  The holographic prescription for computing entanglement entropy requires that
the bulk extremal surface, whose area encodes the amount of entanglement,
satisfies a homology constraint. Usually this is stated as the requirement of a
(spacelike) interpolating surface that connects the region of interest and the
extremal surface. We investigate to what extent this constraint is upheld by
the generalized gravitational entropy argument, which relies on constructing
replica symmetric q-fold covering spaces of the bulk, branched at the extremal
surface. We prove (at the level of topology) that the putative extremal surface
satisfies the homology constraint if and only if the corresponding branched
cover can be constructed for every positive integer q. We give simple examples
to show that homology can be violated if the cover exists for some values of q
but not others, along with some other issues.
