  This paper focuses on a topological version on the Strominger-Yau-Zaslow
mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that
mirror pairs of Calabi-Yau manifolds are related by the existence of dual
special Lagrangian torus fibrations. We explore this conjecture without
reference to the special Lagrangian condition. In this setting, natural
questions include: does there exist a nice class of T^3-fibrations for which
the dual fibration can be constructed? Do such fibrations exist on a manifold
such as the quintic threefold? If so, is the dual fibration the mirror?
  We answer these questions affirmatively. We introduce a class of topological
T^3-fibrations for which duals can be constructed, including over the singular
fibres. We then construct such a fibration on the quintic threefold, and show
that by applying this general dualizing construction to this particular case,
one obtains the mirror quintic. Thus we have constructed the mirror quintic
topologically with no a priori knowledge of the mirror. This shows that in a
non-degenerate (and representative) case, the Strominger-Yau-Zaslow conjecture
correctly explains mirror symmetry.

Algebraic Geometry;Differential Geometry