Given a Heegaard splitting of a closed, connected, oriented three-manifold, Abouzaid and Manolescu defined a perverse sheave on the character variety of the fundamental group as an invariant for the three-manifold. They showed that it is invariant under stabilization and independent of a sequence of moves.

 

Ref:

M. Abouzaid and C. Manolescu “ A sheaf theoretic model for SL(2,C) Floer homology”