Suppose a complex polynomial $f(x,y,z)$ has a zero and an isolated critical
point at the origin. Then the two-dimensional complex hypersurface
$V=\{f=0\}$ has a singular point at $0$. Topologically, it is the cone
over its neighborhood boundary $\Sigma$, the compact $3$-manifold which is
the intersection of $V$ with a small sphere in $\C^3$. $\Sigma$ may be
understood explicitly by resolution of singularities, and its intrinsic
topology gives much (but not enough!) information about geometry of the
singular point. ``Smoothing" of the singularity takes place by considering
$\{f=\delta\}$, and the change in topology as $\delta$ goes to $0$ is
described by the ``Milnor fibre" $M$ (the intersection of $\{f=\delta\}$
with a small ball); it is a compact $4$-manifold with boundary $\Sigma$.
Milnor's classical work proves that $M$ is simply-connected, and describes
its homotopy type.
Our goal is consider the more general (and richer) situation of a
``smoothing of a normal surface singularity $(V,0)$". The local topology
of $V$ is again determined by its neighborhood boundary $\Sigma$, and a
smoothing (if one exists) gives a $4$-manifold $M$ with boundary $\Sigma$.
The questions and techniques which arise cover diverse areas in geometry,
including: Fintushel-Stern'
s ``rational blow-down" in symplectic geometry; construction of new
algebraic surfaces of general type with $p_g=0$; and log-terminal
singularities, introduced by Koll\'ar and Mori. Of special interest in
geometry will be the short list of examples of singular points possessing
smoothings for which the Milnor fibre is particularly ``simple", i.e. has
the same rational homology as a disk.
These lecture will be as self-contained as possible; we will assume
essentially no prior knowledge of the subject.
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