NOTES_ON_BASIC_3_MANIFOLD_T.PDF

Notes on Basic 3-Manifold Topology

Allen Hatcher

Chapter 1. Canonical Decomposition

1. Prime Decomposition.

2. Torus Decomposition.

Chapter 2. Special Classes of 3-Manifolds

1. Seifert Manifolds.

2. Torus Bundles and Semi-Bundles.

Chapter 3. Homotopy Properties

1. The Loop and Sphere Theorems.

These notes, originally written in the 1980’s, were intended as the beginning of a

book on 3 manifolds, but unfortunately that project has not progressed very far since

then. A few small revisions have been made in 1999 and 2000, but much more remains

to be done, both in improving the existing sections and in adding more topics. The

next topic to be added will probably be Haken manifolds in

http://www.math.cornell.edu/˜hatcher

The three chapters here are to a certain extent independent of each other. The

main exceptions are that the beginning of Chapter 1 is a prerequisite for almost ev-

erything else, while some of the later parts of Chapter 1 are used in Chapter 2.

3.2. For any subsequent

updates which may be written, the interested reader should check my webpage:

1.1

Prime Decomposition

Chapter 1. Canonical Decomposition

This chapter begins with the ﬁrst general result on 3 manifolds, Kneser’s theorem

that every compact orientable 3 manifold M decomposes uniquely as a connected

sum M

P 1 ]

P i ]S 3 .

After the prime decomposition, we turn in the second section to the canonical

torus decomposition due to Jaco-Shalen and Johannson.

We shall work in the C 1 category throughout. All 3 manifolds in this chapter

are assumed to be connected, orientable, and compact, possibly with boundary, unless

otherwise stated or constructed.

1. Prime Decomposition

Implicit in the prime decomposition theorem is the fact that S 3 is prime, other-

wise one could only hope for a prime decomposition modulo invertible elements, as

in algebra. This is implied by Alexander’s theorem, our ﬁrst topic.

Alexander’s Theorem

This quite fundamental result was one of the earliest theorems in the subject:

T heorem 1.1. Every embedded 2 sphere in R

bounds an embedded 3 ball.

! R the height function

given by the z coordinate. After a small isotopy of S we may assume h is a morse

function with all its critical points in distinct levels. Namely, there is a small homotopy

of h to such a map. Keeping the same x and y coordinates for S , this gives a small

homotopy of S in

be an embedded closed surface, with h : S

3 . But embeddings are open in the space of all maps, so if this

homotopy is chosen small enough, it will be an isotopy.

Let a 1 <

<a n be noncritical values of h such that each interval

−1

;a 1 ,

contains just one critical value. For each i , h − 1 a i consists of

a number of disjoint circles in the level z

, a n ;

a i . By the two-dimensional Schoenﬂies

Theorem (which can be proved by the same method we are using here) each circle of

h − 1 a i bounds a disk in the plane z

a i . Let C be an innermost circle of h − 1 a i ,

in the sense that the disk D it bounds in z

a i is disjoint from all the other circles

of h − 1 a i . We can use D to surger S along C . This means that for some small "> 0

we ﬁrst remove from S the open annulus A consisting of points near C between the

two planes z

a i

" , then we cap off the resulting pair of boundary circles of S

−

" which these circles bound. The result of

this surgery is thus a new embedded surface, with perhaps one more component than

S ,if C separated S .

This surgery process can now be iterated, taking at each stage an innermost re-

maining circle of h − 1 a i , and choosing " small enough so that the newly introduced

−

A the disks in z

a i

]P n of 3 manifolds P i which are prime in the sense that they can

be decomposed as connected sums only in the trivial way P i

P roof : Let S R

a 1 ;a 2 ,

by adding to S

Canonical Decomposition

1.1

horizontal cap disks intersect the previously constructed surface only in their bound-

aries. See Figure 1.1. After surgering all the circles of h − 1 a i for all i , the original

surface S becomes a disjoint union of closed surfaces S j , each consisting of a number

of horizontal caps together with a connected subsurface S j of S containing at most

one critical point of h .

Figure 1.1

L emma 1.2. Each S j is isotopic to one of seven models: the four shown in Figure 1.2

plus three more obtained by turning these upside down. Hence each S j bounds a

ball.

Figure 1.2

P roof : Consider the case that S j has a saddle, say in the level z a . First isotope

S j in a neighborhood of this level z

a so that for some > 0 the subsurface S j of

S j lying in a

−

is vertical, i.e., a union of vertical line segments, except

int S j of the saddle, where S j has the standard form of the

saddles in the models. Next, isotope S j so that its subsurface S j (the complement of

the horizontal caps) lies in S j . This is done by pushing its horizontal caps, innermost

ones ﬁrst, to lie near z

a , as in Figure 1.3, keeping the caps horizontal throughout

the deformation.

Figure 1.3

After this move S j is entirely vertical except for the standard saddle and the horizontal

caps. Viewed from above, S j minus its horizontal caps then looks like two smooth

circles, possibly nested, joined by a 1 handle, as in Figure 1.4.

in a neighborhood N

1.1

Prime Decomposition

Figure 1.4

Since these circles bound disks, they can be isotoped to the standard position of one

of the models, yielding an isotopy of S j to one of the models.

The remaining cases, when S j has a local maximum or minimum, or no critical

points, are similar but simpler, so we leave them as exercises.

Now we assume the given surface S is a sphere. Each surgery then splits one

sphere into two spheres. Reversing the sequence of surgeries, we start with a collec-

tion of spheres S j bounding balls. The inductive assertion is that at each stage of the

reversed surgery process we have a collection of spheres each bounding a ball. For

the inductive step we have two balls A and B bounded by the spheres @A and @B

resulting from a surgery. Letting the " for the surgery go to 0 isotopes A and B so

that @A

(i) A

D , with pre-surgery sphere denoted @A

(ii) B

A , with pre-surgery sphere denoted @A

−

B .

Since B is a ball, the lemma below implies that A and A

B are diffeomorphic. Since

A is a ball, so is A

B , and the inductive step is completed.

@M, let the manifold N be

obtained from M by attaching a ball B n via an identiﬁcation of a ball B n− 1

@B n

with the ball B n− 1

@M. Then M and N are diffeomorphic.

@B n and using isotopy extension, we

conclude that the pair B n ;B n− 1 is diffeomorphic to the standard pair. So there is an

isotopy of @N to @M in N , ﬁxed outside B n , pushing @N

−

@M across B n to @M

−

@N .

By isotopy extension, M and N are then diffeomorphic.

Existence and Uniqueness of Prime Decompositions

Let M be a 3 manifold and S

M a surface which is properly embedded, i.e.,

@S , a transverse intersection. We do not assume S is connected. Deleting

a small open tubular neighborhood NS of S from M , we obtain a 3 manifold M

which we say is obtained from M by splitting along S . The neighborhood NS is

j j

@B equals the horizontal surgery disk D . There are two cases, up to changes

in notation:

L emma 1.3. Given an n manifold M and a ball B n− 1

P roof : Any two codimension-zero balls in a connected manifold are isotopic. Ap-

plying this fact to the given inclusion B n− 1

Canonical Decomposition

1.1

an interval-bundle over S ,soif M is orientable, NS is a product S

−

";" iff S is

S has two

components, M 1 and M 2 . Let M i be obtained from M i by ﬁlling in its boundary

sphere corresponding to S with a ball. In this situation we say M is the connected

sum M 1 ]M 2 . We remark that M i is uniquely determined by M i since any two ways

of ﬁlling in a ball B 3 differ by a diffeomorphism of @B 3 , and any diffeomorphism

of @B 3 extends to a diffeomorphism of B 3 . This last fact follows from the stronger

assertion that any diffeomorphism of S 2 is isotopic to either the identity or a reﬂection

(orientation-reversing), and each of these two diffeomorphisms extends over a ball.

See [Cerf].

The connected sum operation is commutative by deﬁnition and has S 3

as an

M]S 3 is obtained by choosing the sphere S

to bound a ball in M . The connected sum operation is also associative, since in a

sequence of connected sum decompositions, e.g., M 1 ]M 2 ]M 3 , the later splitting

spheres can be pushed off the balls ﬁlling in earlier splitting spheres, so one may

assume all the splitting spheres are disjointly embedded in the original manifold M .

Thus M

M 1 ]

]M n means there is a collection S consisting of n

−

1 disjoint

S has n components M i , with M i obtained from M i by ﬁlling

in with balls its boundary spheres corresponding to spheres of S .

A connected 3 manifold M is called prime if M

S 3 .

For example, Alexander’s theorem implies that S 3 is prime, since every 2 sphere in S 3

bounds a 3 ball. The latter condition, stronger than primeness, is called irreducibility:

M is irreducible if every 2 sphere S 2

P]Q implies P

S 3

or Q

M bounds a ball B 3

M . The two conditions

are in fact very nearly equivalent:

P roposition 1.4. The only orientable prime 3 manifold which is not irreducible is

S 1

S 2 .

P roof :If M is prime, every 2 sphere in M which separates M into two components

bounds a ball. So if M is prime but not irreducible there must exist a nonseparating

sphere in M . For a nonseparating sphere S in an orientable manifold M the union

of a product neighborhood S

I of S with a tubular neighborhood of an arc joining

to S

in the complement of S

I is a manifold diffeomorphic to S 1

S 2

minus a ball. Thus M has S 1

S 2

as a connected summand. Assuming M is prime,

then M

S 2 .

It remains to show that S 1

S 1

S 2

is prime. Let S

S 1

S 2

be a separating sphere,

S consists of two compact 3 manifolds V and W each with boundary a

2 sphere. We have

S 2

j j

1 W , so either V or W must be simply-

connected, say V is simply-connected. The universal cover of S 1

1 S 1

S 2

1 V

S 2

can be identiﬁed

with

−f

, and V lifts to a diffeomorphic copy V of itself in

−f

. The sphere

orientable.

Now suppose that M is connected and S is a sphere such that M j j

identity since a decomposition M

spheres such that M j j

so S 1

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