  Motivated by physical constructions of homological knot invariants, we study
their analogs for closed 3-manifolds. We show that fivebrane compactifications
provide a universal description of various old and new homological invariants
of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by
the Q-cohomology of the Hilbert space of partially topologically twisted 3d N=2
theory T[M_3] on a Riemann surface with defects. We demonstrate this by
concrete and explicit calculations in the case of monopole/Heegaard Floer
homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter
gives a categorification of Chern-Simons partition function. Some of the new
key elements include the explicit form of the S-transform and a novel
connection between categorification and a previously mysterious role of Eichler
integrals in Chern-Simons theory.
