*Geometry &* *Topology* *GGGG*
*GG*

*GG G GGGGGG*
*T T TTTTTTT*
*TT*

*TT*
*TT*
Volume 1 (1997) 21{40

Published: 30 July 1997

**Canonical Decompositions of 3{Manifolds**

Walter D Neumann Gadde A Swarup

*Department of Mathematics*
*The University of Melbourne*

*Parkville, Vic 3052*
*Australia*

Email: neumann@ms.unimelb.edu.au and gadde@ms.unimelb.edu.au

**Abstract**

We describe a new approach to the canonical decompositions of 3{manifolds along tori and annuli due to Jaco{Shalen and Johannson (with ideas from Wald- hausen) | the so-called JSJ{decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus{torus theorems, and the theory of Seifert brations does not need to be developed in advance.

**AMS Classication numbers** Primary: 57N10, 57M99
Secondary: 57M35

**Keywords:** 3{manifold, torus decomposition, JSJ{decomposition, Seifert
manifold, simple manifold

Proposed: David Gabai Received: 25 February 1997

Seconded: Robion Kirby, Ronald Stern Accepted: 27 July 1997

**1** **Introduction**

In this paper we describe a proof of the so-called JSJ{decomposition theorem for 3{manifolds. This proof was developed as an exercise for ourselves, to conrm an approach that we hope will be useful for JSJ{decomposition in the group-theoretic context. It seems to give a particularly accessible proof of JSJ for 3{manifolds, involving few prerequisites. For example, it does not use the annulus{torus theorems, and the theory of Seifert brations does not need to be developed in advance.

We do not give the simplest version of our proof. We prove JSJ for orientable Haken 3{manifolds with incompressible boundary. This involves splitting along tori and annuli. As we describe later, if one restricts to the case that all bound- ary components are tori, then one only needs to split along tori. With this restriction our proof becomes very much simpler, with many fewer case distinc- tions. The general JSJ{decomposition can then be deduced quite easily from this special case by doubling the 3{manifold along its boundary. We took the more direct but less simple approach because this was an exercise to test a concept: there is no clear analogue of the peripheral structure and that of a double in the group theoretic case and we preferred our approach to be closer to possible group theoretic analogues.

We do not prove all the properties of the JSJ{decomposition, although it seems that this could be done from our approach with a little more eort. The main results we do not prove | the enclosing theorem and Johannson deformation theorem | have very readable accounts in Jaco’s book [4].

**Acknowledgement** This research was supported by the Australian Research
Council.

**2** **Canonical Surfaces and W{Decomposition**

We consider orientable Haken 3{manifolds with incompressible boundary and
consider splittings along essential annuli and tori. Let *S* be an annulus or torus
that is properly embedded in (M; @M) and which is essential (incompressible
and not boundary-parallel).

**Denition 2.1** *S* will be called *canonical* if any other properly embedded
essential annulus or torus *T* can be isotoped to be disjoint from *S*.

Take a disjoint collection *fS*_{1}*; : : : ; S*_{s}*g* of canonical surfaces in *M* such that

no two of the *S**i* are parallel;

the collection is maximal among disjoint collections of canonical surfaces with no two parallel.

A maximal system exists because of the Kneser{Haken niteness theorem. The
result of splitting *M* along such a system will be called a*Waldhausen decom-*
*position* (or briefly*W{decomposition*) of *M*. It is fairly close to the usual JSJ{

decomposition which will be described later. The maximal system of pairwise
non-parallel canonical surfaces will be called a *W{system.*

The following lemma shows that the W{system *fS*_{1}*; : : : ; S*_{s}*g* is unique up to
isotopy.

**Lemma 2.2** *Let* *S*1*; : : : ; S**k* *be pairwise disjoint and non-parallel canonical*
*surfaces in* (M; @M). Then any incompressible annulus or torus *T* *in* (M; @M)
*can be isotoped to be disjoint from* *S*_{1}*[ [S*_{k}*. Moreover, if* *T* *is not parallel*
*to any* *S*_{i}*then the nal position of* *T* *in* *M−*(S_{1}*[ [S** _{k}*)

*is determined up*

*to isotopy.*

**Proof** By assumption we can isotop *T* o each *S** _{i}* individually. Writing

*T*=

*S*0, the lemma is thus a special case of the following stronger result:

**Lemma 2.3** *Suppose* *fS*0*; S*1*; : : : ; S*_{k}*g* *are incompressible surfaces in an irre-*
*ducible manifold* *M* *such that each pair can be isotoped to be disjoint. Then*
*they can be isotoped to be pairwise disjoint and the resulting embedded surface*
*S*0*[: : :[S**k* *in* *M* *is determined up to isotopy.*

Without the uniqueness statement a quick conceptual proof of this uses minimal surfaces (one can also use a traditional cut-and-paste type proof; see below).

Choose a metric on *M* which is product metric near *@M*. Freedman, Hass
and Scott show in [2] that the least area representatives of the *S** _{i}* are pairwise
disjoint or identical. When two least area representatives are identical, we
isotop them o each other in a normal direction (no two of the

*S*

*i*embed to identical one-sided least area surfaces since an embedded one-sided surface can never be isotoped o itself: the transverse intersection with an isotopic copy is a 1{manifold dual to the rst Stiefel{Whitney class of the normal bundle of the surface). There is a slight complication in that the least area representative

*S*

_{i}*!*

*S*

_{i}

^{0}*M*may not be an embedding; it can be a double cover onto an embedded one-sided surface

*S*

_{i}*. But in this case we use a nearby embedding onto the boundary of a tubular neighbourhood of*

^{0}*S*

_{i}*.*

^{0}In [2] surfaces are considered up to homotopy rather than isotopy, so the above argument uses the fact that two homotopic embeddings of an incompressible surface are isotopic (see [9], Corollary 5.5).

For the uniqueness statement assume we have *S*1*; : : : ; S** _{k}* disjointly embedded
and then have two dierent embeddings of

*S*=

*S*

_{0}disjoint from

*T*=

*S*

_{1}

*[: : :[*

*S*

*. Let*

_{k}*f*:

*S*

*I*

*!*

*M*be a homotopy between these two embeddings and make it transverse to

*T*. The inverse image of

*T*is either empty or a system of closed surfaces in the interior of

*SI*. Now use Dehn’s Lemma and Loop Theorem to make these incompressible and, of course, at the same time modify the homotopy (this procedure is described in Lemma 1.1 of [10] for example).

We eliminate 2{spheres in the inverse image of *T* similarly. If we end up with
nothing in the inverse image of *T* we are done. Otherwise each component *T** ^{0}*
in the inverse image is a parallel copy of

*S*in

*SI*whose fundamental group maps injectively into that of some component

*S*

*of*

_{i}*T*. This implies that

*S*can be homotoped into

*S*

*and its fundamental group*

_{i}_{1}(S) is conjugate into some 1(S

*i*). It is a standard fact (see eg [8]) in this situation of two incompressible surfaces having comparable fundamental groups that, up to conjugation, either

_{1}(S) =

_{1}(S

*) or*

_{j}*S*

*is one-sided and*

_{j}_{1}(S) is the fundamental group of the boundary of a regular neighbourhood of

*T*and thus of index 2 in 1(S

*j*). We thus see that either

*S*is parallel to

*S*

*and is being isotoped across*

_{j}*S*

*or it is a neighbourhood boundary of a one-sided*

_{j}*S*

*and is being isotoped across*

_{j}*S*

*. The uniqueness statement thus follows.*

_{j}One can take a similar approach to prove the existence of the isotopy using
Waldhausen’s classication [9] of proper incompressible surfaces in *S* *I* to
show that *S*0 can be isotoped o all of *S*1*; : : : ; S** _{k}* if it can be isotoped o each
of them.

The thing that makes decomposition along incompressible annuli and tori spe- cial is the fact that they have particularly simple intersection with other incom- pressible surfaces.

**Lemma 2.4** *If a properly embedded incompressible torus or annulus* *T* *in*
*an irreducible manifold* *M* *has been isotoped to intersect another properly*
*embedded incompressible surface* *F* *with as few components in the intersection*
*as possible, then the intersection consists of a family of parallel essential simple*
*closed curves on* *T* *or possibly a family of parallel transverse intervals if* *T* *is*
*an annulus.*

**Proof** We just prove the torus case. Suppose the intersection is non-empty. If
we cut *T* along the intersection curves then the conclusion to be proved is that

*T* is cut into annuli. Since the Euler characteristics of the pieces of *T* must add
to the Euler characteristic of *T*, which is zero, if not all the pieces are annuli
then there must be at least one disk. The boundary curve of this disk bounds
a disk in *F* by incompressibility of *F*, and these two disks bound a ball in *M*
by irreducibility of *M*. We can isotop over this ball to reduce the number of
intersection components, contradicting minimality.

**3** **Properties of the W{decomposition**

Let *M*1*; : : : ; M**m* be the result of performing the W{decomposition of *M* along
the W{system *fS*_{1}*[ [S*_{s}*g*.

We denote by *@*1*M**i* the part of *@M**i* coming from *S*1*[ [S**s* and by *@*0*M**i*

the part coming from *@M*. Thus *@*0*M**i* and *@*1*M**i* are complementary in *@M**i*

except for meeting along circles. The components of *@*_{1}*M** _{i}* are annuli and tori.

Both *@*1*M**i* and *@*0*M**i* are incompressible since we started with an *M* with *@M*
incompressible. However, it is possible that *@M**i* is not incompressible.

Lemma 2.2 shows that any incompressible annulus or torus *S* in *M* can be
isotoped into the interior of one of the *M**i*. If this *S* is an annulus its boundary

*@S* is necessarily in *@*_{0}*M** _{i}*. By the maximality of

*fS*

_{1}

*; : : : ; S*

_{s}*g*, if

*S*is canonical it must be parallel to one of the

*S*

*. However, there may be such*

_{j}*S*which are not canonical and not parallel to an

*S*

*j*.

**Denition 3.1** We call *M**i* *simple* if any essential annulus or torus (S; @S)
(M_{i}*; @*_{0}*M** _{i}*) is parallel to

*@*

_{1}

*M*

*. We call*

_{i}*M*

_{i}*special simple*if, in addition, it admits an essential annulus in (M

_{i}*; @*

_{1}

*M*

*) (allowing the possibility that the annulus may be parallel to*

_{i}*@*0

*M*

*i*). This will, in particular, be true if some component of

*@*

_{0}

*M*

*is an annulus.*

_{i}**Proposition 3.2** *If* *M*_{i}*is non-simple then* (M_{i}*; @*_{0}*M** _{i}*)

*is either Seifert bred*

*or an*

*I{bundle.*

*If* *M*_{i}*is special simple then either* (M_{i}*; @*_{0}*M** _{i}*)

*is Seifert bred of one of the*

*types illustrated in Figure 1 or*(M

*i*

*; @*0

*M*

*i*) = (T

^{2}

*I;;)*

*(in which case*

*M*

*is a*

*T*

^{2}

*{bundle over*

*S*

^{1}

*with holonomy of trace*

*6*=2

*).*

A consequence of this proposition is that an irreducible manifold *M* that has
an essential annulus or torus but no canonical one is an *I*{bundle or Seifert
bered.

1

2

3

4

5

6

7

8

Figure 1: The eight Seifert bred building blocks. In each case we have drawn the base
surface with the portion of the boundary in *@*0 resp. *@*1 drawn solid resp. dashed. The
dots in cases 4,5,6 represent singular bres and in case 8 it represents a bre that may
or may not be singular. These cases thus represent innite families.

**Proof** We rst consider the non-simple case. There are a few cases that will
be exceptional from the point of view of our argument and that we will need to
treat specially as they come up. They are the cases

*M* is an *I*{bundle over a torus or Klein bottle | then *M* also bres as
an *S*^{1}{bundle over annulus or M¨obius band respectively;

*M* is an *S*^{1}{bundle over torus or Klein bottle.

These are, in fact, among the non-simple manifolds that admit more than one
bred structure. In these cases it is not hard to see that no essential annulus
or torus is canonical, so the W{decomposition of *M* is trivial, so *M* =*M*1.
We drop the index and denote (N; @_{0}*N; @*_{1}*N*) = (M_{i}*; @*_{0}*M*_{i}*; @*_{1}*M** _{i}*).

Consider a maximal disjoint collection of pairwise non-parallel essential annuli
and tori *fT*_{1}*; : : : ; T*_{r}*g* in (N; @_{0}*N*). Split *N* along this collection into pieces
*N*1*; : : : ; N**n*. We still denote by*@*0*N**i* the part coming from *@M*, that is, *@*0*N**i*=

*@*_{0}*N* *\N** _{i}*=

*@M*

*\N*

*, and we denote by*

_{i}*@*

_{1}

*N*

*the rest of*

_{i}*@N*

*.*

_{i}We shall show that if *M* is not one of the exceptions listed above, then the
pieces *N**i* are bred either as *I*{bundles or Seifert brations, and, moreover,
these brations match up when we glue the *N** _{i}* together to form

*N*. In fact:

**Claim 1** *Each* *N**i* *is either an* *I{bundle over a twice punctured disk, a M¨obius*
*band, or a punctured M¨obius band or is Seifert bred of one of the types in*
*Figure 1.*

*@*_{1}*N** _{i}* consists of annuli and tori, some of which may come from the

*S*

*, but at least one of which comes from a*

_{j}*T*

*j*. For this

*T*

*j*we let

*T*

_{j}*be an essential annulus or torus that intersects*

^{0}*T*

*j*essentially and we assume the intersection

*T*

_{j}*\T*

_{j}*is minimal.*

^{0}The proof is a case by case analysis of this situation. Since the proofs in the dierent cases are fairly similar, we give a complete argument in only a couple of typical cases.

**Case A** *@T**j* *\@T*_{j}^{0}*6*= *;*. We shall see that (N*i**; @*0*N**i*) is an (I; @I){bundle
over a surface and *@*_{1}*N** _{i}* is the part of the bundle lying over the boundary of
the surface. The surface in question is either a twice punctured disk, a M¨obius
band, or a punctured M¨obius band (except when

*N*=

*M*was itself an

*I*{bundle over a torus or Klein bottle).

This will be proved case by case. Denote a component of *T*_{j}^{0}*\N**i* which intersects
*T**j* by *P* and let *s* be a component of the intersection *P\T**j*. Since *T*_{j}^{0}*\*(T1*[*
* [T** _{t}*) will be a nite number of segments crossing

*T*

_{j}*, there will be another segment*

^{0}*s*

*in*

^{0}*@P\(T*1

*[ [T*

*t*) and the rest of

*@P*will consist of two segments in

*@*

_{0}

*N*

*. Let*

_{i}*s*

*be in*

^{0}*T*

*. We distinguish two subcases.*

_{k}**Case A1** *T*_{j}*6*= *T** _{k}*. Note that

*P*is an

*I*{bundle over an interval with

*s*and

*s*

*the bres over the ends of this interval, and*

^{0}*T*

*j*and

*T*

*k*are

*I*{bundles over circles. Cutting

*T*

*and*

_{j}*T*

*along*

_{k}*s*and

*s*

*, pushing them just inside*

^{0}*N*

*, and then pasting parallel copies of*

_{i}*P*gives an annulus (see Figure 2).

It cannot be parallel into *@*0*N**i* because then *T**j* would be inessential (note
*P*

*T**j* *T**k*

Figure 2: Case A1

that our assumption of boundary-incompressibility implies that an annulus is inessential if an arc across the annulus is isotopic into the boundary). It can

not be isotopically trivial since then *T**j* and *T**k* would be parallel. It is thus
parallel into *@*1*N**i*. Thus *N**i* is an *I*{bundle over a twice punctured disk.

**Case A2** *T** _{j}* =

*T*

*. Since*

_{k}*P*is an

*I*{bundle over an interval with

*s*and

*s*

*the bres over the ends of this interval, and*

^{0}*T*

*is an*

_{j}*I*{bundle over a circle, we may orient the bres of these bundles so that

*s*is oriented the same in each.

Then *s** ^{0}* may or may not be oriented the same in each. Moreover,

*P*may meet

*T*

*at*

_{j}*s*

*at the same side of*

^{0}*T*

*as it meets it at*

_{j}*s*or at the opposite side.

**Case A2.1.1** *P* meets *T** _{j}* both times from the same side and

*s*

*has the same orientation in both*

^{0}*I*{bundles. Then if we cut

*T*

*j*along

*s[s*

*and glue two parallel copies*

^{0}*P*

*and*

^{0}*P*

*of*

^{00}*P*we get two annuli in

*N*

*(see Figure 3). If either*

_{i}*P*

*T**j*

Figure 3: Case A2.1.1

of these annuli is trivial in *N**i* we could have removed the intersection *s[s** ^{0}* of

*T*

*and*

_{j}*T*

_{j}*by an isotopy. Thus they are both non-trivial and must be parallel to components of*

^{0}*@*

_{1}

*N*

*or*

_{i}*@*

_{0}

*N*

*. If either is parallel into*

_{i}*@*

_{0}

*N*

*then*

_{i}*T*

*would have been inessential. They are thus both parallel into*

_{j}*@*1

*N*

*i*and we see that

*N*

*is an*

_{i}*I*{bundle over a twice punctured disk.

**Case A2.1.2** *P* meets *T** _{j}* both times from the same side and

*s*

*has opposite orientations in the two*

^{0}*I*{bundles. Then cutting

*T*

*as above and gluing two copies*

_{j}*P*

*and*

^{0}*P*

*of*

^{00}*P*yields a single annulus (see Figure 4) which may be isotopically trivial or parallel into

*@*

_{1}

*N*

*(as before, it cannot be parallel into*

_{i}*@*0*N**i*). This gives an *I*{bundle over a M¨obius band or punctured M¨obius band.

**Case A2.2** *P* meets *T**j* from opposite sides (and *s** ^{0}* has the same or opposite
orientations in the two

*I*{bundles). After cutting open along

*T*

*we have two copies*

_{j}*T*

_{j}^{(1)}and

*T*

_{j}^{(2)}of

*T*

*j*and this becomes similar to case A1: cutting

*T*

_{j}^{(1)}and

*T*

_{j}^{(2)}along

*s*and

*s*

*and pasting parallel copies of*

^{0}*P*gives an annulus. As

*P*

*T**j*

Figure 4: Case A2.1.2

in Case A1, this annulus cannot be parallel into *@*0*N**i*. It may be parallel into

*@*1*N**i*, and then *N**i* is an *I*{bundle over a twice punctured disk. Unlike case
A1, it may also be isotopically trivial, in which case *N** _{i}* is an

*I*{bundle over an annulus, so

*N*=

*M*is an

*I*{bundle over the torus or Klein bottle, giving two of the exceptional cases mentioned at the start of the proof.

**Case B** *@T*_{j}*\@T*_{j}* ^{0}* =

*;*. In this case we will see that

*N*

*is Seifert bred of one of eight basic types of Figure 1 (in the exceptional cases mentioned at the start of the proof | where the W{decomposition is trivial and*

_{i}*N*=

*M*is itself an

*S*

^{1}{bundle over an annulus, M¨obius band, torus, or Klein bottle | there is just one piece

*N*

_{1}after cutting and it does not occur among the eight types).

Denote a component of *T*_{j}^{0}*\N**i* which intersects *T**j* by *P* and let *s* be a
component of the intersection *P\T** _{j}*. This

*P*is an annulus and

*s*is one of its boundary components. Denote the other boundary component by

*s*

*. It either lies on*

^{0}*@*0

*N*

*i*or on some

*T*

*.*

_{k}**Case B1** *s** ^{0}* lies on

*@*

_{0}

*N*

*. There are two subcases according as*

_{i}*T*

*is an annulus or torus.*

_{j}**Case B1.1** *T**j* is an annulus (and *s** ^{0}* lies on

*@*0

*N*

*i*; here, and in the following, subcases inherit the assumptions of their parent case). Cutting

*T*

*along*

_{j}*s*and pasting parallel copies of

*P*to the resulting annuli gives a pair of annuli which must both be parallel into

*@*1

*N*

*i*, since if either were parallel into

*@*0

*N*

*i*we could have removed the intersection

*s*of

*T*

*and*

_{j}*T*

_{j}*by an isotopy. This gives Figure 1.1 (ie case 1 of Figure 1).*

^{0}**Case B1.2** *T**j* is a torus. Cutting *T**j* along *s* and pasting parallel copies of
*P* to the resulting annulus gives an annulus which must be parallel into *@*_{1}*N** _{i}*,

since if it were parallel into *@*0*N**i* then *T**j* would be boundary parallel. This
gives Figure 1.2.

**Case B2** *s** ^{0}* lies on

*T*

*and*

_{k}*T*

_{k}*6*=

*T*

*. There are three subcases according as both, one, or neither of*

_{j}*T*

*j*and

*T*

*are annuli.*

_{k}**Case B2.1** *T** _{j}* and

*T*

*both annuli. Cutting*

_{k}*T*

*and*

_{j}*T*

*along*

_{k}*s*and

*s*

*and pasting parallel copies of*

^{0}*P*to the resulting annuli gives a pair of annuli. They cannot both be parallel into

*@*

_{1}

*N*

*, for then we would get a picture like Figure 1.1 but based on an octagon rather than a hexagon and an annulus joining opposite*

_{i}*@*0 sides of this would contradict the fact that the*T**i*’s formed a maximal family.

They cannot both be parallel into *@*_{0}*N** _{i}* for then

*T*

*and*

_{i}*T*

*would be parallel.*

_{j}Thus one is parallel into *@*_{1}*N** _{i}* and the other into

*@*

_{0}

*N*

*. This gives Figure 1.1.*

_{i}again.

**Case B2.2** One of *T** _{j}* and

*T*

*an annulus and the other a torus. Cut and paste as before gives a single annulus which may be parallel into*

_{k}*@*0, giving Figure 1.2. It cannot be parallel into

*@*1, since then we would get a picture like Figure 1.2 but with two

*@*

_{0}annuli in the outer boundary rather than just one. One could span an essential annulus from one of these components of

*@*0

to itself around the torus, contradicting maximality of the family of *T** _{j}*’s.

**Case B2.3** Both of *T**j* and *T**k* tori. Cut and paste as before gives a single
torus. It cannot be parallel into *@*0 since then the annulus that can be spanned
across *P* from this torus to itself would contradict the fact that the *T** _{j}*’s form
a maximal family. It may be parallel into

*@*

_{1}leading to Figure 1.3 or it may also bound a solid torus which gives 1.6.

**Case B3** *s** ^{0}* lies on

*T*

*. There two subcases according as*

_{j}*T*

*is an annulus or torus, and each subcase splits into three subcases according to whether*

_{j}*P*meets

*T*

*from the same or opposite sides at*

_{j}*s*and

*s*

*, and if the same side, then whether bre orientations match at*

^{0}*s*and

*s*

*. We shall describe them briefly and leave details to the reader.*

^{0}**Case B3.1** *T** _{j}* an annulus.

**Case B3.1.1** *P* meets the annulus *T**j* both times from the same side.

**Case B3.1.1.1** The orientations of *s* and *s** ^{0}* match. Cut and paste as before
gives an annulus and a torus. The annulus cannot be parallel into

*@*1 since otherwise one can nd an annulus that contradicts maximality of the family of

*T*

*’s, so it is parallel into*

_{j}*@*

_{0}. Similarly, the torus cannot be parallel into

*@*

_{0}. This leads to the cases of Figure 1.2 or 1.4 according to whether the torus is parallel into

*@*

_{1}or bounds a solid torus.

**Case B3.1.1.2** Orientations of *s*and *s** ^{0}* do not match. Cut and paste as before
gives an annulus. Whether it is parallel into

*@*0 or

*@*1,

*N*

*i*is a circle bundle over a M¨obius band with part of its boundary in

*@*

_{0}and an annulus from

*@*

_{0}to

*@*0 running once around the M¨obius band contradicts the maximality of the family of

*T*

*j*’s. Thus this case cannot occur.

**Case B3.1.2** Assumptions of Case B3.1 and *P* meets the annulus *T** _{j}* from
opposite sides. This gives the same as Case B2.1, except for one extra possibility,
since both annuli of Case B2.1 being parallel into

*@*

_{0}is not ruled out. This extra case leads to

*N*=

*M*being an

*S*

^{1}{bundle over the annulus or M¨obius band.

**Case B3.2** Assumptions of Case B3 with *T**j* a torus.

**Case B3.2.1** *P* meets the torus *T** _{j}* both times from the same side.

**Case B3.2.1.1** Orientations of *s* and *s** ^{0}* match. Cut and paste gives two tori,
neither of which is parallel to

*@*

_{0}. This leads to Figure 1.3, 1.5, or 1.6.

**Case B3.2.1.2** Orientations of *s* and *s** ^{0}* do not match. Cut and paste gives
one torus which may be parallel to

*@*

_{1}or bound a solid torus. This gives Figure 1.7 or 1.8.

**Case B3.2.2** Assumptions of Case B3.2 and *P* meets the torus *T**j* from
opposite sides. This gives the same as Case B2.3, except that in the situation
of Figure 1.6 the \singular" bre need not actually be singular, in which case
*N* =*M* is an *S*^{1}{bundle over the torus or Klein bottle (these are among the
special manifolds listed at the start of the proof).

This completes the analysis of the pieces *N**i* and thus proves Claim 1. We must
now verify that the bred structures on the pieces *N** _{i}* match up in

*N*.

Suppose two pieces *N**i*1 and *N**i*2 meet across *T**j* and let *T*_{j}* ^{0}* be as above. The
above argument showed that a bred structure can be chosen on these two
pieces to match the bred structure of the

*T*

_{j}*and hence to match each other across*

^{0}*T*

*. Thus if every piece*

_{j}*N*

*has a unique bred structure then the bred structures must match across every*

_{i}*T*

*j*.

**Claim 2** *The only pieces* *N*_{i}*for which the bration is not unique up to isotopy*
*are:*

**1)** *I{bundle over M¨obius band, which also admits a bration as*
**2)** *the Seifert bration of Figure 1.4 with degree* 2 *exceptional bre;*

**3)** *the Seifert bration of Figure 1.5 with both exceptional bres of degree 2,*
*which also admits the structure of*

**4)** *the circle bundle of Figure 1.8 with no exceptional bre.*

If we grant this claim then the matching of brations on the various *N**i* follows:

since in the above non-unique cases *N**i* has only one boundary component *T**j*,
we simply choose the bration on this *N** _{i}* that is forced by

*T*

_{j}*and it will then match the neighbouring piece.*

^{0}Claim 2 follows by showing that the bration is unique in all other cases. It is
clear that the only *N** _{i}* that is both an

*I*{bundle and Seifert bred is the one of cases 1 and 2, so we need only consider non-uniqueness of Seifert bration.

All we really need for the above proof is that the Seifert bration is unique up
to isotopy when restricted to the boundary, which is clear for cases 1, 2, 4 of
Figure 1 and follows from the fact that a circle bre generates an innite cyclic
normal subgroup of the fundamental group in the other cases. The uniqueness
of the bration on the whole of *N**i* follows, if desired, by a standard argument
once at least one bre has been determined.

To complete the proof of the proposition we must consider the case that *M** _{i}*
is special simple, so it is simple but has an essential annulus in (M

*i*

*; @*1

*M*

*i*).

If we call this annulus *P* we can repeat word for word the arguments of Case
B above to see that *M** _{i}* is of one of the eight types listed in Figure 1 or is
an

*S*

^{1}{bundle over an annulus or M¨obius band with boundary belonging to

*@*_{1}*M** _{i}*. Since

*S*

^{1}{bundle over M¨obius band is included in the eight types and

*S*

^{1}{bundle over annulus is

*T*

^{2}

*I*, the statement of the proposition follows.

Note that the *T*^{2}*I* case can only occur if the two boundary tori are the same
in *M*, in which case *M* is a torus bundle over the circle. The holonomy of this
torus bundle cannot have trace 2 since then the torus would not be canonical
(in fact *M* would be an *S*^{1}{bundle over torus or Klein bottle).

It is worth noting that the above proof shows also that the only non-simple
Seifert bred manifolds with non-unique Seifert bration are those with trivial
W{decomposition mentioned at the beginning of the above proof (circle bun-
dles over annulus, M¨obius band, torus, or Klein bottle) and the one that comes
from matching the non-unique cases 3 and 4 of the above Claim 2 together,
giving the Seifert bration over the projective plane with unnormalized Seifert
invariant *f−*1; (2;1);(2;*−*1)*g* (two exceptional bres of degree 2 and rational
Euler number of the bration equal to zero). This manifold has two distinct -
brations of this type. Note that matching two of case 3 or two of case 4 together
gives the tangent circle bundle of the Klein bottle, which is one of the exam-
ples with trivial W{decomposition already mentioned, but we see this way its
Seifert bration over *S*^{2} with four degree two exceptional bres (unnormalized
invariant *f*0; (2;1);(2;1);(2;*−*1);(2;*−*1)*g*).

We shall classify the pieces *M** _{i}* with

*@*

_{1}

*6*=

*;*into three types:

*M**i* is*strongly simple, by which we mean simple and not special simple;*

*M** _{i}* is an

*I*{bundle;

*M** _{i}* is Seifert bred.

We rst discuss which *M** _{i}* belong to more than one type.

**Proposition 3.3** *The only cases of an* *M**i* *having more than one type are:*

*The* *I{bundle over M¨obius band is also Seifert bered.*

*The* *I{bundles over twice punctured disk and once punctured M¨obius band*
*are also strongly simple.*

**Proof** If *M** _{i}* admits both a Seifert bration and an

*I*{bundle structure, it is an

*I*{bundle over an annulus or M¨obius band (since we are assuming

*@*

_{1}

*6*=

*;*).

If *M**i* is *I*{bundle over annulus then *@*1*M**i* consists of two annuli. Since they
are parallel in *M** _{i}* they must have been equal in

*M*, so

*M*is obtained by pasting these two annuli together, that is,

*M*is an

*I*{bundle over torus or Klein bottle. But the annulus is then not canonical (in fact, such

*M*has trivial W{decomposition), so this cannot occur. Thus the only case is that

*M*is

*I*{bundle over M¨obius band.

If *M**i* is both strongly simple and an *I*{bundle then we have already pointed
out that *M** _{i}* cannot be

*I*{bundle over annulus. The

*I*{bundle over M¨obius band is Seifert bred and is therefore special simple (see below). An

*I*{bundle over a bounded surface of Euler number less than

*−*1 will not be simple. Thus the only cases are those of the proposition.

Suppose *M** _{i}* is Seifert bred and simple. If some component of

*@*

_{0}

*M*

*is an annulus then*

_{i}*M*

*i*is special simple. Otherwise,

*@*1

*M*

*i*consists of tori and

*M*

*i*is Seifert bred with at least two singular bres if the base is a disk and at least one if it is an annulus. It is thus easy to see that for each component of

*@*

_{1}

*M*

*there will be an essential annulus with both boundaries in this component, so*

_{i}*M*

*is again special simple. Thus*

_{i}*M*

*cannot be both Seifert bred and strongly simple.*

_{i}We now want to investigate the possibility of adjacent bred pieces in the W{

decomposition having matching bres where they meet. We therefore assume
that the W{decomposition is non-trivial, so *M** _{i}* has non-empty

*@*

_{1}.

Suppose the pieces *M**i* and *M** _{k}* on the two sides of a canonical annulus

*S*

*j*are both

*I*{bundles. If

*i6*=

*k*then in each of

*M*

*and*

_{i}*M*

*we can nd an essential*

_{k}*I* *I* from *S**j* to itself. Gluing these gives an essential annulus crossing *S**j*

essentially. Thus *S**j* was not canonical. If *i*=*k* the argument is similar. Thus
*I*{bundles cannot be adjacent in the W{decomposition.

It remains to consider the case that both pieces on each side of a canonical
annulus or torus *S**j* are Seifert bred. Suppose the brations on both sides
match along *S** _{j}*.

If a piece *M** _{i}* adjacent to

*S*

*is not simple we shall decompose it as in the proof of Proposition 3.2 and, for the moment, just consider the simple piece of*

_{j}*M*

*i*

adjacent to *S**j*. We will call it *N**i* for convenience. We rst assume that *N**i* is
not pasted to itself across *S** _{j}*.

This *N**i* has an embedded essential annulus compatible with its bration of one
of the following types:

an essential annulus from *S** _{j}* to a component of

*@*

_{0}

*N*

*(\essential" here means incompressible and not parallel into*

_{i}*@*1

*N*

*i*by an isotopy that keeps the one boundary component in

*@*

_{0}and the other in

*@*

_{1}, but of course allows the rst to isotop to

*@*

_{0}

*\@*

_{1});

an essential annulus from *S** _{j}* to

*S*

*in*

_{j}*N*

*(incompressible and not parallel to*

_{i}*@*

_{1}

*N*

*).*

_{i}This can be seen on a case by case basis by considering the eight cases of Figure 1.

If we had an essential annulus of the rst type in *N**i* then, depending on the
situation on the other side of *S** _{j}*, we can either glue it to a similar annulus on
the other side to obtain an essential annulus in (M; @M) crossing

*S*

*in a circle, or glue two parallel copies to an essential annulus from*

_{j}*S*

*j*to itself on the other side, to obtain an essential annulus crossing

*S*

*in two circles. Either way we see that*

_{j}*S*

*was not canonical, so this cannot occur.*

_{j}Thus the only possibility is that we have essential annuli from *S** _{j}* to itself on
both sides of

*S*

*j*, which we can then glue together to get a torus or Klein bottle crossing

*S*

*in two circles. If it were a Klein bottle, then the boundary of a regular neighbourhood would be an essential torus crossing*

_{j}*S*

*in four circles, contradicting that*

_{j}*S*

*j*is canonical. If it is a torus it will be essential unless it is parallel into

*@M*. The only way this can happen is if the pieces on each side of

*S*

*are of the type of case 2 or 4 of Figure 1 and*

_{j}*S*

*is the annular part of*

_{j}*@*

_{1}of these pieces. Moreover, we see that the

*M*

*i*on each side of

*S*

*j*is simple, for

*N*

*would otherwise have to be case 2 of Figure 1 with another Seifert building*

_{i}block pasted along the inside torus component of *@*1*N**i* and we could nd an
annulus in *M**i* from *S**j* to itself that goes through this additional building block
and is therefore not parallel to *@M*.

We must nally consider the case that *N** _{i}* is pasted to itself across

*S*

*. An an- nulus connecting the two boundary components of*

_{j}*N*

*i*that are pasted becomes a torus or Klein bottle in

*M*. If it were a Klein bottle, then the boundary of a regular neighbourhood would be an essential torus crossing

*S*

*in two circles, contradicting that*

_{j}*S*

*is canonical. If it is a torus it is essential unless it is parallel into*

_{j}*@M*. By considering the cases in Figure 1 one sees that the only possible

*N*

*is Figure 1.1 pasted as in the bottom picture of Figure 5. More- over, as in the previous paragraph we see that there cannot be another Seifert building block pasted to the remaining free annulus.*

_{i}Summarising, we have:

**Lemma 3.4** *If two bred pieces of the W{decomposition are adjacent in* *M*
*with matching brations then they are each of type 2 or 5 of Figure 1 matched*
*along the annular part of* *@*_{1}*. If a bred piece is adjacent to itself then it is of*
*type 1 of Figure 1. The possibilities are drawn in Figure 5.*

Figure 5: Matched annuli. In each case we have drawn the base surface of the Seifert
bration with the portion of the boundary in *@*0 drawn solid. The matched annulus
lies over the dashed line in the interior.

**Denition 3.5** We shall call an annulus separating two matched bred pieces
or a bred piece from itself as in Lemma 3.4 a*matched annulus. If we delete all*
matched annuli from the W{system then the remaining surfaces will be called
the *JSJ{system* and the decomposition of *M* along this JSJ{system will be
called the *JSJ{decomposition of* *M*.

Let *fS*1*; : : : ; S**t**g* be the JSJ{system of canonical surfaces, so that splitting along
this system gives the JSJ{decomposition. We recover the W{decomposition as
follows: for each piece *M** _{i}* of the JSJ{decomposition as in Figure 5 we add an
annulus as in that gure.

**4** **Other versions of JSJ{decomposition**

The JSJ{decomposition is the same as the canonical splitting as described by
Jaco and Shalen in [3] Chapter V, section 4. That decomposition is charac-
terised by the fact that it is a decomposition along a minimal family of essen-
tial annuli and tori that decompose *M* into bred and simple non-bred pieces.

The JSJ{system of surfaces satises this minimality condition by construction.

Jaco and Shalen’s \characteristic submanifold" is a bred submanifold of
*M* which is essentially the union of the bred parts of the JSJ{splitting except
that:

wherever two bred parts of the JSJ{decomposition meet along an essen-
tial torus or annulus, thicken that torus or annulus and add the resulting
*T*^{2}*I* or *AI* to the complementary part *M−* ;

wherever two non-bred pieces meet along an annulus or torus, thicken
that surface and add the resulting *AI* or *T*^{2}*I* to .

The special case that *M* has Euler characteristic 0, so its boundary consists
only of tori, deserves mention. If there is an annulus in the maximal system of
canonical surfaces, then any adjacent piece *M** _{i}* of the W{decomposition has an
annular component in its

*@*

_{0}and is therefore Seifert bred by Proposition 3.2.

Moreover, the brations on these adjacent pieces match along this annulus, so it is a matched annulus. Hence:

**Proposition 4.1** *If* *M* *has Euler characteristic* 0 *(equivalently,* *@M* *consists*
*only of tori) then the JSJ{system consists only of tori.*

This proposition seems less well-known than it should be. It is contained in Proposition 10.6.2 of [5] and is used extensively without reference in [1].

There is another modication of the JSJ{decomposition that is often useful,
called the*geometric decomposition, since it is the decomposition that underlies*
Thurston’s geometrisation conjecture (or rather \geometrisation theorem" since

Thurston proved it in the Haken case). The geometric decomposition may have incompressible M¨obius bands and Klein bottles as well as annuli and tori in the splitting surface. It is obtained from the JSJ{decomposition by eliminating all pieces which are bred over the M¨obius band as follows:

delete the canonical annulus that bounds any piece which is an *I*{bundle
over a M¨obius band and replace it by the *I*{bundle over the core circle
of the M¨obius band (which is itself a M¨obius band), and

delete the canonical torus that bounds any piece which is an *S*^{1}{bundle
over a M¨obius band and replace it by the *S*^{1}{bundle over the core circle
of the M¨obius band (which is a Klein bottle).

One advantage of the geometric decomposition is that it lifts correctly in nite covering spaces.

We may also consider only essential annuli or only essential tori and restrict our denition of \canonical" to the one type of surface. Thus we dene an essential annulus to be \annulus-canonical" if it can be isotoped o any other essential annulus, and an essential torus is \torus-canonical" if it can be isotoped o any other essential torus.

**Proposition 4.2** *An essential annulus is annulus-canonical if and only if it is*
*canonical or is as in Figure 6. An essential torus is torus-canonical if and only*
*if it is canonical or is parallel to a torus formed from a canonical annulus and*
*an annulus in* *@M* *(Figure 7).*

Figure 6: The annulus over the dashed interval is annulus-canonical although it has an essential transverse torus. The dashed portion of the M¨obius band boundary represents a canonical annulus.

**Proof** The \if" is clear. For the only if, note that any annulus-canonical
annulus that is not canonical will intersect an essential torus and must therefore
occur inside a Seifert bred piece. It will thus play a similar role to the matched
canonical annuli discussed above. But the argument we used to classify matched
canonical annuli shows that any matched annulus-canonical annulus is as in

Figure 7: The torus over the inner circle is always torus-canonical, unless it bounds a solid torus. The dashed portion of the outer boundary represents a canonical annulus.

Lemma 3.4 and is thus canonical, except that the argument using the boundary of a regular neighbourhood of a Klein bottle no longer eliminates the case of Figure 6. The argument in the torus case is similar and left to the reader.

**5** **Only torus boundary components**

If *M* only has torus boundary components then we can do W{decomposition
from the start only using torus-canonical tori. This leads to a much simpler
proof. There are many fewer cases to consider and the issue of \matched annuli"

disappears | Seifert brations never match across a canonical torus, so the W{

decomposition one gets by this approach is exactly the JSJ{decomposition. The general case as described by Jaco and Shalen can then be deduced from this case by a doubling argument. If one is only interested in 3{manifold splittings this is therefore probably the best approach.

We sketch the argument. If *M* is not an *S*^{1}{bundle over torus or Klein bottle
we cut a non-simple piece *N* =*M** _{i}* of this \toral" W{decomposition into pieces

*N*

*along a maximal system of disjoint non-parallel essential tori, analogously to the proof of Proposition 3.2. These pieces turn out to be of nine basic types, namely what one gets by taking the examples of Figure 1 that have no annuli in*

_{j}*@*

_{0}(cases 3,5,6,7,8), and then allowing some but not all of the boundary components to be in

*@*

_{0}(so, for example, Case 3 splits into three types according to whether 0, 1, or 2 of the boundary components are in

*@*0.) One then shows, as before, that the brations match up to give a Seifert bration of

*N*.

The classication of pieces *M** _{i}* that are both simple and Seifert bered is well
known and can also easily be extracted from our discussion. The ones with
boundary are precisely cases 3,5,6,7,8 of Figure 1 but with any number of
boundary components in

*@*

_{0}

*M*=

*@M*. The closed ones are manifolds with nite

*H*1(M) that ber over

*S*

^{2}with at most three exceptional bers or over R

*P*

^{2}with at most one exceptional ber.

**6** **Bibliographical Remarks**

The theory of characteristic submanifolds of 3{manifolds for Haken manifolds with toral boundaries was rst outlined by Waldhausen in [11]; see also [12] for his later account of the topic. It was worked out in this form by Johannson [5].

The description of the decomposition in terms of annuli and tori was given in
Jaco{Shalen’s memoir [3] (see also Scott’s paper [7] for this description as well as
a proof of the Enclosing Theorem). The idea of canonical surfaces is suggested
by the work of Sela and Thurston; a similar idea was used independently by
Leeb and Scott in [6] in the context of non-positively curved manifolds. General-
izations using submanifolds of the boundary are discussed in both Jaco{Shalen
and Johannson; the characteristic submanifold that we described here is with
respect to the whole boundary (T =*@M*, in the terminology of [4]).

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[2] **M Freedman,** **J Hass,** **P Scott,** *Least area incompressible surfaces in 3{*

*manifolds, Invent. Math. 71 (1983) 609{642*

[3] **W H Jaco,** **P B Shalen,** *Seifert bered spaces in 3{manifolds, Memoirs of*
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[4] **W H Jaco,***Lectures on 3{manifold topology, AMS Regional Conference Series*
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[5] **K Johannson,** *Homotopy equivalences of 3{manifolds with boundary, Lecture*
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[6] **B Leeb,** **P Scott,** *A geometric characteristic splitting in all dimensions,*
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[7] **G P Scott,** *Strong annulus and torus theorems and the enclosing property of*
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