Contemporary Mathematics
Minimal Page-Genus of Milnor Open Books on Links of
Rational Surface Singularities
Selma Altınok and Mohan Bhupal
1. Introduction
In light of the fundamental theorem of Giroux [9] relating contact structures on
a 3-manifold M to open book decompositions of M, it is interesting to ask what the
minimal page-genus of an open book supporting a given contact structure on a 3-
manifold is. In general, not much is known in answer to this question. However, for
some special classes of contact structures there are some results (see, for example,
[6], [15], [7], [5], [2]). Here we consider the class of Milnor fillable contact structures
on links of rational surface singularities. Since the associated Milnor open books
all support the given contact structure (see [4]), one may consider minimizing the
page-genus over this smaller class of open books. Here we prove the following result.
Theorem 1.1. Let (X, x) be a germ of a normal complex analytic surface
having a rational singularity at x. The page-genus of the associated Milnor open
books OB(f) is minimized when f is taken to be the restriction of a generic linear
form on CN to (X, x) for some/any local embedding of (X, x) in (CN , 0).
A related problem is to try to minimize the sum of the page-genus plus the
number of binding components. Here we have the following result.
Theorem 1.2. With the same hypotheses as Theorem 1.1, the page-genus plus
the number of binding components of OB(f) is minimized when f is taken to be the
restriction of a generic linear form on CN to (X, x) for some/any local embedding
of (X, x) in
(CN
, 0).
2. Preliminaries
2.1. Contact structures and open books. All our manifolds are assumed
to be oriented.
A contact structure on a 3-manifold M is a 2-plane field ξ T M that can be
written as the kernel of a 1-form α
Ω1(M)
such that
α 0.
2000 Mathematics Subject Classification. 57R17, 53D10,32S25, 32S55.
Key words and phrases. Contact structure, Milnor open book, minimal page-genus, rational
singularity, Tjurina component, non-Tjurina component.
1
Contemporary Mathematics
Volume 475, 2008
1
c 0000 (copyright holder) c 2008 American Mathematical Society
http://dx.doi.org/10.1090/conm/475/09272
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