LECTURE I

TRANSC. EDWIN SPARK

The primary objects of study in this course are manifolds. The title

specifically calls 3-manifolds, although the actual content will not be

quite so restricted. Manifolds are in some sense general, natural and

homogeneous objects, which grants them a certain niceness.

In contrast, our main tool for studying them, namely piece-wise lin-

ear (PL) topology, appears to have none of these properties. However,

it will allow us to learn and know things about the more general ob-

jects, and as a set of techiniques has two nice properties: finiteness,

which among other things allows induction, and a kind of tameness,

manifesting as a niceness of intersection.

Other tools relevant to the content of this course include techniques

from hyperbolic geometry, which we won’t use much of, but will provide

some inspiration for our work, and algebra.

Note that the Riemannian Geometry course is concerned with similar

objects, but with very different tools and perspective.

1. Basic Definitions

Recall that a topological space is second countable if it has a count-

able basis of open sets, and separable if there exists a dense set that is

countable.

Write Rn+ for the half-space {x ∈ Rn | xn ≥ 0}.

Definition 1.1. A (topological) n-manifold, M is a

(1) second-countable Hausdorff topological space,

or

(2) second-countable metric space,

or

(3) separable metric space

such that each point of M lies in an open neighbourhood homeomorphic

to Rn or Rn+.

Note that (1), (2) and (3) are equivalent, each ensuring that the space

in some sense is not “too big”, so when working form this definition,

Date: 25th July, 2016.1

2 TRANSC. EDWIN SPARK

one can use whichever is most convenient. That (1) implies either of the

other statements is a highly non-trivial metrization result from point-

set topology (and hence very much not of interest in this course). The

other implications are much easier.

Remark 1.2. If we wish to emphasise that M is an n-manifold we write

Mn. Note that this does not indicate an n-fold product.

Example 1.3. Some 1-manifolds are the circle S1, a closed line segment,

a half-open line segment and an open line segment. Note that the last

two are homeomorphic to R1+ and R1, respectively.

Example 1.4. Examples of 2-manifolds, or surfaces include the 2-sphere

S2, and a “pair of pants”, homeomorphic to a cylinder with the interior

of a disc removed from its side (See Figure 1.1).

Figure 1.1. A “pair of pants”, with boundary indicated

Definition 1.5. Let M be an n-manifold. Then:

(1) The boundary of M is ∂M , the set of all points x ∈ M such

that no neighbourhood of x is homeomorphic to Rn.

(2) The interior of M is int (M) = M \ ∂M .

(3) M is closed if M is compact and ∂M = ∅.

Note that the boundary consists of those points that locally look like

the half-space, and coincides with the definition in point-set topology.

In contrast, the definition of “closed” for manifolds is quite different

from the definition in point-set topology.

Exercise 1. Prove:

(1) either ∂Mn = ∅ or ∂Mn is an (n− 1)-manifold.

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(2) for all manifolds M , ∂∂M = ∅.

Example 1.6 (n-ball, n-sphere). This example is really a definition,

presenting some basic n-manifolds, the so-called n-balls. We present

two definitions, the “smooth version”, Bn = {x ∈ Rn | |x| ≤ 1} and

the “PL version”, Dn = {x ∈ Rn | |xi| ≤ 1} (See Figure 1.2).

Figure 1.2. B2, D2 and S1

These are the unit balls with respect to the Euclidean norm and the

1-norm. The latter is equipped with a natural product structure, which

is very useful as we shall see in section 4 below.

One can check that for each n, Bn and Dn are homeomorphic.

Furthermore, from this we can define the n-sphere, as Sn = ∂Bn+1.

It is easily seen that this coincides with the definitions of S1 and S2

given above.

Exercise 2. Prove that Sn is homeomorphic with:

(1) The one-point compactification of Rn.

Rn ∪ {∞}, topologised so that for all compact sets K ⊂ Rn,

(Rn \K) ∪ {∞} is an open neighbourhood of ∞.

(2) The double of Bn (See Figure 1.3).(Bn × 0) ∪ (Bn × 1)�∼, where (x, 0) ∼ (x, 1)∀x ∈ ∂Bn.

Note that in (2) we have adopted a notational convention where

one-element sets are written without braces.

Exercise 3. Prove that for manifolds Mm and Nn, M × N is an

(m+ n)-manifold.

Example 1.7. S1×S1 is a 2-manifold, known as the torus. Furthermore,

since fundamental group distributes over direct products, π1(S1×S1) ∼=

π1(S1)× π1(S1) ∼= Z⊕ Z.

4 TRANSC. EDWIN SPARK

Figure 1.3. The doubles of B1 and B2

In contrast, π1(S2) = 0. Fundamental group is preserved under

homeomorphism, so this tells us that S1 × S1 is not homeomorphic to

S2.

Solving problems of this type is one of the tasks of manifold theory.

2. Motivating Problems of Manifold Theory

The following problems provide the most basic motivation for man-

ifold theory.

Problem 1 (Homeomorphism Problem). Given two n-manifolds, are

they homeomorphic?

This is a decision problem, with two answers: “yes”, or “no”.

Problem 2 (Classification Problem). Determine a complete list of n-

manifolds up to homeomorphism without duplicates, along with an al-

gorithm which locates any given n-manifold on the list.

Note that this encompasses the Homeomorphism Problem, since ap-

plying the algorithm twice will answer the question.

This problem is clearly a very difficult one, and indeed is sometimes

impossible, so we can also consider a weaker version:

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Problem 3 (Algorithmic Classification Problem). Determine an algo-

rithm that enumerates all n-manifolds up to homeomorphism without

duplication.

2.1. What is known. In the case where n = 2, all three problems are

solved. We will state part of the answer to the Classification Problem

in the next section.

When n ≥ 4, all three problems are unsolvable. In fact, in the next

lecture we will see that the Homeomorphism Problem is unsolvable on

a very nice subclass of manifolds. As a result, manifold theory in these

dimensions is interested in restrictions of these problems, and finding

restricted subclasses upon which the problems become tractable.

In the case of 3-manifolds, the Homeomorphism Problem and the

Algorithmic Classification Problem were solved by Perelman in around

2003 (in the process, proving the special case known as the Poincare

conjecture). There are, however, no good ideas about how to solve the

Classification Problem itself.

The n = 3 case is thus in a certain sense the most interesting, and as

suggested by the title of this course, where we will be most concerned.

3. Classification of Compact Surfaces

In this section, we provide a partial answer to the Classification

Problem in the case n = 2, restricting ourselves to compact surfaces.

Usually in this course we will consider only the further restricted case of

compact and orientable manifolds. The fact that non-compact surfaces,

a quite general class of objects, can be classified at all is amazing, but

we will say no more of that here.

We first provide some examples of compact surfaces, developing no-

tation at the same time.

Example 3.1 (Closed orientable surfaces). Let Fg denote S2 with g ≥ 0

pair-wise disjoint discs (that is, copies of D2) removed, with a “handle”

(that is, a copy of S1×S1 \D2) attached to each boundary component

(See Figure 3.1).

Example 3.2 (Closed non-orientable surfaces). Let Nh denote S2 with

h ≥ 1 pair-wise disjoint discs removed, with a Mobius Band (See Figure

3.2) attached to each boundary component (See Figure 3.3).

Example 3.3 (Non-closed surfaces). For Fg (resp. Nh) with the interior

of b ≥ 1 pair-wise disjoint discs removed, we write Fg,b (resp. Nh,b).

6 TRANSC. EDWIN SPARK

Figure 3.1. Constructing F3

Figure 3.2. A Mobius Band; note that it is non-

orientable and has only 1 boundary component

Figure 3.3. Constructing N2

We can now state the following classification theorem, which answers

the first part of the Classification Problem. It was first understood by

Jordan and Mobius in the 1860s, although the first written proof did

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not appear until 1907, published by Dehn and Heegaard. The first

rigorous proof was published by Brahana in 1921.

For modern and quick proofs, two are recommended; the first is

“Conway’s ZIP” proof, written by Francis and Weeks [1], the second,

by Hatcher, uses the “Kirby torus trick” [2].

Theorem 3.4. (Classification of compact surfaces) Each compact 2-

manifold is homeomorphic to exactly one of Fg,b or Nh,b, where g ≥ 0,

h ≥ 1 and b ≥ 0.

In some ways, this is the most disappointing theorem in topology,

since it tells us that in the case of surfaces, there is nothing surprising.

It’s worth noting, however, that this theorem is a classification up to

homeomorphism. The embeddings of surfaces can be complicated and

surprising, and form the subject of knot theory.

3.1. An Algorithm. This theorem solves the first part of the Classi-

fication Problem, but not the second. However, its neatness suggests

that knowing three pieces of information are sufficient to classify any

given surface, suggesting the following algorithm:

(1) Check if the surface is orientable, or not (giving F or N , resp.).

(2) Count the number of boundary components (which is b).

(3) Compute the Euler characteristic (determining the remaining

number uniquely).

Remark 3.5. The Euler characteristic has a number of equivalent for-

mulations. We recall some:

(1) χ(S) = V − E + F , which we can consider the “PL version”.

(2) χ(S) = 12π

(∫SKdA+

∫∂Sκds), where K is the Gaussian cur-

vature, and κ the geodesic ∂-curvature.

(3) χ(S) =∑

indicies of critical points of smooth vector fields with

isolated singularities. This is the Poincare-Hopf Theorem.

In this course, we will restrict ourselves to the PL formulation, leav-

ing the other two to the Riemannian Geometry course.

The questions from Section 2 are not the only sources of motivation

in manifold theory. Another, is the search for ways to collect and sort

manifolds according to real, structural differences.

In the case of orientable surfaces, we can see qualitative differences

between the following three groups:

8 TRANSC. EDWIN SPARK

Fg g = 0 g = 1 g ≥ 2

π1 finite infinite, abelian infinite, non-abelian

growth rate1 of π1 trivial polynomial exponential

χ > 0 0 < 0

metric spherical Euclidean hyperbolic

curvature 1 0 −1Note that those above the line are algebraic descriptors, whereas those

below it have a combinatorial and/or geometric flavour. These are the

sorts of tools we wish to develop.

4. Handle Decompositions

We observed earlier that given an embedding of a surface, the em-

bedding can be quite complicated, making it difficult to work out where

to start. One way is to study the surface via “essential” circles, whose

complement can be construted from discs. The “circles” in this context

are actually pieces of surface, and as “thickened” circles are themselves

copies of D2.

An analogous tchnique is useful in all dimensions. We now introduce

these so-called handle decompositions.

Definition 4.1. Let M be an n-manifold. An n-manifold V is M with

a handle h of index k attached if:

(1) h = V \M ∼= Dk × Dn−k (that is, the handle is Dn with an

imposed product structure),

and

(2) h ∩M ∼= ∂Dk × Dn−k (this is the attaching region and what

distinguishes the index),

and

(3) h ∩M = h ∩ ∂M (that is, the handle is attached only to the

boundary of M).

We will call such a handle an n-dimensional k-handle, written as hk.

Given such a k-handle h (see Figure 4.1), we furthermore define its:

core: Dk × 0

attaching sphere: ∂Dk × 0, which is the boundary of core(h),

co-core: 0×Dn−k, and

belt sphere: 0× ∂Dn−k, the co-core’s boundary.

1The growth rate of a group is a concept from geometric group theory, and

considers in some sense the growth rate of balls in a metrized group.

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Figure 4.1. Some 1-handles

In some sense, adding a k-handle to a manifold is adding a k-dimensional

“disc”, which is the core, which we need to “thicken” by the second

term in the product in order to still have an n-manifold.

This can be made more precise by observing that there is a defor-

mation retract of the handle onto the core that respects the structure.

In particular, if ϕ is the homeomorphism from (2) in Definition 4.1,

and ϕ is its restriction to the attaching sphere, there is a deformation

retarct of M ∪ h = M ∪ϕ(Dk ×Dn−k) onto M ∪ϕ

(Dk × 0

).

Definition 4.2. A handle decomposition of a manifold Mn is an ex-

pression:

M =

(((⋃i

h0i

)∪⋃j

h1j

)∪ . . . ∪

⋃k

hnk

),

where each ∪ means handle addition and handles of the same index are

pair-wise disjoint in M .

10 TRANSC. EDWIN SPARK

The expression given in the definition implies a nice ordering of the

handle additions, building up from 0-handles one dimension at a time.

In standard presentations, a more general definition is given, and then

a theorem is presented proving that such a nice ordering is possible.

Since this is not a course in handle theory, we bundle this result into

the definition.

Theorem 4.3 (Kirby-Siebenmann (1970s)). Every compact n-manifold

has a handle decomposition.

This result is one of the reasons handle decompositions are as pow-

erful as they are, and in fact, they form the cornerstone of higher

manifold theory. For example, the fact that every compact manifold

has a CW-complex structure is proven via this result.

There are also connections to Morse theory, because Morse functions

give handle structures on smooth manifolds. This will be discussed in

slightly more detail in the next lecture.

Example 4.4. Sn can be decomposed as a 0-handle attached to an n-

handle. Note that this is a rephrasing of Exercise 2, part (2).

The torus, S1 × S1, or F1 in the notation of Example 3.2, can be

decomposed as a 0-handle, two 1-handles and a 2-handle. This can be

seen fairly easily from the flat representation (See Figure 4.2).

Figure 4.2. Handle decomposition for F1, with cores

(in purple) and co-cores (in red)

Exercise 4. Determine handle decompositions for:

(1) the 3-Torus, T 3 = S1 × S1 × S1, and

(2) S2 × S2.

References

[1] G. K. Francis and J. R. Weeks, “Conway’s ZIP Proof ”, The American Mathe-

matical Monthly 106, 1999: 393–9.

[2] A. Hatcher, The Kirby Torus Trick for Surfaces, https://www.math.cornell.

edu/~hatcher/Papers/TorusTrick.pdf