Trivial cocycles and invariants of homology 3-spheres (Q957982)

The author studies the relationship between trivial 2-cocycles on the Torelli groups \(T_{g,1}\) of surfaces \(\Sigma_g\) with genus \(g \geq 3\) and invariants of oriented integral homology 3-spheres. A necessary and sufficient condition is given for such a family of cocycles satisfying certain conditions on the Torelli groups to provide a compatible family of trivializations of the Torelli groups which produces invariants of homology 3-spheres. This result is applied to give a purely algebraic construction of the Casson invariant, and a new torsion cohomology class in \(H^2(T_{g,1};\mathbb Z)\) is obtained by the pull back of a certain unique 2-cocycle \(J_g\) on \(\Lambda^3H_1(\Sigma_g;\mathbb Z)\) via the Johnson homomorphism, for which the Casson invariant is the invariant associated with the cocycles \(-2J_g\).