Church–Farb–Putman high-dimensional vanishing conjecture for mapping class groups

Determine whether, for integers g and i with g sufficiently large relative to i, the rational cohomology groups H^{4g−5−i}(MCG(Σ_g); Q) and H^{4g−5−i}(M_g; Q) vanish.

Background

The mapping class group MCG(Σ_g) is a virtual duality group with vcd(MCG(Σ_g)) = 4g−5, and its rational cohomology matches that of the moduli space M_g. Church–Farb–Putman proposed an analogue of their SL_n(Z) conjecture in this setting, predicting vanishing of high-dimensional cohomology in low codimension.

While the codimension i = 0 case was proved, subsequent work constructed many non-trivial high-dimensional classes for M_g, showing the conjecture is false in numerous codimensions; the quoted formulation records the original conjecture.