  Suppose that X is a Fano manifold that corresponds under Mirror Symmetry to a
Laurent polynomial f, and that P is the Newton polytope of f. In this setting
it is expected that there is a family of algebraic varieties over the unit disc
with general fiber X and special fiber the toric variety defined by the
spanning fan of P. Building on recent work and conjectures by
Corti--Hacking--Petracci, who construct such families of varieties, we
determine the topology of the general fiber from combinatorial data on P. This
provides evidence for the Corti--Hacking--Petracci conjectures, and verifies
that their construction is compatible with expectations from Mirror Symmetry.
