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

TL;DR

This paper explores invariants of -homology 3-spheres derived from the abelianization of the level- mapping class group, providing new tools and disproving a conjecture related to the Casson invariant extension.

Contribution

It introduces a criterion for constructing -homology 3-spheres from level- mapping class groups and develops invariants from 2-cocycles on their abelianizations.

Findings

Classified all invariants of -homology 3-spheres from abelianized level- groups for prime p.

Disproved the conjecture extending the Casson invariant modulo p to rational homology 3-spheres.

Provided a new method to construct invariants using trivial 2-cocycles.

Abstract

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 the canonical maps from the mapping class group of an oriented surface to the symplectic group with coefficients in $\mathbb{Z}/d\mathbb{Z}$. We formulate a criterion to decide whenever a $\mathbb{Z}/d$-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 $\mathbb{Z}/d$-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 $\mathbb{Z}/p$-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 invariant modulo a prime $p$ to rational homology $3$-spheres due B. Perron.