z}/p$-Homology 3-Spheres from the Abelianization of the Level-p Mapping Class Group