  In 1996, Strominger, Yau and Zaslow made a conjecture about the geometric
relationship between two mirror Calabi-Yau manifolds. Roughly put, if X and Y
are a mirror pair of such manifolds, then X should possess a special Lagrangian
torus fibration $f:X\to B$ such that Y is obtained by dualizing the fibration
f. This leaves a huge amount to be done to verify such conjectures. This paper
takes a speculative point of view, in that it assumes that a special Lagrangian
torus fibration exists on X. We address a number of questions of a topological
nature: what is the relationship between the cohomology of X and the cohomology
of the dual fibration? what kind of information does the Leray spectral
sequence for f contain? what is the relationship between the topological (1,1)
couplings of the dual of f and the (1,n-1)-couplings of X in the large complex
structure limit? These questions are shown to have nice answers if a key
conjecture about the monodromy diffeomorphisms about a large complex structure
limit point holds. Roughly put, this conjecture says that monodromy about a
large complex structure limit point can be described as a very natural
generalization of a Dehn twist for an elliptic curve. Given this conjecture, we
show, among other results, that the large complex radius limit of the (1,n-1)
couplings on X coincide with the topological (1,1) couplings on Y, and if dim
X=3, the Leray filtration and weight filtrations of the mixed Hodge structure
coincide, as conjectured by myself and P.M.H. Wilson, and independently by D.
Morrison.
