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Contents
Preface
v
1 Topological Manifolds
1.1 Topological Manifolds . . . . .
1.2 Quotient Spaces and Connected
1.3 Triangulations . . . . . . . . .
1.4 Partition of Unity . . . . . . .
1.5 Imbeddings and Immersions . .
1.6 Manifolds with Boundary . . .
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Sums
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2 Local Theory of Smooth Functions
2.1 Differentiability . . . . . . . . . . . . . . . . . .
2.2 Tangent Vectors . . . . . . . . . . . . . . . . .
2.3 Smooth Maps . . . . . . . . . . . . . . . . . . .
2.4 Diffeomorphisms and Maps of Constant Rank .
2.5 Smooth Submanifolds in Rn . . . . . . . . . . .
2.6 Smooth Functions . . . . . . . . . . . . . . . .
2.7 Smooth Vector Fields . . . . . . . . . . . . . .
2.8 Local Flows . . . . . . . . . . . . . . . . . . . .
2.9 Differentiation of Vector Fields by Vector Field
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3 Global Theory of Smooth Functions
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3.1 Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 50
Bibliography
53
Index
54
vii
Differentiable Manifolds
Sungpyo Hong
Pohang University of Science and Technology
Chapter 1
Topological Manifolds
1.1
Topological Manifolds
Definition 1.1 A topological space X is locally Euclidean if for any x ∈
X there is an n ∈ N, and an open neighborhood U ⊆ X of x, an open subset
W ⊆ Rn and a homeomorphism ϕ : U → W , which is called a coordinate
chart at x ∈ X and denoted by (U, ϕ).
The number n, determined uniquely by x, is called the local dimension
of x, denoted by n = d(x).
Example 1.1 (1) Any open subset X ⊆ Rn is locally Euclidean, Hausdorff
and 2-nd countable with d(x) = n for all x ∈ X.
˙
(2) X = R∪{∗},
disjoint union of R and {∗}: The basis for the topology is
the usual open subsets in R, and the open neighborhoods of ∗ are (V −{0})∪
{∗} for any open neighborhood V of 0 ∈ R. Then X is locally Euclidean,
2-nd countable, but not Hausdorff since 0 and ∗ can not be separated by
disjoint open sets. d(x) = 1 for all x ∈ X.
Facts: (1) Each connected component of a locally Euclidean space X is an
open subset of X.
(2) A connected locally Euclidean space X is path connected.
(3) A locally compact (in particular, a locally Euclidean), Hausdorff
space is regular: any proper closed set C and a point not in C are separated
by disjoint open neighborhood of each.
(4) [Urysohn] 2-nd countable regular spaces are metrizable, so that manifolds are metrizable: Thus continuity, closure, compactness, etc, on manifolds can be reduced in terms of sequentially.
1
2
CHAPTER 1. TOPOLOGICAL MANIFOLDS
(5) [Invariance of domain by L. Brouwer] If U ⊆ Rn is open and f :
U → Rn is continuous and 1-1, then f (U ) is open in Rn . Thus, if open
sets U → Rn and V → Rm are homeomorphic, then n = m. Moreover, if
X is connected locally Euclidean, then the local dimension d : X → N is a
constant function on X, called the dimension of X.
Definition 1.2 A topological space X is a topological manifold of dimension n (or, an n-manifold) if X is locally Euclidean, Housdorff, 2-nd
countable with d(x) = n = dim X for all x ∈ X.
Open subsets in Rn are manifolds. A connected, compact, locally Euclidean (so that, 2-nd countable), Housdorff (so that, metrizable) space is
an n-manifold for some n ∈ N.
Example 1.2 (1) If N and M are n and m-manifolds, respectively, then
N × M is an (n + m)-manifold. Thus the n-Torus T n ≡ S1 × · · · × S1 is an
n-manifold.
(2) Sn ≡ {x ∈ Rn+1 | kxk = 1} is an n-manifold. A vector y ∈ Rn+1 is
tangent to Sn at x ∈ Sn if x ⊥ y, denoted by (x, y) ∈ Sn × Rn+1 , and
T (Sn ) ≡ {(x, y) ∈ Sn × Rn+1 | x ⊥ y, x ∈ Sn , y ∈ Rn+1 } ⊆ Rn+1 × Rn+1 .
The continuous map p : T (Sn ) → Sn defined by p(x, y) = x is called the
tangent bundle of Sn , T (Sn ) is the total space, Sn is the base space,
and p is called the natural projection map. For each x ∈ Sn , the set
Tx (Sn ) = p−1 (x)
becomes an n-dimensional vector space under the operations:
a · (x, y) ≡ (x, a · y),
(x, y1 ) + (x, y2 ) ≡ (x, y1 + y2 ).
For an open subset U ⊆ Sn , let pU : T (U ) = p−1 (U ) ⊆ T (Sn ) → U
be the tangent bundle of U . A continuous map s : U → T (U ) such that
pU ◦ s = idU is called a vector field, or a section, on U .
Then one can show that, for x ∈ Sn , there is an open neighborhood
U ⊆ Sn of x and vector fields si : U → T (U ), 1 ≤ i ≤ n, such that
{s1 (u), . . . , sn (u)} form a basis of the vector space Tu (Sn ), for u ∈ U . Thus,
ϕ : U × Rn → T (U )
P
defined by ϕ(u, y) = (u, y i si (u)) is a homeomorphism, so that T (U ) is a
2n-manifold. Moreover, for each u ∈ U , ϕu (y) = ϕ(u, y) defines an isomorphism ϕu : Rn → Tu (Sn ).
1.2. QUOTIENT SPACES AND CONNECTED SUMS
3
Let r(n) be the maximum number of vector fields si : Sn → T (Sn ),
1 ≤ i ≤ r(n), that are everywhere linearly independent.
Definition 1.3 The sphere Sn is parallelizable if r(n) = n.
Thus, for any n, Sn is locally parallelizable so that T (U ) = U × Rn .
However, if Sn is parallelizable, then T (Sn ) = Sn × Rn . It has long been
known that r(2n) = 0: that is, every vector field on S2n is somewhere zero.
It also has been known for some time that S1 , S3 , S7 are parallelizable.
Theorem 1.1 (R. Bott and J. Milnor, M. Kervaire) The sphere Sn is
parallelizable if and only if n = 0, 1, 3, or 7.
For n = 0, S0 = {±1} ⊆ R has 0 independent fields.
Theorem 1.2 There is a multiplication on Rn+1 without zero divisor if and
only if n = 0, 1, 3, or 7.
Indeed, R1 = R, R2 = C, R4 = H (the quaternions), R8 = Ca(the Cayley
numbers).
1.2
Quotient Spaces and Connected Sums
Let X be a topological space, and let “∼” denote an equivalence relation on
X such that:
x ∼ x, ∀ x ∈ X;
x ∼ y ⇒ y ∼ x;
x ∼ y and y ∼ z ⇒ x ∼ z.
Definition 1.4 For x ∈ (X, ∼), the equivalence class of x is the set
[x] ≡ {y ∈ X | y ∼ x}. The set of such equivalence classes partition X,
denoted by X/ ∼ and called the quotient space of X modulo ∼.
The surjective map π : X → X/ ∼ that assigns to each x ∈ X its
equivalence class π(x) ≡ [x] ∈ X/ ∼ is called the quotient map. The
topology on X/ ∼ is given by open sets U ⊆ X/ ∼ such that π −1 (U ) ⊆ X
is open in X, which is called the quotient topology and makes π to be
continuous.
4
CHAPTER 1. TOPOLOGICAL MANIFOLDS
Facts: (1) Even if X is Hausedorff, X/ ∼ needs not be Haudorff: For
example, X = R × {0, 1}, and (x, α) ∼ (y, β) if and only if either α = β and
x = y, or α 6= β and x = y 6= 0. Thus [(0, 0)] 6= [(0, 1)] and [(x, 0)] = [(x, 1)]
for all x(6= 0) ∈ R. Then π : X → X/ ∼ is an open map and X/ ∼ is locally
Euclidean, but not Hausdorff, while X is Hausdorff.
(2) When π : X → X/ ∼ is an open map, X/ ∼ is Haosdorff if and only
if R ≡ {(x, y) ∈ X × X | x ∼ y} is closed subset in X × X (See Theorem
13.12 in Willard’s General Topology). Hence, if π : X → X/ ∼ is an open
map and R is a closed subset in X × X, X is a manifold if and only if X/ ∼
is a manifold.
(3) In general, X/ ∼ needs not be 2-nd countable even if X is. However,
it is if π : X → X/ ∼ is an open map and X is 2-nd countable.
(4) X/ ∼ is compact, or connected, as the continuous image of X if X
is compact, or connected, respectively.
(5) For any f : X → Y such that f (x) = f (y) whenever x ∼ y (i.e., f
respects ∼), there is the induced map f : X/ ∼→ Y such that f (π(x)) =
f (x): that is, the following diagram commutes:
X
∀f
π R
X/ ∼
- Y
>
∃f
Since the quotient map π is continuous, it is easy to see that f is continuous if and only if f is.
Example 1.3 (1) Let X = [0, 1]. Define x ∼ y on X if x = y or {x, y} =
{0, 1}. Define f : X → S1 by f (x) = e2πix , which is a continuous surjection,
and respects ∼ since [0] = [1] and f (0) = f (1). Hence, there exists a
continuous map f : X/ ∼→ S1 such that f ◦ π = f by Fact (5). Clearly,
f is bijective. Since X is compact, X/ ∼ is also compact by Fact (4).
Since S1 is Hausdorff, a 1-1 continuous map f from a compact space to a
Hausdorff space is a homeomorphism, so that X/ ∼ and S1 are canonically
homeomorphic.
(2) For X = Sn ⊆ Rn+1 , define u ∼ v if u = ±v. The quotient space
Sn / ∼≡ Pn is called the projective n-space. The quotient map π : Sn →
Pn is given by π(u) = [u] = {±u}.
(3) Let X = S1 × [0, 1] and Y = D2 the unit circle and disk in C. On X,
define (x, s) ∼ (y, t) if x = y and s = t 6= 0, or s = t = 0 for all x, y ∈ S1 .
Define p : X → D2 by p(x, t) = tx. Then p respects ∼ since [(x, 0)] = [(y, 0)]
and p(x, 0) = p(y, 0) so that p is 1-1 on S1 × (0, 1] and collapses S1 × {0} to
1.2. QUOTIENT SPACES AND CONNECTED SUMS
5
a point 0 ∈ D2 . By the same argument as (1), p : X/ ∼→ D2 is a canonical
homeomorphism.
In general, if A ⊆ X is a subset of X, we can define ∼A by x ∼A y if
either x = y, or x, y ∈ A. Then X/ ∼A is the quotient by collapsing A to a
single point in X, denoted by X/ ∼A ≡ X/A.
Example 1.4 (1) Note that Sn−1 = ∂Dn is the boundary of the unit n-disk
(or, n-ball). Then Dn /Sn−1 ≃ Sn is a canonical homeomorphism.
(2) Consider Rn as an abelian topological group under vector addition.
The Zn is a normal sub group of Rn , so that
Tn ≡ Rn /Zn ≃ S1 × · · · × S1
is the n-torus.
Definition 1.5 Let M1 and M2 be 2-manifolds. Let Di ⊆ Mi be imbedded
disks, and Mi′ = Mi −int(Di ), i = 1, 2. Then, ∂Mi′ in Mi are homeomorphic
to ∂Di as they are circles, so that they can be glued together by those
homeomorphisms. The resulting 2-manifold, denoted by M1 #M2 , is called
the connected sum of M1 and M2 .
Example 1.5 (1) The cylinder C is obtained by identifying the left side
with the right side in the same direction of the square D = [0, 1] × [0, 1]. It
is also called a handle sometimes.
(2) The Möbius band is demoted by M2 , which is obtained by identifying
the left side with the right side in opposite direction of the square D =
[0, 1] × [0, 1]. It is also called a cross cap sometimes.
(3) If the top and the bottom of D are identified and the left and the right
are identified in the same orientation, the resulting space is the 2-torus T2 .
(4) If the top and the bottom of D are identified in the same direction,
and the left and the right are identified in orientation-reversing flip, the
resulting 2-manifold is called the Klein bottle, denoted by K2 .
(5) If the pairs of opposite side of D are both identified in the reverse
directions, the resulting 2-manifold is called the real projective plane,
denoted by P2 .
Example 1.6 One can easily show that
(1) P2 = S2 #M2 = D/ ∼ where ∼ is by identifying the boundary circle
of D in antipodal points.
(2) K2 = P2 #P2 = S2 #M2 #M2 .
(3) T22 ≡ T2 #T2 . In general, the g holed torus T2g ≡ T2 # · · · #T2 .
(4) T2 #P2 = P2 #K2 = P2 #P2 #P2 . P2g ≡ P2 # · · · #P2 .
6
CHAPTER 1. TOPOLOGICAL MANIFOLDS
Theorem 1.3 Every compact, connected 2-manifold without boundary is
either T2g , or P2g .
The number g in the connected sum representation of T2g , or P2g is called
the genus of T2g , or P2g .
1.3
Triangulations
The standard 2-simplex in R2 is the closed triangular region
△ ≡ {(x, y) ∈ R2 | x, y ≥ 0, and x + y ≤ 1},
which is the convex hull of the points v0 = (0, 0), v1 = (1, 0), v2 = (0, 1)
with the three edges
e0 = {(x, y) ∈ △ | x + y = 1},
e1 = {(0, y) ∈ △ | 1 ≥ y ≥ 0},
e2 = {(x, 0) ∈ △ | 0 ≤ x ≤ 1}.
Let X be a compact 2-manifold. Let
ϕi : △ → △i ≡ ϕi (△) ⊆ X, 1 ≤ i ≤ r,
be homeomorphisms of △ onto the images △i in X such that △i and △j
are either disjoint or have just one vertex ϕi (vk ) = ϕj (vℓ ), or have one edge
ϕi (ek ) = ϕj (eℓ ), in common.
Definition 1.6 A triangulation of X is a decomposition
T ≡ {△i ⊆ X | ϕi : △ → △i , 1 ≤ i ≤ r}
of X into a finite number of such △i ’s, 1 ≤ i ≤ r such that X =
Sr
1 △i .
Theorem 1.4 (T. Radó) Every compact 2-manifold X admits a triangulation.
We usually take the standard counterclockwise orientation given by the
ordering
e1
e0
e2
v0 ) ≡ (v0 , v1 , v2 ).
v2 →
v1 →
(v0 →
Then the homeomorphism α : △ → △ defined by α(x, y) = (y, x) gives
orientation reversing.
1.3. TRIANGULATIONS
7
Let T be a triangulation of a compact surface X. Each triangle △i ≡
ϕi (△) has the standard orientation (ϕi (v0 ), ϕi (v1 ), ϕi (v2 )). Two triangles △i
and △j with a common edge e are coherently oriented if their standard
orientations along e are in opposite directions. Since the number of triangles
in a triangulation is finite, one can try to make all the orientations coherent:
Starting with one triangle, the next triangle can be made to be coherent
with the first one by using the orientation reversing map α, if necessary.
Continuing this for all the triangles, one can either make all the pairs are
coherently oriented or be lead to conflicting orientations on at least one
simplex.
Definition 1.7 A triangulation of X is said to be orientable, or nonorientable, if every pair of the triangles in the triangulation is coherently
oriented, or there is at least one conflicting orientation.
A compact surface X is orientable if a triangulation is orientable. Otherwise, it is nonorientable.
Note that it can be proven that, if a triangulation of X is orientable,
then so are any other triangulations.
Example 1.7 (1) The orientable surfaces are T2g for g ≥ 0 with T20 ≡ S2
and T21 = T2 .
(2) The nonorientable surfaces are P2g for g ≥ 1.
Let T be a triangulation of a compact surface X. Let V denote the
number of vertices, E the number of edges, and F the number of triangular
faces.
Theorem 1.5 The number F − E + V is independent of the choice of a
triangulation T of X, so that it depends only on X.
Definition 1.8 The Euler characteristic of X is χ(X) ≡ F − E + V .
Example 1.8 (1) For two compact connected surfaces M1 and M2 , one can
show that
χ(M1 #M2 ) = χ(M1 ) + χ(M2 ) − 2.
(2) χ(T2g ) = 2(1 − g) for g ≥ 0. Thus S2 only has χ(S2 ) = 2 > 0 and
χ(T2g ) ≤ 0 for all g ≥ 1.
(3) χ(P2 ) = 1, χ(K2 ) = χ(P2 #P2 ) = χ(P2 ) + χ(P2 ) − 2 = 0, and in
general χ(P2g ) = 2 − g for g ≥ 1.
8
CHAPTER 1. TOPOLOGICAL MANIFOLDS
1.4
Partition of Unity
Recall that a topological manifold X is locally Euclidean, Hausdorff, and
2-nd countable space. The first property, in particular, implies locally compactness, which together with the second property imply regularity of X.
One can also show that the third property together with the regularity and
locally compactness imply paracompactness of X. This paracompactness
also imply the existence of the partition of unity on X, which is the most
basic building block of the manifold theory. For instance, it is essential to
extend a locally defined function to global function on X, or to integrate a
function on a manifold, etc.
Definition 1.9 An open cover U ≡ {Uα }α∈A of open subsets Uα of a manifold X is locally finite if each x ∈ X has an open neighborhood W such
that W ∩ Uα 6= ∅ for only finitely many indices α ∈ A.
Definition 1.10 An open cover V = {Vβ }β∈B of X is a refinement of
another open cover U = {Uα }α∈A if for any β ∈ B there is α ∈ A such that
V β ⊆ Uα .
Definition 1.11 A Hausdorff space X is paracompact if it is regular and
if every open cover of X has a locally finite refinement.
Definition 1.12 A partition of unity on X is a collection {λα : X →
R+ }α∈A of continuous functions such that
(1) {supp(λα ) | α ∈ A} is a locally finite closed cover of X, where
+
supp(λα ) = closure{λ−1
α (R − {0})} = Cl{x ∈ X | λα (x) > 0}.
P
(2)For all x ∈ X, α∈A λα (x) = 1, which is a finite sum.
Definition 1.13 A partition of unity {λα } on X is subordinate to an open
cover V = {Vβ }β∈B if for all α ∈ A there is a β ∈ B such that supp(λα ) ⊆ Vβ .
Theorem 1.6 Every locally compact 2-nd countable Hausdorff space X is
paracompact.
Corollary 1.7 Every manifold is paracompact.
Lemma 1.8 If X is paracompact and U ≡ {Uα }α∈A is an open cover of X,
then there is a locally finite refinement V = {Vα }α∈A such that Vα ⊆ Uα for
all α ∈ A.
1.5. IMBEDDINGS AND IMMERSIONS
9
Lemma 1.9 A paracompact space is normal.
Theorem 1.10 (Uryshon’s lemma) If X is a normal space and if A and
B are closed disjoint subsets, then there is a continuous function f : X →
[0, 1] such that f |A ≡ 1 and supp(f ) ⊆ X − B (or f |B ≡ 0).
Theorem 1.11 If X is a paracompact space and U ≡ {Uα }α∈A is an open
cover of X, then there exists a partition of unity subordinate to U.
Theorem 1.12 (Stronger form) If X is a manifold and U ≡ {Uα }α∈A is
an open cover of X, then there exists a countable partition of unity {λi }i∈B
subordinate to U with compact supp(λi ) for all i.
Corollary 1.13 Every open cover of a manifold admits a subordinate partition of unity.
Corollary 1.14 If U is open and K is closed in X such that K ⊆ U , then
there is a continuous function f : X → R such that
(1) 0 ≤ f (x) ≤ 1 for all x ∈ X,
(2) f (x) = 1 for all x ∈ K,
(3) f (x) = 0 for all x ∈ X − U , or supp(f ) ⊆ U .
1.5
Imbeddings and Immersions
Definition 1.14 Let N n and M m be topological manifolds of dimensions
n and m, respectively.
(1) A topological imbedding of N into M is a continuous map i : N →
M which is a homeomorphism of N onto i(N ) ⊆ M in the relative
topology in M .
(2) A topological immersion of N into M is a continuous map j : N →
M such that ∀ x ∈ N there is an open neighborhood W of x in N such
that j|W : W → M is a topological imbedding.
(3) The imbedded image X = i(N ) ⊆ M of a manifold N in a manifold
M is called a submanifold of M .
(4) The image of a 1-1 immersion is called an immersed submanifold
of M .
In the following figure, the left is an immersed submanifold, the center is a submanifold, and the right is an immersion but not an immersed
submanifold of R2 .
10
CHAPTER 1. TOPOLOGICAL MANIFOLDS
R
9
i
?
-
jz
?
6
Not imbedding
But immersion
R2
Imbedding
Not imbedding
But immersion
Example 1.9 Consider f : S2 → R4 defined by the formula
f (x, y, z) = (yz, xz, xy, x2 + 2y 2 + 3z 2 ).
Then one can show that f ((x, y, z) = f (−(x, y, z)), and so f induces a well
defined topological imbedding f : P2 ֒→ R4 .
Theorem 1.15 If M is a compact n-manifold, then there exists an integer
k > n and an imbedding i : M ֒→ Rk .
Proof: Since M is compact, we can take a finite open cover {Ui }ri=1 of M
with homeomorphisms ϕi : Ui → Wi ⊆ Rn , and let {λi }ri=1 be a partition of
unity subordinate to {Ui }ri=1 . Take k = r(n + 1) and define an imbedding
i : M ֒→ Rn × · · · × Rn × R × · · · × R = Rk
given by
i(x) = (λ1 (x)ϕ1 (x), . . . , λr (x)ϕr (x), λ1 (x), . . . , λr (x)),
where 0 · ϕi (x) = 0, ∀ x ∈ M . Since supp(λi ) ⊆ Ui and domain of ϕi is Ui ,
λi (x)ϕi (x) = 0 on M − Ui . This implies i : M → Rk is continuous. Thus
i(M ) is compact and Hausdorff as M is.
To show i is 1-1 (that is, i is a homeomorphism). Suppose that i(x) =
i(y). There is j such that λj (x) 6= 0. Then (nr + j)-th coordinates of
i(x) and i(y) are λj (x) = λj (y) 6= 0 so that x, y ∈ supp(λj ) ⊆ Uj . Also,
λj (x)ϕj (x) = λj (y)ϕj (y) so that ϕj (x) = ϕj (y). Since ϕj is 1-1, x = y. It is well known that a differentiable n-manifold M can be imbedded in
R2n+1 , which is the best possible in the sense that there exist n-manifolds
that can not be imbedded in R2n , due to H. Whitney.
1.6. MANIFOLDS WITH BOUNDARY
1.6
11
Manifolds with Boundary
Note that the closed n-ball Dn = {x ∈ Rn | kxk ≤ 1} is not a manifold
in our sense, since a point on the boundary ∂Dn = Sn−1 does not have a
neighborhood homeomorphic to an open subset of Rn .
Definition 1.15
(1) The Euclidean half space of dimension n is
Hn ≡ {(x1 , . . . , xn ) ∈ Rn | x1 ≤ 0}.
(2) The boundary of Hn is
∂Hn ≡ {(x1 , . . . , xn ) ∈ Rn | x1 = 0} ≃ Rn−1 .
(3) The interior of Hn is
int(Hn ) ≡ {(x1 , . . . , xn ) ∈ Rn | x1 < 0}.
Definition 1.16 (1) A topological space X is an n-manifold with boundary if X is Hausdorff and 2-nd countable, and for each x ∈ X there is an
open neighborhood Ux of x in X and a homeomorphism
ϕ : Ux −→ Wx (= ϕ(Ux )) ⊆ Hn
which is an open subset in Hn .
(2) A point x of X is a boundary point, or interior point, if ϕ(x) ∈
∂Hn , or ϕ(x) ∈ int(Hn ), respectively.
(3) The boundary of X is the set of all boundary points of X, denoted
by ∂X.
Facts: (1) X = int(X) ∪˙ ∂X.
(2) If ∂M = ∅ and ∂N 6= ∅, then ∂(N × M ) = ∂N × M .
(3) Let M1 and M2 be compact 3-manifolds with nonempty connected
boundaries ∂Mi 6= ∅. For closed imbedded 2-disks Di ⊆ Mi , choose a
homeomorphism φ : D1 → D2 , and define an equivalence relation on X ≡
M1 ∪˙ M2 by x ∼φ y if either x = y, y = φ(x), or x = φ(y). Then the
quotient space is a compact 3-manifold, denoted by X/ ∼φ ≡ M1 ∪φ M2
whose boundary is
∂(M1 ∪φ M2 ) = ∂M1 #∂M2 .
Thus, if M and N are compact, connected 2-manifolds that bound suitable
compact 3-manifolds, then M #N is also the boundary of some compact
3-manifold.
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CHAPTER 1. TOPOLOGICAL MANIFOLDS
(4) Note that every compact, connected orientable surfaces are S2 and
= T2 × · · · T2 , which are the boundaries of some compact 3-manifolds.
(5) Compact, connected nonorientable surface are P2g = P2 × · · · P2 .
Note that the Klein bottle K2 is boundary of a 3-manifold X: Construct
X from the solid cylinder D2 × [0, 1] identify D2 × {1} with D2 × {0} by
orientation reversing flip about a diameter. Since K2 = P2 #P2 , by (3) above,
K2 is the boundary of a compact 3-manifold X. In general, the connected
sum of even number of P2 is the boundary of a compact 3-manifold.
It is known that a connected sum of odd number of projective planes fail
to bound a compact 3-manifold, as P2 fails to do.
T2g
Example 1.10 (1) Note that the unit open ball B n = {x ∈ Rn | kxk < 1}
is int(Dn ) and Sn−1 = ∂Dn . Thus Dn = Sn−1 ∪˙ B n .
(2) The solid torus is Tn ≡ D2 × Tn−1 with
∂(D2 × Tn−1 ) = ∂D2 × Tn−1 = S1 × Tn−1 = S1 × (S1 × · · · × S1 ) = Tn . Facts: (1) Any connected compact 1-manifold is homeomorphic to either
S1 ) or [0, 1].
(2) The boundary of any compact 1-manifold consists of even number of
points.