A question on the topological change of dualizing a SLAG fibration.  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). 


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*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. 


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*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. 
 A: The crucial point for the second question is the following. In arbitrary dimension, it is not true that $\pi^{-1}(B_0)$ and $\pi^{-1}(B_0)^{\vee}$ are homeomorphic as fibre bundles. This is essentially
because there is no canonical isomorphism between a torus and its dual. If we think of
an $n$-torus as $V/\Lambda$, with $V$ an $n$-dimensional vector space and $\Lambda$
a lattice in $V$, then the dual torus is $V^\vee/\Lambda^\vee\cong H^1(V/\Lambda,R/Z)$, while $V/\Lambda\cong H_1(V/\Lambda,R/Z)$. But in two dimensions, Poincare
duality gives a canonical isomorphism between $H_1(V/\Lambda,R/Z)$ and
$H^1(V/\Lambda,R/Z)$. This canonical isomorphism relativizes, so that if $X_0\rightarrow
B_0$ is a two-torus bundle with section, there is a canonical fibrewise isomorphism
$X_0^{\vee}\cong X_0$.
For the first question as to what "compactiifcation" means, we assume given a torus bundle
$X_0\rightarrow B_0$ with $B_0\subseteq B$ and $B\setminus B_0=\Delta$ relatively nice
(perhaps codimension two). One would then like to extend $X_0\rightarrow B_0$ to a proper
map $X\rightarrow B$ which is sufficiently "nice". Typically our ability to do this depends on the monodromy of the torus bundle $X_0\rightarrow B_0$ near $\Delta$. For example, in the two-dimensional case, if one has monodromy around a point of $\Delta$ (monodromy
acting on $H^1$ of a fibre) which takes the form  $\pmatrix{1&0\cr 1&1\cr}$ in some
basis, we can compactify by adding a pinched torus over the singular point. To do this,
one needs to glue in a local model: see e.g., my paper http://arxiv.org/abs/math/9909015
for details on this kind of thing in dimensions two and three. In general, there is
no canonical compactification, but if the monodromy is sufficiently nice, then there
is a reasonable choice of compactification.
One can also ask for compactifications in various categories, e.g., not just topological
compactifications, but symplectic or complex compactifications if $X_0$ is symplectic
or complex. The symplectic case has been done in dimension three by Castano-Bernard
and Matessi, while the complex case is far more subtle. Discussing this probably goes a bit too far afield for this question.
A: I am not completely sure what is meant by this statement since the topology of the torus fibration certainly changes if the fibration doesn't have a section. If on the other hand we follow your prescription, dualize the smooth fibers, and compactify to a smooth space, then the total space of this new family of tori is a K3 again. And all K3 surfaces are diffeomorphic so the topology does not change. 
Up to a hyper-Kaehler rotation the dualization is an algebraic process - it is the minimal resolution of the relative Jacobian of a holomorphic genus one fibration. It is unique and automatically a K3 hence diffeomorphic to the original K3.
