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GRAPH 3-MANIFOLDS, SPLICE DIAGRAMS, SINGULARITIES GRAPH 3-MANIFOLDS, SPLICE DIAGRAMS, SINGULARITIES
WALTER D. NEUMANN
A
BSTRACT
. We describe how a coarse classication of graph manifolds can
give clearer insight into their structure, and we relate this particularly to the man-
ifolds that can occur as the links of points in normal complex surfaces. We relate
this discussion to a special class of singularities; those of splice type, which
turn out to play a central role among singularities of complex surfaces.
An appendix gives a brief introduction to classical 3-manifold theory.
This paper was written to serve as notes for a short course at ICTP Trieste.
1. I
NTRODUCTION
The early study of 3-manifolds and knots in 3-manifolds was motivated to a
large extent by the theory of complex surfaces. For example, Poul Heergaards
1898 thesis [
7
], in which he introduced the fundamental tool of 3-manifold theory
now called a Heegaard splitting, was on the topology of complex surfaces. For
a thread from Heergaards thesis through knot theory to the splice diagrams that
will play a central role in this paper, see the survey [
23
] on topology of complex
surface singularities.
The local topology of a normal complex surface (normal roughly means that
any inessential singularities have been removed) at any point is the cone on a
closed oriented 3-manifold. The manifold is called the link of the point. We call
it a singularity-link, even though we allow S
3
, which can only be the link of a
non-singular point (Mumford [
16
]).
Singularity links and other 3-manifolds that arise in the study of complex sur-
faces are of a special type, namely graph manifolds. Graph manifolds were de-
ned and classied by Waldhausen in his thesis [
40
]. The motivation was certainly
that the set of graph manifolds includes all singularity-links, and Waldhausens
work together with Grauerts criterion effectively gave a description of exactly
what 3-manifolds are singularity-links. This description was put in a more conve-
nient algorithmic form in [
17
]. More elegant versions have emerged since, which
depend on taking a coarser look at the classication of graph manifolds. These
coarse classications are a central theme of this paper. They will also lead us to a
special class of singularities, the singularities of splice type which encompasses
several important classes of singularities.
2000 Mathematics Subject Classication. 32S50, 14B05, 57M25, 57N10.
Key words and phrases.
graph manifold, surface singularity, rational homology sphere, complete
intersection singularity, abelian cover.
Research supported under NSF grant no. DMS-0083097 and DMS-0206464.
1 2
WALTER D. NEUMANN
An appendix to this paper provides a convenient reference for some of the basic
3-manifold theory that we use.
This paper was written to serve as notes for a short course at ICTP Trieste. It is
based in part also on lectures the author gave at CIRM (Luminy) in March 2005.
2. T
HE PLACE OF GRAPH MANIFOLDS IN
3-
MANIFOLD THEORY
Throughout this paper, 3-manifolds will be compact and oriented unless other-
wise stated. They will also be prime not decomposable as a non-trivial con-
nected sum. One forms the connected sum of two 3-manifolds by removing the
interior of a disk from each and then gluing the resulting punctured 3-manifolds
along their S
2
boundaries. Kneser and Milnor [
12
,
15
] showed that any oriented
3-manifold has an essentially unique decomposition into prime 3-manifolds. Sin-
gularity links are always prime ([
17
]).
Denition 2.1. A graph-manifold is a 3-manifold M that can be cut along a family
of disjoint embedded tori to decompose it into pieces S
i
× S
1
, where each S
i
is a
compact surface (i.e., 2-manifold) with boundary.
The JSJ-decomposition is a natural decomposition of any prime 3-manifold into
Seifert bered and simple non-bered pieces (see the appendix for relevant deni-
tions and more detail). Its existence was proved in the mid-1970s independently by
by Jaco and Shalen [
9
] and by Johannson [
11
], although it had been sketched earlier
by Waldhausen [
41
]. From the point of view of JSJ-decomposition, a graph man-
ifold is simply a 3-manifold which has no non-Seifert-bered JSJ-pieces. There
are various modications of the JSJ decomposition, depending on the intended ap-
plication, and they differ in essentially elementary ways (see e.g., [
25
]). One ver-
sion is the geometric decomposition a minimal decomposition along tori and
Klein bottles into pieces that admit geometric structures in the sense of Thurston
(nite volume locally homogeneous Riemannian metrics). The relevant geometry
for simple non-Seifert-bered pieces is hyperbolic geometry
1
. From this geomet-
ric point of view, graph manifolds are manifolds that have no hyperbolic pieces in
their geometric decompositions.
In summary, a graph manifold is a 3-manifold that can be glued together from
pieces of the form (surface)×S
1
, or more efciently, from pieces which are Seifert
bered. Both points of view will be useful in the sequel.
3. S
EIFERT MANIFOLDS
Let M
3
F be a Seifert bration of a closed 3-manifold. It is classied up
to orientation preserving homeomorphism (or diffeomorphism) by the following
data:
1
The existence of the hyperbolic structure when M is simple non-Siefert bered and the JSJ de-
composition is trivial was still conjectural until recently; although proved in many cases by Thurston,
it is probably now proved in general by Perelmans work. GRAPH 3-MANIFOLDS, SPLICE DIAGRAMS, SINGULARITIES
3
The homeomorphism type of the base surface F , which we can encode by
its genus g. We use the convention that g < 0 refers to a non-orientable
surface, so g = 1, 2, . . . means F is a projective place, Klein bottle,
etc.
A collection of rational numbers 0 < q
i
/p
i
< 1, i = 1, . . . , n, that en-
code the types of the singular bers. Here p
i
is the multiplicity of the i-th
singular ber and q
i
encodes how nearby bers twist around this singular
ber.
A rational number e = e(M F ) called the Euler number of the Seifert
bration. Its only constraint is that e +
n
i=1
q
i
p
i
should be an integer.
It is most natural to think of the base surface F as an orbifold rather than a
manifold, with orbifold points of degrees p
1
, . . . , p
n
. As such, it has an orbifold
Euler characteristic orb
(F ) =
g i
(1 1
p
i
)
where
g
is the Euler characteristic of the surface of genus g: g
=
2 2g ,
g 0 ,
2 + g ,
g < 0 .
Note that an oriented 3-manifold M
3
may be Seifert bered with non-orientable
base. However, we do not need to consider this for links of singularities: a Seifert
bered 3-manifold is a singularity link if and only if it has a Seifert bration over
an orientable base and the Euler number e(M F ) is negative.
From the point of view of geometric structures and geometric decomposition,
there are exactly six geometries that occur for Seifert bered manifolds and the
type of the geometry is determined by whether
orb
(F ) is > 0, = 0, < 0 and
whether e(M F ) is = 0 or = 0 ([
19
,
34
]). These two invariants, which we will
abbreviate simply as and e, are thus fundamental invariants for a Seifert bered
M
3
. If e = 0 then M
3
has a unique orientation that makes e < 0, and we call this
its natural orientation, since it is the orientation that makes it (or a double cover
of it if the base surface is non-orientable) into a singularity link.
The above discussion was for a closed 3-manifold M
3
. If M
3
is allowed to have
boundary (but is still compact) then the Euler number e is indeterminate unless one
has extra data. The additional data consists of a choice of a simple closed curve in
each boundary torus of M
3
, transverse to the bers of the Seifert bration.
Denition 3.1. We call this collection of curves a system of meridians for M .
Given a system of meridians, we can form a closed Seifert bered manifold ¯
M
3
by gluing a solid torus onto each boundary component, matching a meridian of the
solid torus with the chosen meridian on the boundary T
2
. The Euler invariant
e( ¯
M ) is called the Euler invariant of M with its system of meridians. 4
WALTER D. NEUMANN
4. D
ECOMPOSITION GRAPHS
,
DECOMPOSITION MATRICES
We now return to a general graph manifold M , considering it from the point of
view of JSJ-decomposition. So M can be cut along tori so that if breaks into pieces
that are Seifert bered 3-manifolds. JSJ decomposition means that no smaller
collection of cutting tori will work (see the Appendix for a proof of existence and
uniqueness of JSJ decomposition).
If M bers over the circle with torus ber or is double covered by such a mani-
fold then M admits a geometric structure, so the geometric version of JSJ decom-
position would not decompose it, even though the standard JSJ usually cuts it along
a torus. Such manifolds are completely understood (for a discussion close to the
current point of view see [
20
]) so:
Assumption. From now on we assume that M cannot be bered over S
1
with T
2
ber.
Each piece M
i
in the JSJ decomposition com