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 dθ 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 dα 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

2

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 dθ 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 dα 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

2