Let $S$ be a K3 surface and $\pi:S\rightarrow B$ be a SLAG $T^2$-fibration. I am struggling with a statement "fiberwise dualization does not change the topology of $S$". Why is this true? Here by fiberwise dualization I mean dualizing smooth fibers and compactifying the dual fibration (is a compactification unique?).

This statement is certainly not true in dimension three or higher. Otherwise SYZ mirror conjecture does not produce a mirror Calabi--Yau manifold.

Let $S$ be a K3 surface and $\pi:S\rightarrow B$ be a SLAG $T^2$-fibration. I am struggling with a statement that

Fiberwise dualization does not change the topology of $S$.

Here by fiberwise dualization I mean dualizing smooth fibers and compactifying the dual fibration (I think this is the standard definition).

1. My first question is, what do people mean by "compactification"?

Here is what I naively thought. Let $X$ be a Calabi--Yau n-fold and $\pi:X\rightarrow B$ a SLAG $T^n$-fibration. Let $B_0 \subset B$ be the set on which $\pi$ is smooth. Then $\pi^{-1}(B_0)$ and its dual $\pi^{-1}(B_0)^\vee$ is topologically the same, so we may compactify $\pi^{-1}(B_0)^\vee$ to get the original $X$. This should not happen in the SYZ picture. Of course this only holds at the topological level and ignores complex structure etc.

2. My second question is, what makes difference between dimension two and higher?

The statement above is certainly not true in dimension three or higher. Otherwise SYZ mirror conjecture does not produce a mirror Calabi--Yau manifold, which in general has different topological type from the original one.