Homotopy 最近の更新履歴 KSakaiIDTopology

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トポロジー _{III [講義録]} _{2010 年版} An Introduction to Algebraic Topology

Homotopy Groups

— A Short Course

Katsuro Sakai

Institute of Mathematics University of Tsukuba

代数的トポロジー入門

ホモトピー群

短期課程

筑波大学・数学系

酒井克郎

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Preface

In Algebraic Topology, topological properties of topological spaces and continuous maps are studied by using algebraic methods (e.g., groups, homomor- phisms, etc.). Homology Theory and Homotopy Theory are typical and clas- sical. As an introduction to Algebraic Topology, this course is provided. Our purpose is knowing how to associate Topology (spaces) with Algebra (groups) by learning the process of defining the homotopy groups, and how to use Al- gebra in Topology by applying the homotopy groups. The 1-dimensional homotopy group is also called the fundamental group, which is not commutative (i.e., non-Abelian), whereas higher dimensional homotopy groups are commutative. We use more than half of this lecture to discuss the fundamental groups of spaces. In the last two sections, CW-complexes are studied, whose concept generalizes one of simplicial complexes. It will be proved that a continuous map between CW-complexes is homotopy equivalence if it induces the isomorphisms between the homotopy groups.

The reader is required to familiar with basic language of sets and maps, elementary point-set topology of Euclidean spaces and elementary group theory. One can study this course independently from the course of Homology Theory.

Tsukuba, March, 2010 Katsuro Sakai

2010 K. Sakaic

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Homotopy Groups

— A Short Course

In this course, we study Homotopy Groups, which is a sequence of groups associated with a topological space. These groups are commutative except for the first Homotopy Group. The first is also called the Fundamental Group.

As the most important fact, it is shown that homeomorphic spaces have isomorphic homotopy groups. Some applications of homotopy groups are given. The homotopy groups of several surfaces are also calculated.

Notations

We use the following notations:

• N = 1, 2, . . . — the set of natural numbers; ^{自然数全体}

• Z = 0, ±1, ±2, . . . — the set of integers; ^整数全体

• ω = N ∪ {0} = 0, 1, 2, . . .

— the set of non-negative integers; ^{非負整数全体}

• R = (−∞, ∞) — the real line; (実) 数直線

• R₊ = [0, ∞) — the non-negative half line; ^半数直線

• I = [0, 1] — the unit closed interval, ∂I = {0, 1}; ^{単位閉区間}

• Xⁿ = X × · · · × X

n times

, ∀x ∈ Xⁿ, x = (x(1), . . . , x(n)),

x(i) ∈ X — the i-th coordinate of x, ^{i 座標} pr_i : Xⁿ→ X (pr_i(x) = x(i)) — the projection; ^射影

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• R^k — the k-dimensional Euclidean space, ^{k 次元}

ユークリッド空間

∀x ∈ R^k, x =^k_i=1x(i)² — (Euclidean) norm, ^{ノルム}

∀x, y ∈ R^k, ∀t ∈ R,

x + y = (x(1) + y(1), . . . , x(k) + y(k)) — sum, ^和 tx = (tx(1), . . . , tx(k)) — scalar product, ^{スカラー倍} ei ^{∈ R}^k (i = 1, . . . , k) defined by

ei(j) =

1 if i = j, 0 if i = j;

• B^k =

def

x ∈ R^k x ≤ 1 — the unit k-ball; k 次元単位球_(体)

• S^k−1 = ∂B^k =

def

x ∈ R^k x = 1 — the unit (k − 1)-sphere, k − 1 次元単位球面

• id_X = id — the identity map of X; ^恒等写像

• ca — the constant map with {a} the image; ^定値写像

• int_XA = int A — the interior of A in X; ^内部

bdXA = bd A — the boundary of A in X; ^境界

clXA = cl A — the closure of A in X; ^閉包

• card A — the cardinality of A (i.e., the number of elemnents of A); ^濃度

• X ≈ Y — X is homeomorphic to Y ; ^同相

• C(X, Y ) — the set of all continuous maps from X to Y . For a metric space X = (X, d), we denote

• diam A =

defsup{d(x, y) | x, y ∈ A} — the diameter of A; ^直径

• dist(A, B) =

definf{d(x, y) | x ∈ A, y ∈ B}

— the distance between A and B; ^距離

• d(x, A) =

def^inf^y∈Ad(x, y) (= dist({x}, A));

• B(x, ε) =

def{y ∈ X | d(x, y) < ε} — the ε-neighborhood of x. ε 近傍

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A pair (X, A) of spaces means a pair of a space X and its subspace A. ^空間の対 A map (or a homeomorphism) f : (X, A) → (Y, B) from a pair (X, A) of ^{対の間の写像} spaces to another pair (Y, B) means a map (or a homeomorphism) f : X → Y 同相写像

with f (A) ⊂ B (or f (A) = B), whence f |A : A → B is also a map (or a homeomorphism). When there is a homeomorphism f : (X, A) → (Y, B), we

say that (X, A) is homeomorphic to (Y, B) and denote (X, A) ≈ (Y, B). ^{空間の対の同相} A pair (X, x0) of a space (or a set) X and a point x0 ∈ X is called a

pointed space (pointed set) with x0 the base point. ^{基点を持つ空間}^(集合)

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1 Homotopy and Paths

It is said that continuous maps f, g : X → Y are homotopic (in Y ) and ^{ホモトピック} denoted by f ≃ g if there is a continuous map h : X × I → Y such that

h0 = f and h1 = g, where ht: X → Y , t ∈ I, is defined by ht(x) = h(x, t) for

x ∈ X. In this case, h : X × I → Y is called a homotopy from f to g. We ^{ホモトピー} denote h : f ≃ g. Given a subset A ⊂ X, when ht|A = f |A = g|A for every

t ∈ I, it is said that f and g are homotopic relative to A and denoted by f ≃ g rel. A or h : f ≃ g rel. A. When A is a singleton {x0}, we write f ≃ g rel. x0 (or h : f ≃ g rel. x0). Note that f ≃ g rel. ∅ is none other than f ≃ g.

1.1 Proposition Let A ⊂ X, B ⊂ Y , f, f^′, f^′′ ∈ C(X, Y ) and g, g^′ ∈ C(Y, Z). Then, the following hold:

(i) f ≃ f rel. A;

(ii) f ≃ f^′ rel. A ⇒ f^′ ≃ f rel. A;

(iii) f ≃ f^′ rel. A, f^′ ≃ f^′′ rel. A ⇒ f ≃ f^′′ rel. A;

(iv) f ≃ f^′ rel. A, g ≃ g^′ rel. B and f (A), f^′(A) ⊂ B ⇒ gf ≃ g^′f^′ rel. A. Proof. (i) The homotopy h : f ≃ f rel. A is defined by h(x, t) = f (x). (ii) If h : f ≃ f^′ rel. A then a homotopy h^′ : f^′ ≃ f rel. A can be defined by h^′(x, t) = h(x, 1 − t). (iii) If h : f ≃ f^′ rel. A and h^′ : f^′ ≃ f^′′ rel. A then a homotopy h^′′ : f → f^′′ rel. A can be defined by h^′′_t = h2t for 0 t ¹₂ and h^′′_t = h^′_2t−1) for ¹₂ t 1 (i.e., h^′′(x, t) = h(x, 2t) for 0 t ¹₂ and h^′′(x, t) = h^′(x, 2t − 1) for ¹₂ t 1). (iv) If h : f ≃ f^′ rel. A and h^′ : g ≃ g^′ rel. B then a homotopy h^′′ : gf → gf^′ rel. A can be defined by h^′′_t = h^′_tht (i.e., h^′′(x, t) = h^′(h(x, t), t)).

By Proposition 1.1 above, ≃ (or ≃ rel. A) is an equivalence relation on the set C(X, Y ). The quotient set C(X, Y )/ ≃ (or C(X, Y )/ ≃ rel. A) is denoted by [X, Y ]. The equivalence class of f ∈ C(X, Y ) is called the

homotopy class of f and denoted by [f ]. By (iv) in Proposition 1.1, we ^{ホモトピー類} can define [g][f ] = [gf ] for each f ∈ C(X, Y ) and g ∈ C(Y, Z).

A continuous map f : X → Y is called a homotopy equivalence if ^{ホモトピー同値写像} there is a continuous map g : Y → X such that gf ≃ idX and f g ≃ idY,

where g is called a homotopy inverse of f . When there exists a homotopy ^{ホモトピー逆写像} equivalence f : X → Y , it is said that X is homotopy equivalent to Y

(they are homotopy equivalent or they have the same homotopy type) ^{ホモトピー同値} and denoted by X ≃ Y . In this case, we also say that X has the homotopy ^{ホモトピー型}

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type of Y . Obviously, every homeomorphism is a homotopy equivalence,

hence X ≈ Y implies X ≃ Y . Although M¨obius band is not homeomorphic ^{メビウスの帯} to the annulus, they have the same homotopy type (the homotopy type of ^円環帯 the circle S¹).

Annulus M¨obius band

Figure 1: The M¨obius band and the annulus

A continuous map f : X → Y is said to be null-homotopic (denoted ^{ナル・ホモトピック} by f ≃ 0) if f is homotopic to a constant map. A space X is said to be

contractible if the identity map idX of X is null-homotopic, that is, there ^可縮 is a homotopy h : X × I → X such that h0 = id and h1(X) is a singleton,

where h is called a contraction. ^収縮

1.1 Exercise – Show that every convex set in Rⁿ is contractible. ^演習 1.2 Proposition A space X is contractible if and only if X ≃ {0}.

1.2 Exercise – Prove the above proposition. ^演習

1.3 Proposition If X or Y is contractible, then every continuous map f : X → Y is null-homotopic.

Hint. See Proposition 1.1 (iv).

A continuous map α : I → X is called a path (in X) from α(0) to α(1) ^パス^{, 弧} (or connecting α(0) and α(1)). For paths α, β : I → X with α(1) = β(0), we

define a path α ∗ β : I → X from α(0) to β(1) as follows:

α ∗ β(t) =

def

α(2t) for 0 t ¹₂, β(2t − 1) for ¹₂ t 1.

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α ^β α(1) = β(0)

α(0) β(1)

0 ¹ 1

2

0 1

α ∗ β(¹₂)

α ∗ β(1) α ∗ β(0)

Figure 2: The join of two paths

This path α ∗ β is called the join of α and β. ^{ジョイン}^{, 結び} In general, given paths α₁, . . . , α_n : I → X such that α_i(1) = α_i+1(0) for each

i = 1, . . . , n − 1, we define a path α1∗ · · · ∗ αn : I → X as follows: α1∗ · · · ∗ α_n(t) =

def^αⁱ(nt − i + 1) for ^{i − 1}

n ^t i

n, i = 1, . . . , n.

1

n−1n 2

n 1

0 n

α₁ ^α² α_n

α₁(0)

α₁(1) = α₂(0)

α₂(1) = α₃(0)

α_n−1(1) = α_n(0)

α_n(1)

Figure 3: The join of several paths

The reverse α^← of a path α : I → X is define by ^逆 α^←(t) = α(1 − t) for every t ∈ I,

which is a path from α(1) to α(0). By the definitions, we have the following: (α ∗ β)^←= β^←∗ α^← for every paths α, β : I → X with α(1) = β(0). For each x ∈ X, we denote by c_x : I → X the constant path with c_x(I) = {x}. 1.4 Proposition Let α, α^′, β, β^′, γ : I → X be paths such that α(0) = α^′(0), α(1) = α^′(1) = β(0) = β^′(0) and β(1) = β^′(1) = γ(0). Then, the following hold:

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(i) α ≃ α^′ rel. ∂I and β ≃ β^′ rel. ∂I ⇒ α ∗ β ≃ α^′ ∗ β^′ rel. ∂I; (ii) (α ∗ β) ∗ γ ≃ α ∗ (β ∗ γ) ≃ α ∗ β ∗ γ rel. ∂I;

(iii) α ≃ α^′ rel. ∂I ⇒ α^←≃ α^′← rel. ∂I;

(iv) α ∗ α^←≃ cα(0) rel. ∂I and α^←∗ α ≃ cα(1) rel. ∂I; (v) α ∗ cα(1) ^{≃ c}α(0)∗ α ≃ α rel. ∂I.

Proof. (i) For h : α ≃ α^′ rel. ∂I and h^′ : β ≃ β^′ rel. ∂I, define h^′′ : α ∗ β ≃ α^′ ∗ β^′ rel. ∂I by h^′′_t = ht ∗ h^′_t (i.e., h^′′(s, t) = h(2s, t) for 0 s ¹₂ and h^′′(s, t) = h^′(2s − 1, t) for ¹₂ s 1).

(ii) We can define h : (α ∗ β) ∗ γ ≃ α ∗ (β ∗ γ) rel. ∂I as follows:

h(s, t) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ α

4s 1 + t

for 0 s ^{1 + t} 4 ^, β(4s − (1 + t)) for ^{1 + t}

4 ^s 2 + t

4 ^, γ

4s − (2 + t) 2 − t

for ^{2 + t}

4 ^{s 1.}

α _β γ

γ α β

t

s = ^{2 + t} 4 s = ^{1 + t}

4

1 4 ¹2

0 1

1 2 ³4

Figure 4: (α ∗ β) ∗ γ ≃ α ∗ (β ∗ γ) rel. ∂I

Similarly to the above, we can show that (α ∗ β) ∗ γ ≃ α ∗ β ∗ γ rel. ∂I. (iii) Let h : α ≃ α^′ rel. ∂I. Then, we have h^← : α^← ≃ α^′← rel. ∂I defined by h^←(s, t) = h(1 − s, t).

(iv) We can define h : α ∗ α^←≃ cα(0) rel. ∂I as follows:

h(s, t) =

⎧⎪

⎨

⎪⎩

α(2s) for 0 s ¹₂(1 − t), α(1 − t) for ¹₂(1 − t) s ¹₂(t + 1), α^←(2s) = α(2 − 2s) for ¹₂(t + 1) s 1.

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α _α^r t

s = ¹₂(t + 1) s = ¹₂(1 − t)

0 ¹ 1

2

Figure 5: α ∗ α^←≃ c_α(0) rel. ∂I By replacing α and α^←, the latter half can be obtained.

(v) We can define h : α ∗ cα(1) ≃ α rel. ∂I and h^′ : cα(0)∗ α ≃ α rel. ∂I as follows:

h(s, t) =

⎧⎪

⎨

⎪⎩ α

2s t + 1

for 0 s ¹₂(t + 1), cα(1)(s) = α(1) for ¹₂(t + 1) s 1;

h^′(s, t) =

⎧⎪

⎨

⎪⎩

c_α(0)(s) = α(0) for 0 s ¹₂(1 − t), α

2s + t − 1 t + 1

for ¹₂(1 − t) s 1.

α c_α(1) t

s = ¹₂(t + 1)

0 1 1

2

c_α(0) α t

s = ¹₂(1 − t)

0 1 1

2

Figure 6: α ∗ c_α(1) ≃ α rel. ∂I and c_α(0) ∗ α ≃ α rel. ∂I

Given paths α₁, . . . , α_n : I → X so that α_i(1) = α_i+1(0) for each i = 1, . . . , n − 1, the following can be obtained by induction:

α1∗ · · · ∗ α_n≃ (α₁∗ · · · ∗ α_n−1) ∗ αn rel. ∂I.

A space X is path-connected if every points x, y ∈ X can be connected ^弧状連結 by a path, that is, there is a path α : I → X from x to y (i.e., α(0) = x,

α(1) = y). On the other hand, X is connected if X is not a union of disjoint ^連結 non-empty open sets.

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1.5 Proposition If X is path-connected then X is connected.

Hint. Suppose that X is not connected. Then, a contradiction occur because I is connected.

1.6 Proposition Let f : X → Y be a continuous map. If X is path- connected then f (X) is also path-connected.

1.5 Exercise – Show the above proposition. ^演習

1.7 Proposition A space X is contractible then X is path-connected.

1.6 Exercise – Give a proof of the above proposition. ^演習 It is said that X is locally connected (resp. locally path-connected) ^局所連結 if each x ∈ X has a neighborhood basis consisting of connected (resp. path- ^{局所弧状連結} connected) neighborhoods.

In usual, the local path-connectedness of X is defined by the following condition:

• Every neighborhood U of each x ∈ X contains a neighborhood V of x such that each pair of points y, z ∈ V are connected by a path in U , that is, there is a path f : I → U with f (0) = y and f (1) = z.

In the condition above, let V^∗= {f (1) | f ∈ C(I, U ), f (0) = x} ⊂ U . Then, V^∗ is a neighborhood of x because V ⊂ V^∗. Moreover, it is easy to see that each pair of points y, z ∈ V^∗ can be connected by a path in V^∗, that is, V^∗ is path-connected. Thus, this condition is equivalent to the above definition of local path-connectedness.

A (connected) component of X is defined as a maximal connected

set of X. Then, every connected component of X is a closed set because the ^{(連結) 成分} closure of a connected set in X is also connected. A maximal path-connected

set in X is called a path-component of X. Note that a path-component of ^弧状(連結) 成分

X is not closed in general.

1.8 Example Let X be the closure of the following set in Euclidean plane R²_:

X1 =(x, y) ∈ R² y = sin(1/x), 0 < x 1.

Then, X is connected but not path-connected. Moreover, X has two path- components, the above set X1 and {0} × [−1, 1]. This space X is called the

sin(1/x)-curve. sin(1/x) カーブ (曲線)

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Proof. Since X is the closure of the connected set X1≈ I, it is also connected. Assume that there exists a path f : I → X with f (0) ∈ {0} × [−1, 1] and f (1) ∈ X1. Let t0 = sup(pr₁f )⁻¹(0). Since (pr₁f )⁻¹(0) is closed in I, t0∈ (pr₁f )⁻¹(0), that is, f (t0) = 0. Then, t0 < 1. By the continuity of f , we have 0 < δ < 1 − t0such that t0< t t0+ δ implies f (t) − f (t0) < 1. Choose n ∈ N so that 1/2nπ < pr₁f (t0+ δ). By the intermediate value theorem, we have t0< t1, t2t0+ δ such that pr₁f (t1) = 1/(2n +¹₂)π and pr₁f (t2) = 1/(2n +³₂)π. Then,

f (t1) − f (t2) f (t1) − f (t0) + f (t2) − f (t0) < 2 but

f (t1) − f (t2) | pr₂f (t1) − pr₂f (t2)| = 2, which is a contradiction.

Remark. One should note that the image of a path is very diﬀerent from I. In fact, the following is well known:

Hahn-Mazurkiewiecz Theorem. A space X is the image of ハーン・マツールキエヴィッツの定理

a path if and only if X is connected, locally connected compact metrizable space.

In particular, there is a continuous surjection f : I → I². Such a first exmaple was constructed by Peano. Then, the image of a path is called a

Peano continuum. ^{ペアノ連続体}

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2 Fundamental Groups

A path α : I → X is called a loop (at x₀ ∈ X) if α(0) = α(1) (= x₀). Given ^ループ a point x0 ∈ X, let Ω(X, x0) be the set of all loop at x0. We define

π1(X, x0) =

def^{Ω(X, x}⁰)/ ≃ rel. ∂I

= [(I, ∂I), (X, x0)].

The homotopy class of a loop α ∈ Ω(X, x0) is denoted by [α], that is, [α] =β ∈ Ω(X, x0) α ≃ β rel. ∂I.

By Proposition 1.4 (i), we can define the maltiplication on π1(X, x0) as follows:

[α][β] =

def[α ∗ β] for α, β ∈ Ω(X, x₀).

Due to Proposition 1.4 (ii), this is associative. It follows from Proposition 1.4 (v) that the homotopy class [cx0] is the unit element of π1(X, x0) with respect to this multiplication. Moreover, by Proposition 1.4 (iii), (iv), we can define the inverse of [α] as follows:

[α]⁻¹ =

def^[α

←] for α ∈ Ω(X, x0).

Thus, π1(X, x0) is a group with this multiplication, which is called the fun-

damental group of X at x0. ^基本群

Let ω : I → X be a path from x₀ to x₁ (i.e., ω(0) = x₀ and ω(1) = x₁). We define ω_∗ : π1(X, x1) → π1(X, x0) as follows:

ω_∗([α]) = [ω ∗ α ∗ ω^←] for each α ∈ Ω(X, x1).

x₁ x₀ _ω←

ω

α Figure 7: ω ∗ α ∗ ω^←

2.1 Proposition Let ω, ω^′, ω^′′ : I → X be paths such that ω(0) = ω^′(0) = x0, ω(1) = ω^′(1) = ω^′′(0) = x1 and ω^′′(1) = x2. Then, the following hold:

(i) ω ≃ ω^′ rel. ∂I ⇒ ω_∗ = ω^′_∗ : π1(X, x1) → π1(X, x0);

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(ii) (cx0)_∗ = idπ1(X,x0) : π1(X, x0) → π1(X, x0); (iii) (ω ∗ ω^′′)_∗ = ω_∗ω_∗^′′ : π1(X, x2) → π1(X, x0);

(iv) ω_∗ : π₁(X, x₁) → π₁(X, x₀) is an isomorphism with ω⁻¹_∗ = ω^←_∗; Proof. By Proposition 1.4, it is easy to see (i), (ii) and (iii).

(iv): For each α, β ∈ Ω(X, x₁),

ω ∗ (α ∗ β) ∗ ω^← ≃ ω ∗ α ∗ c_x₀ ∗ β ∗ ω^←

≃ (ω ∗ α ∗ ω^←) ∗ (ω ∗ β ∗ ω^←) rel. ∂I and ω ∗ α^←∗ ω^← = (ω ∗ α ∗ ω^←)^←. This means that

ω_∗([α][β]) = ω_∗([α])ω_∗([β]) and ω_∗([α]⁻¹) = ω_∗([α])⁻¹.

Thus, ω_∗ is a homomorphism. Moreover, since ω ∗ ω^← ≃ c_ω(0) rel. ∂I and ω ∗ ω^←≃ cω(0) rel. ∂I, it follows from (i), (ii) and (iii) that ω_∗ is bijective and ω_∗⁻¹ = ω^←_∗.

2.1 Exercise – Prove (i), (ii) and (iii) in Proposition 2.1 above. ^演習 By Proposition 2.1, we have the following:

2.2 Corollary If X is path-connected then π1(X, x0) ∼= π1(X, x1) for each x0, x1 ^{∈ X.}

When X is path-connected, the isomorphism class of π1(X, x0) is uniquely determined, it is independent from choice of x0 ∈ X. Then, we may write π₁(X) omitting a point x₀ ∈ X.

Let f : X → Y be a continuous map such that f (x0) = y0 (i.e., f : (X, x0) → (Y, y0)). Then, we define f_∗ : π1(X, x0) → π1(Y, y0) as follows:

f_∗([α]) = [f ◦ α] for every α ∈ Ω(X, x0).

2.3 Proposition Let f, f^′ : (X, x₀) → (Y, y₀) and g : (Y, y₀) → (Z, z₀) be continuous maps. Then the following hold:

(i) f ≃ f^′ rel. x₀ ⇒ f_∗ = f_∗ : π₁(X, x₀) → π₁(Y, y₀); (ii) (idX)_∗ = idπ1(X,x0) : π1(X, x0) → π1(X, x0);

(iii) (gf )_∗ = g_∗f_∗ : π1(X, x0) → π1(Z, z0);

(iv) f_∗ : π1(X, x0) → π1(Y, y0) is homomorphism.

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Proof. The assertions (ii) and (iii) are trivial and (i) follows from Proposition 1.1 (iv). It is straightforward to show (iv).

2.2 Exercise – Prove (iv) in Proposition 2.3 above. ^演習 2.4 Corollary Every homotopy equivalence f : (X, x0) → (Y, y0) induces

the isomorphism f_∗ : π1(X, x0) → π1(Y, y0), where if g is a homotopy inverse of f then g_∗ = f_∗⁻¹. Consequenctly, (X, x0) ≃ (Y, y0) implies π1(X, x0) ∼= π1(Y, y0)

In the above corollary, gf ≃ idX rel. x0 and gf ≃ idY rel. y0. However, this condition can be weakened as gf ≃ idX and gf ≃ idY. To show this, we need the following:

2.5 Proposition Let f, f^′ : X → Y be continuous maps with f ≃ f^′. Then, for each x0 ∈ X, there is a path ω : I → Y from f (x₀) to f^′(x0) such that f_∗ = ω_∗f_∗^′ : π1(X, x0) → π1(Y, f (x0)), that is, the following diagram commutes:

π1(Y, f^′(x0))

ω∗

π₁(X, x₀)

f∗^′ _g₃₃

gg gg gg g

f∗ ^W⁺⁺

WW WW WW W

π1(Y, f (x0)).

Proof. Let h : f ≃ f^′ and define ω : I → Y by ω(t) = h(x0, t) for t ∈ I. Then, ω is a path from f (x0) to f^′(x0). For each α ∈ Ω(X, x0), we can define h^′ : f α ≃ ω ∗ (f^′α) ∗ ω^← rel. ∂I as follows:

h^′(s, t) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ω(3s) for 0 s ^t 3^, htα

3s − t 3 − 2t

for ^t

3 ^s 3 − t

3 ^, ω(3 − 3s) for ^{3 − t}

3 ^{s 1.} Thus, we have f_∗([α]) = ω_∗f_∗^′([α]).

2.6 Corollary Every null-homotopic continuous map f : X → Y induces the trivial homomorphism f_∗ : π₁(X, x₀) → π₁(Y, f (x₀)) for each x₀ ∈ X, that is, f_∗(π1(X, x0)) = {e}.

Proof. In Proposition 2.5 let f^′ is the constant map with f^′(X) = {f (x0)}. Since f_∗^′ is trivial, we have the result.

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f α

ω f^′α ω

t

s = ^{3 − t} 3

s = ^t 3

1

3 ²3

0 1

ω

f^′(x₀)

f (x₀)

f^′α

f α

h_tα

ω(t)

Figure 8: f α ≃ ω ∗ (f^′α) ∗ ω^← rel. ∂I

2.7 Corollary Every homotopy equivalence f : X → Y induces the isomorphism f_∗ : π1(X, x0) → π1(Y, f (x0)) for each x0 ∈ X.

Proof. Let g : Y → X be a homotopy inverse of f , that is, gf ≃ idX and f g ≃ idY. By Proposition 2.5, we have a path ω : I → X from gf (x0) to x0

and ω^′ : I → Y from f gf (x0) to f (x0) such that

(gf )_∗ = ω_∗ : π₁(X, x₀) → π₁(X, gf (x₀)) and (f g)_∗ = ω^′_∗ : π₁(Y, f (x₀)) → π₁(Y, f gf (x₀)).

Due to Proposition 2.3 (iii), (gf )_∗ = g_∗f_∗ and (f g)_∗ = f_∗g_∗, where the second f_∗ is diﬀerent from the first f_∗ (the second is f_∗ : π1(X, gf (x0)) → π1(Y, f gf (x0))) but both g_∗ are the same. Thus, we have the following commutative diagram:

π1(X, x0) ^ω^∗ ^//

f∗

π1(X, gf (x0))

f∗

π1(Y, f (x0))

g∗ _m_mm^m ^66m mm mm mm mm

ω^′∗

//_π₁(Y, f gf (x0)),

Since ω_∗ and ω^′_∗ are isomorphisms by Proposition 2.1, it follows that g_∗ : π1(Y, f (x0)) → π1(X, gf (x0)) is an isomorphism, hence so is f_∗ = g_∗⁻¹ω_∗ : π1(X, x0) → π1(X, f (x0)).

Combining Corollaries 2.2 and 2.7, we have the following:

2.8 Corollary Let X and Y be path-connected. Then, X ≃ Y implies π1(X, x0) ∼= π1(Y, y0) for every x0 ^{∈ X and y}0 ^{∈ Y .}

A space X is said to be simply connected if X is path-connected and ^単連結 any continuous map f : S¹ → X is null-homotopic.

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2.9 Proposition For a path-connected space X, the following are equivalent:

(a) X is simply connected;

(b) Every continuous map f : S¹ → X extends to a continuous map ˜f : B² → X;

(c) For each loop α : I → X, α ≃ cx0 rel. ∂I, where α(∂I) = {x0}; (d) π1(X, x0) ∼= {e} for some x0 ^{∈ X.}

Proof. (a) ⇒ (b): For each continuous map f : S¹ → X, we have h : f ≃ cx0^,

x0 ∈ X. Then, we can extend f to a continuous map ˜f : B² → X as follows:

f (z) =˜

⎧⎨

⎩ h

z

z^{, 1 − z}

if z = 0,

x0 if z = 0.

(b) ⇒ (c): Note that (I², ∂I²) ≈ (B², S¹), where ∂I² = (∂I × I) ∪ (I × ∂I). For each loop α : I → X at x0 ∈ X, we define f : ∂I² → X as follows:

f ((∂I × I) ∪ (I × {1}) = {x0} and f (t, 0) = α(t) for each t ∈ I. By (b), f extends to a continuous map h : I² → X, which is a homotopy giving α ≃ cx0 rel. ∂I.

(c) ⇒ (a): Since I/∂I ≈ S¹, we have a quotient map q : I → S¹ such that q⁻¹(e₁) = ∂I and q|I \ ∂I is a homeomorphism. For each continuous map f : S¹ → X, f q is a loop at f (e1), hence we have h : f q ≃ cf (e1) rel. ∂I by (c). Note that h(∂I × I) = {f (e1)}. Then, there is a homotopy ˜h : S¹ × I → X such that ˜h(q × id^I) = h. Since ˜h0q = f q and ˜h1(S¹) = h1(I) = {f (e1)}, it follows that ˜h0 = f and ˜h1 = c_{f (e}₁₎, hence f ≃ 0.

(c) ⇔ (d): The condition (c) means that π₁(X, x₀) = {e} for every x₀ ∈ X. Then, this equivalence follows from Corollary 2.2.

2.3 Exercise – In the proof of (a) ⇒ (b) in the above, verify the ^演習 continuity of ˜f at 0 ∈ B².

2.10 Proposition Every contractible space X is simply connected.

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3 Covering Spaces

Let p : X → X be a continuous surjection. It is said that an open set U ⊂ X

is evenly covered by p if p⁻¹(U ) can be written as the union of disjoint ^{均一に覆われる} open sets Uλ ^⊂X, λ ∈ Λ, i.e., p⁻¹(U ) =_λ∈ΛUλ, and each p|Uλ : Uλ ^{→ U}

is a homeomorphism. When each x ∈ X has an open neighborhood U which

is evenly covered by p, we call p : X → X a covering (projection) and ^被覆写像 X a covering space of X. In case card p ⁻¹(x) = n for every x ∈ X, p (or ^被覆空間 X) is called an n-fold covering. Recall that a contnuous map f : X → Y n 重被覆

is quotient provided R ⊂ Y is open (or closed) in Y if f⁻¹(R) is open (or ^商写像 closed) in X.

3.1 Proposition Every covering projection p : X → X is quotient.

Proof. For R ⊂ X, assume that p⁻¹(R) is open in X. Each x ∈ R has an open neighborhood Ux which is evenly covered by p. In particular, we have an open set U_x^′ in X such that p|U_x^′ : U_x^′ → U_x is a homeomorphism. Since (p|U_x^′)⁻¹(U_x∩R) = U_x^′∩p⁻¹(R) is open in U_x^′, it follows that U_x∩R is open in Ux, hence it is open in X. Then, R =_x∈R(Ux∩ R) is open in X. Therefore, p is quotient.

3.2 Example In the following, we regard S¹ = {z ∈ C | |z| = 1}. (1) exp : R → S¹ — the exponential map defined by exp(t) = e^2πit.

(We can define exp(t) = (cos 2πt, sin 2πt) where S¹= {x ∈ R²| x = 1}.)

R

S¹

exp exp

0

−1 1

1 1

0 1

R −1

Figure 9: The exponential map

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(2) rn: S¹ → S¹ — the n-fold coverling defined by rⁿ(z) = zⁿ.

S¹

r₃ S¹

1 1

√−1 3 ⁺

√2 3ⁱ

√−1 3 ⁻

√2 3ⁱ

1 1

r₃

S¹ S¹

√−1 3 ⁺

√2i 3

√−1 3 ⁻

√2i 3

Figure 10: The 3-fold coverling of S¹

(3) Let T² = S¹ × S¹ be the torus. Then, we have a covering q : R² → T² where q(s, t) = (e^2πis, e^2πit). This is none other than the natural quotient map.

S¹

S¹ q

R²

S¹× S¹ = T²

0 1

1

U

q⁻¹(U )

Figure 11: The coverling q of the torus

The torus T² can be obtained from the square I² by identifying (x, 0) with (x, 1) and (0, y) with (1, y). This can be regarded as the quotient space R²/ ∼, where the equivalence relation ∼ is defined as follows:

(x, y) ∼ (x^′, y^′) ⇐⇒

def ^{x − x}

′_{, y − y}′ _{∈ Z.}

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Considering R² as an additive group and Z² as its subgroup, T² is the quotient group R²/Z².

(4) We can define an nm-fold covering p : T² → T² as follows: p = rn^{× r}m : T² = S¹× S¹ → T² = S¹× S¹,

where rn, rm : S¹ → S¹ are the n-fold and the m-fold coverings defined in (2) above.

(5) Let K² be Klein bottle. Then, the natural quotient map q : R² → K² is a covering.

q R²

0 1

1

q⁻¹(U )

K² U

Figure 12: The coverling q of Klein bottle

Klein bottle T² can be obtained from the square I² by iden- ^{クラインの壺} tifying (x, 0) with (x, 1) and (0, y) with (1, 1 − y). This can be

regarded as the quotient space R²/ ∼, where the equivalence relation ∼ is defined as follows:

(x, y) ∼ (x^′, y^′) ⇐⇒

def ^{x − x}

′_{, y − (−1)}x−x^′_y′ _{∈ Z.}

(6) Let P²be the projective plane, that is, the quotient space of R³\{0}/ ∼, where the equivalence relation ∼ is defined as follows:

x ∼ y ⇐⇒

def ∃t ∈ R \ {0} such that y = tx (i.e., Rx = Ry). For each x ∈ R³\ {0}, the equivalence class of x is denoted by [x], that is, [x] = Rx \ {0}. We can regard P² as a quotient space of the sphere S², where every antipodal points x, −x ∈ S² are identified. Then, the quotient map q : S² → P² is a 2-fold covering.

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The projective plane P² can be also obtained from the disk B² ^射影平面 by identifying every antipodal points of the bounday circle S¹.

In the picture, the upper half is called the cross cap, which is ^{クロス・キャ}^ップ homeomorphic to M¨obius band. This meas that the projective

plane P² can be also obtained by pasting M¨obius band to the disk B² around their boundary circles.

paste _fold

v v

v v v

x, −x ∈ S¹ are identified

◦0 ^x²

x3

R³

◦

◦0 ^x²

x3 _[x]

x ^R³

v = [e₁] [x]

q

≈ x

◦

x₁ x₁

Figure 13: The 2-hold coverling q of the projective plane

The following is a generalization of Example 3.2 (3) and (4):

3.3 Proposition If p : X → X and q : Y → Y are covering projections then p × q : X × Y → X × Y is also a covering.

3.4 Proposition Let p : X → X be a covering projection. Then, for each A ⊂ X, p|p⁻¹(A) : p⁻¹(A) → A is a covering projection.

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Recall that X is locally connected if every neighborhood U of each ^局所連結 x ∈ X contains a connected neighborhood V of x. Here, V can be taken to

be open in X.¹ Indeed, let V be the connected component of int U containing x. Since X is locally connected, each y ∈ V has a connected neighborhood W such that W ⊂ int U . Note that x ∈ V ∪ W ⊂ int U and V ∪ W is connected. Then, it follows that V ∪ W ⊂ V , which means W ⊂ V . Hence, V is open in X.

3.5 Proposition Let p_i : X_i → X, i = 1, 2, be coverings over a locally connected space X. If p : X1 → X2 is a continuous surjection such that p1 = p2p, then p is also a covering projection.

X1 p1

@ @

@@

@

p //_X

2 p2

~~~~^~~

~~^~

X

Proof. For each x ∈ X2, we can choose a connected open neighborhood U of p₂(x) in X which is evenly covered by both p₁ and p₂, that is, p⁻¹₁ (U ) =

λ∈Λ^U^λ ^{and p}⁻¹² ^{(U ) =}

γ∈Γ^V^γ ^{where U}^λ ^⊂^X¹ ^{and V}^γ ^⊂^X² ^{are open,}

Uλ ^{∩ U}λ^′ = ∅ for λ = λ^′ and Vγ ^{∩ V}γ^′ = ∅ for γ = γ^′, each p1^|Uλ : Uλ ^{→ U}

and each p1|Vγ : Vγ → U are homeomorphisms. Observe that each Uλ is connected, hence so is p(Uλ). Then, p(Uλ) ∩ Vγ = ∅ implies p(U_λ) ⊂ Vγ. In this case, we have (p2^|Vγ)(p|Uλ) = p1^|Uλ, hence p|Uλ : Uλ ^{→ p(U}λ) = Vγ

is a homeomorphism. Choose γ ∈ Γ so that x ∈ Vγ. Then, Vγ is an open neighborhood of x evenly covered by p.

In Proposition 3.5 above, the local connectedness of X is essential, that is, Proposition 3.5 does not hold without the local connectedness of X. The following example shows this fact:

3.6 Example Let X = {0} ∪ {n⁻¹ | n ∈ N} ⊂ R, X₁ = X × N and X₂ = X × {0, 1}. The projections p1 = pr_X : X1 ^{→ X and p}2 = pr_X : X2 ^{→ X are}

coverings. We define a map p : X₁ → X₂ as follows:

p|X × {1} = id, p(n⁻¹, n) = (n⁻¹, 1) for each n > 1 and p(x, n) = (x, 0) for n > 1 and x = n⁻¹.

Then, p1 = p2p but p is not a covering. Indeed, p⁻¹(0, 1) is a singleton but each p⁻¹(n⁻¹, 1) has two points if n > 1.

1It should be noted that int V is not connected in general.

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In Example 3.2 (2), the composition of rn and rm is also a covering. In fact, rnm = rnrm. More general, we have the following proposition concerning composition of coverings:

3.7 Proposition Let p : Y → X and q : Z → Y be coverings. If each p⁻¹(x) is finite then pq : Z → X is a covering.

In Proposition 3.7 above, we cannot replace the finiteness of p⁻¹(x) with the one of q⁻¹(y) even if spaces are locally connected.

3.8 Example Let f : R² → R² be the homeomorphism defined by f (x, y) = (¹₂x +¹₂, y). Then, fⁿ(x, y) = (2⁻ⁿx + 1 − 2⁻ⁿ, 2⁻ⁿy) for each n ∈ N. Let

X =

n∈

fⁿ⁻¹(S¹) ⊂ R² and Y = X × N,

where X is called the Hawaiian earring. Then, the projection p = pr_X : ハワイアン・イヤリング

Y → X is a covering.

S¹ f (S¹)

f²(S¹)

e1

X

Figure 14: Hawaiian earring

Now, let g, h, k : R² → R² be the homeomorphisms defined as follows: g(x, y) = (x,¹₂y), h(x, y) = (2 − x, y) and k(x, y) = (−x, −y).

Then, gⁿ(x, y) = (x, 2⁻ⁿy) for each n ∈ N. Moreover, h² = id and k² = id. We define

Z =

n∈

_n

i=1

gⁱ⁻¹(S¹) ∪

i n

hfⁱ(S¹) ∪ khfⁱ(S¹)

× {n}.

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S¹ S¹ g(S¹) hf (S¹)

hf²(S¹)

khf (S¹) ^khf

2_(S1₎

= −hf (S¹) Z

Figure 15: The space Z

Let r₂ : S¹ → S¹ be the 2-fold covering defined in Example 3.2 (2). We define a map q : Z → Y as follows:

q|gⁱ⁻¹(S¹) × {n} =fⁱ⁻¹r2g⁻ⁱ⁺¹|gⁱ⁻¹(S¹)× id, q|hfⁱ(S¹) × {n} =h|hfⁱ(S¹)× id and

q|khfⁱ(S¹) × {n} =hk|khfⁱ(S¹)× id .

Then, q is also a 2-fold covering. However, any neighborhood of e1 ∈ X is not evenly covered by pq, that is, pq is not a covering projection. Indeed, it contains some fⁿ(S¹), hence (pq)⁻¹(fⁿ(S¹)) ⊃ gⁿ(S¹) × {n}.

q Z

Y

q

Figure 16: The 2-hold covering q

3.9 Theorem Let p : X → X be a covering over X such that X is Hausdorﬀ. For two continuous maps f, g : Y → X of a connected space Y , if pf = pg and f (y0) = g(y0) for some y0 ∈ Y then f = g.

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Proof. Let Y0 = {y ∈ Y | f (y) = g(y)}. Then, Y0 = ∅ because y0 ∈ Y0. We shall show that Y0 is open and closed in Y . Then, since Y is connected, it follows that Y0 = Y , which means f = g.

To see that Y0 is open in Y , let y ∈ Y0. Then, pf (y) = pg(y) has an open neighborhood U in X which is evenly covered by p, hence f (y) = g(y) has an open neighborhood V in X such that p|V : V → U is a homeomorphism. We denote s = (p|V )⁻¹ : U → V . By the continuity of f and g, y has an open neighborhood W in Y such that f (W ) ∪ g(W ) ⊂ V . Since pf |W = pg|W and s(p|V ) = id, we have f |W = spf |W = spg|W = g|W , hence W ⊂ Y0. Thus, Y0 is open in Y .

To see that Y \ Y₀ is open in Y , let y ∈ Y \ Y₀. Since X is Hausdorﬀ and f (y) = g(y), X has disjoint open sets U and V such that f (y) ∈ U and g(y) ∈ V . Then, W = f⁻¹(U ) ∩ g⁻¹(V ) is an open neighborhood of y. Observe that f (y^′) = g(y^′) for every y^′ ∈ W , hence W ⊂ Y \ Y0. Thus, Y \ Y0

is open in Y . The proof is completed.

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4 Homotopy Lifting Property

Given a continuous map p : E → B, a lift of a continuous map f : X → B ^リフト(持上げ)

is a continuous map ˜f : X → E such that p ˜f = f , where f : X → B is said

to lift to ˜f . It is said that a continuous map p : E → B has the homotopy ^{持上がる}

lifting property with restect to a space X if the following condition holds: ホモトピー・リフティング・プロパティー

(HL) For each homotopy h : X × I → B, if h0 lifts to a continuous map f : X → E then h lifts to a homotopy ˜h : X × I → E with ˜h0 = f .

X

×0

f //_E

p

X × I

∃_˜_h

o 77o o o o o o

h ^//^B

The following is a direct consequence of the definition:

4.1 Lemma Let p : E → B be a continuous map having the homotopy lifting property with respect to a space X. If a continuous map f : X → B has a lift then every continuous map g : X → B homotopic to f has also a lift.

It is said that a continuous map p : E → B has the absolute homotopy ^{絶対ホモトピー}^{・リフティ}

ング・プロパティー

lifting property if it has the homotopy lifting property with restect to

an arbitrary space. A continuous map p : E → B is called a Hurewicz ^{フレヴィ}^{ッツ・ファイブレ} fibration if p has the absolute homotopy lifting property, where E is called イション

a total space, B a base space and each p⁻¹(b) is the fiber over b ∈ B. ^全空間^{, 底空間}

ファイバー

4.2 Proposition The projection pr_B : B × F → B is a Hurewicz fibration for every space F .

Proof. Given a continuous map f : X → B × F and a homotopy h : X × I → B with h₀ = pr_Bf , we can define a homotopy ˜h : X × I → B × F by

˜h(x, t) = (h(x, t), pr_F f (x)), which is a lift of h with ˜h0 = f .

A continuous map f : E → B is called a trivial fibration if there exist a space F and a homeomorphism ϕ : E → B × F such that f = pr_Bϕ.

E

f ^@@

@@

@

ϕ //_{B × F}

pr_B

{{ww^ww ww^ww

w

B

The following is an easy consequence of Proposition 4.2:

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4.3 Corollary Every trivial fibration is a Hurewicz fibration.

4.4 Proposition The composition of Hurewicz fibrations is also a Hurewicz fibration.

4.5 Theorem Every covering p : X → X is a Hurewicz fibration.

Proof. Let h : Y × I → X be a homotopy of an arbitrary space Y with f : Y → X a lift of h0. Let y ∈ Y be fixed. For each t ∈ I, h(y, t) has an open neighborhood Ut which is evenly covered by p. Then, y has an open neighborhood Vt with δt> 0 such that

Vt× ((t − δt, t + δt) ∩ I) ⊂ h⁻¹(Ut). Since I is compact, we have 0 t₁ < · · · < t_k 1 such that

I ⊂

k i=1

(ti^{− δ}ti, ti+ δti).

Note that t1^{− δ}t1 < 0 and tk+ δtk > 1. Choose 0 = s0 < s1 < · · · < sk = 1 so that t_i+1− δ_t_i+1 < s_i < t_i+ δ_t_i. Then,