2 SELMA ALTINOK AND MOHAN BHUPAL
An open book decomposition of a 3-manifold M is a pair (N, θ), where N is a link,
that is, a not necessarily connected 1-dimensional submanifold, and θ : M \N S1
is a smooth fibration over the circle which agrees with the angular coordinate in
a tubular neighbourhood D2 × N of N. The link N is called the binding of the
open book and the compact surfaces θ−1(t) N, for t S1, are called the pages of
the open book. The pages are naturally co-oriented by and hence are naturally
oriented (since we are assuming M is oriented). We assume that the boundary
orientation induced by the pages on N coincides with the given orientation on N.
An open book decomposition (N, θ) of a 3-manifold M is said to support a
contact structure ξ on M if there is a contact 1-form α for ξ such that α is positive
on N and is positive on the pages of (N, θ). The following fundamental result
is due to Giroux (see [9]).
Theorem 2.1. Every contact structure ξ on a 3-manifold M is supported by
an open book decomposition of M. If two contact structures on a 3-manifold are
supported by the same open book decomposition, then they are isotopic.
For an open book OB, we will denote by genus(OB) the genus of a page, or
page-genus, of the open book. (Recall that the genus of a compact surface S with
boundary is defined to be the genus of the associated closed surface
ˆ
S obtained
from S by sewing a disc onto each boundary circle.)
Remark 2.2. In [8], the minimal page-genus of a supporting open book was
already defined as a possible complexity of a contact structure.
2.2. The semigroup E +.
Let (X, x) be a germ of a normal analytic surface
having a singularity at x. Denote by X a sufficiently small representative of (X, x).
Fix a resolution π :
˜
X X of (X, x) and denote the irreducible components of the
exceptional divisor E = π−1(x) by
n
i=1
Ei.
As in [11], let E
+
denote the set of nonzero effective divisors Y =

miEi
supported on E such that Y · Ei 0 for all i. By Zariski (see [19]), this set is
nonempty. If Y =

miEi, Y =

miEi are elements of E
+,
then one has the
following:
(i) Y + Y E
+,
and therefore E
+
is a semigroup under addition;
(ii) min(Y, Y ) :=

min(mi, m )Ei E
+.
In general, for a divisor Y =
∑i
miEi, say that mi is the multiplicity of Ei in Y
and write mi = multY Ei. The set E
+
is partially ordered by Y Y if multY Ei
multY Ei for all i. The least element of E
+,
which exists by (ii), is called the
fundamental cycle of E and will be denoted Z(E). As any connected reduced
divisor F whose support is included in E is again the exceptional divisor of a
resolution of a normal surface singularity, one can also define its fundamental cycle
Z(F ).
Definition 2.3. The singularity at x of the germ (X, x) is called rational if
each irreducible component Ei of the exceptional divisor E is isomorphic to P1 and
Z · Z +
∑n
i=1
zi(−Ei 2 2)
2
+ 1 = 0,
where Z =

ziEi is the fundamental cycle of E (compare [16], Ch. I, Def. 1.9).
Remark 2.4. The initial definition of a rational singularity is that h1(O
˜
X
) = 0
for some resolution of singularities π :
˜
X X. In [1], M. Artin showed that this
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