  We consider bilayer graphene in the presence of spin orbit coupling, to
assess its behavior as a topological insulator. The first Chern number $n$ for
the energy bands of single and bilayer graphene is computed and compared. It is
shown that for a given valley and spin, $n$ in a bilayer is doubled with
respect to the monolayer. This implies that bilayer graphene will have twice as
many edge states as single layer graphene, which we confirm with numerical
calculations and analytically in the case of an armchair terminated surface.
Bilayer graphene is a weak topological insulator, whose surface spectrum is
susceptible to gap opening under spin-mixing perturbations. We also assess the
stability of the associated topological bulk state of bilayer graphene under
various perturbations. Finally, we consider an intermediate situation in which
only one of the two layers has spin orbit coupling, and find that although
individual valleys have non-trivial Chern numbers, the spectrum as a whole is
not gapped, so that the system is not a topological insulator.
