Lecture
§ 47. Homotopy equivalence. Contractible spaces.
47.1. Definition. Spaces X1 and X2 are called homotopy equivalent
(notation X1 ∼ X2), if there exist continuous mappings f : X1 → X2, g :
X2 → X1, such that the compositions g ◦ f : X1 → X1, f ◦ g : X2 → X2 are homotopic to the identity
mappings idX1
, idX2
respectively.
The mappings f and g are called mutually homotopy-inverse homotopy equivalences.
If the mappings g ◦ f and f ◦ g are not merely homotopic to the identity mappings, but
are in fact equal to them, then f and g are mutually inverse homeomorphisms. Thus, the relation
of homotopy equivalence is a generalization of the relation of homeomorphism. Homotopy equivalent
spaces need not be homeomorphic.
47.2. Examples. 1. The closed ball B2
, the open ball D2
, are homotopy equivalent
to a point, but are pairwise non-homeomorphic.
2. The circle S
1 and the annulus K = {(x, y) ∈ R
2
: 1 ≤ x
2 + y
2 ≤ 4} are homotopy equivalent,
but not homeomorphic.
47.3. Theorem. The relation of homotopy equivalence is an equivalence relation on the class of all topological spaces.
Proof. In fact only transitivity needs to be checked. Let X1 ∼ X2
and X2 ∼ X3. There exist mappings f1 : X1 → X2, g1 : X2 → X1, f2 : X2 → X3,
g2 : X3 → X2, such that
g1 ◦ f1 ∼h idX1
, f1 ◦ g1 ∼h idX2
; (47.1)
g2 ◦ f2 ∼h idX2
, f2 ◦ g2 ∼h idX3
. (47.2)
Set
f = f2 ◦ f1, g = g1 ◦ g2.
Then g ◦ f = g1 ◦ g2 ◦ f2 ◦ f1 = g1 ◦ (g2 ◦ f2) ◦ f1 ∼h
by Lemma 46.6 of Lecture 14 and (47.2)
∼h
g1 ◦ idX2 ◦ f1 = g1 ◦ f1 ∼h (47.1) ∼h idX1
. In the same way we show that f ◦ g ∼h idX3
. Hence,
the mappings f and g are homotopy mutually inverse and X1 ∼ X3.
47.4. A class of homotopy equivalent spaces is called a homotopy type.
A space X is called contractible, if the identity mapping id : X → X
is homotopic to a constant (i.e. to the constant mapping constx0
: X → X, sending all of X to a point
x0 ∈ X).
47.5. Example. 1. A space is contractible if and only if it has the homotopy type of a point.
2. A segment (a convex subset of R
n) is contractible.
§ 48. Fundamental group.
48.1. Definition. Paths ϕ, ψ : I → X are called homotopic (notation ϕ ∼ ψ), if
ϕ(0) = ψ(0) = x0, ϕ(1) = ψ(1) = x1 (their starting and end points coincide) and there exists a homotopy Φ : I × I → X, fixed on the set {0, 1},
connecting ϕ and ψ.
The relation of homotopy of paths is an equivalence relation.
48.2. Multiplication of homotopy classes of paths. If for paths ϕ, ψ : I → X ϕ(1) =
ψ(0), then
[ϕ] · [ψ] = [ϕψ].
Inverse homotopy class of a path. For a path ϕ : I → X
[ϕ]
−1 = [ϕ
−1
].
48.3. Lemma. (correctness of multiplication and taking the inverse) If ϕ0 ∼ ϕ1 and ψ0 ∼ ψ1, then
ϕ0ψ0 ∼ ϕ1ψ1. If ϕ ∼ ψ, then ϕ
−1 ∼ ψ
−1
.
Proof. Let F : I × I → X be a homotopy fixed on {0,1}, connecting ϕ0 and ϕ1, and
G : I × I → X a homotopy fixed on {0,1}, connecting ψ0 and ψ1. ϕ0(0) = ϕ1(0) = x0, ϕ0(1) =
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ϕ1(1) = ψ0(0) = ψ1(0) = x1, ψ0(1) = ψ1(1) = x2. Define a homotopy Φ : I ×I → X as
follows:
Φ(s, t) =
F(2s, t) : 0 ≤ s ≤
1
2
;
G(2s − 1, t) : 1
2 ≤ s ≤ 1.
It is clear that this homotopy connects ϕ0ψ0 with ϕ1ψ1. For any t ∈ I we have Φ(0, t) = F(0, t) = x0,
Φ(1, t) = G(1, t) = x2. Thus it is fixed on {0,1}.
The restriction of the mapping Φ to [0,
1
2
] × I and [
1
2
, 1] × I coincides with the compositions of products
of similarities of the segment with the identity mapping and the mappings F and G respectively.
Hence they are continuous. Since for any t ∈ I Φ({
1
2
} × t) = F(1, t) = G(0, t) = x1, then the continuity
of Φ follows from item 13.4.6 of Lecture 3.
Let F : I ×I → X be a homotopy fixed on {0,1}, connecting ϕ and ψ. Then Φ(s, t) = F(1−s, t) is a
homotopy fixed on {0,1}, connecting ϕ
−1 ∼ ψ
−1
. Indeed, Φ(s, 0) = F(1−s, 0) = ϕ(1−s) =
ϕ
−1
(s), Φ(s, 1) = F(1 − s, 1) = ψ(1 − s) = ψ
−1
(s). For any t ∈ I we have Φ(0, t) = F(1, t) = x0,
Φ(1, t) = F(0, t) = x0. Thus it is fixed on {0,1}. Its continuity is obvious.
48.4. Theorem. The operations of multiplication of homotopy classes and taking the inverse homotopy class satisfy the following properties.
a) (Associativity) If the class [ϕ] · ([ψ] · [χ]) is defined, then the class ([ϕ] · [ψ]) · [χ] is defined,
and they coincide.
b) (Left and right units) For x ∈ X denote by constx : I → X the constant mapping to the point x. If ϕ : I → X, ϕ(0) = x0, ϕ(1) = x1, then
[ϕ] · [constx1
] = [ϕ], [constx0
] · [ϕ] = [ϕ].
c) (Inverse class) If ϕ : I → X, ϕ(0) = x0, ϕ(1) = x1, then
[ϕ] · [ϕ]
−1 = [constx0
], [ϕ]
−1
· [ϕ] = [constx1
].
Proof. a) Let ϕ(1) = ψ(0), ψ(1) = χ(0). Then
F(s, t) =
ϕ
4s
t+1
, if 0 ≤ s ≤
t+1
4
,
ψ(4s − t − 1), if t+1
4 ≤ s ≤
t+2
4
,
χ
4s−t−2
2−t
, if t+2
4 ≤ s ≤ 1,
is a homotopy fixed on {0,1}, connecting the paths (ϕψ)χ and ϕ(ψχ) (verify independently).
b) We give the homotopy (verify independently) proving the second equality
F(s, t) =
x0, t ≤ 1 − 2s;
ϕ(
2s+t−1
1+t
), t ≥ 1 − 2s.
Hence the homotopy class [constx0
] is a left unit. The first equality is proved analogously.
c) We show only that ϕϕ−1 ∼ constx0
. Indeed,
F(s, t) =
ϕ(2s), if 0 ≤ s ≤
1−t
2
,
ϕ(1 − t), if 1−t
2 ≤ s ≤
1+t
2
,
ϕ(2 − 2s), if 1+t
2 ≤ s ≤ 1,
is a homotopy fixed on {0,1}, connecting the path ϕϕ−1
with the constant path constx0
(verify
independently).
48.5. Spaces with a marked point. In topology one often considers spaces with a marked (base) point, i.e. one assumes that a base point is chosen
in every space
and all mappings send base points to base points. A space X with a base
point x0 is denoted (X, x0). Identical spaces with different base points are considered different.
48.6. Definition of the set π1(X, x0). We consider loops of the space X with the base point x0, i.e. paths ϕ : I → X such that ϕ(0) = ϕ(1) = x0, where x0 is the base
point.
The fundamental group of the space X with the base point x0 is the set
of homotopy classes of loops ϕ : I → (X, x0) with the multiplication operation. Notation π1(X, x0).
The set π1(X, x0) is a group by Theorem 48.4.
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A loop, regarded as a mapping ϕ : I → X with the condition ϕ(0) = ϕ(1) = x0, is equivalent to
a mapping ϕb : S1 → X, sending the point 0 = (cos(0),sin(0)) to the base point x0 ∈
X. Therefore the set π1(X, x0) can be regarded as the set of homotopy classes
π(S
1
, X) of mappings of the pointed spaces (S
1
, 0) and (X, x0).
48.7. Definition. Let α : I → X be a path with beginning x0 and end x1. The equality
α#([ϕ]) = [α
−1
] · [ϕ] · [α]
defines a mapping
α# : π1(X, x0) → π1(X, x1).
48.8. Theorem. The mapping α# is a group isomorphism.
Proof. Let us show that α# is a homomorphism. Take elements [ϕ1], [ϕ2] ∈ π1(X, x0).
Then
α#
[ϕ1]·[ϕ2]
= [α
−1
]·[ϕ1]·[ϕ2]·[α] = [α
−1
]·[ϕ1]·[constx0
]·[ϕ2]·[α] = [α
−1
]·[ϕ1]·[α]·[α
−1
]·[ϕ2]·[α] =
α#([ϕ1]) · α#([ϕ2]).
To prove the bijectivity of the homomorphism α# let us show that the homomorphism
α
−1
# : π1(X, x1) → π1(X, x0),
α
−1
# ([ϕ]) = [α] · [ϕ] · [α
−1
]
is inverse to α#. Indeed,
α
−1
#
α#([ϕ])
= α
−1
# ([α
−1
] · [ϕ] · [α]) = [α] · [α
−1
] · [ϕ] · [α] · [α
−1
] = [ϕ].
Similar reasoning establishes the equality
α#
α
−1
# ([ϕ])
= [ϕ].
48.9. Corollary (Dependence of the fundamental group on the base point).
If the space X is path connected, then the groups π1(X, x0) and π1(X, x1) are isomorphic for any
points x0, x1 ∈ X.
48.10. Remark. From Corollary 48.9 it follows that for a path connected space X
the groups π1(X, x0) at different points x0 ∈ X are isomorphic to each other and can be regarded
as a single group π1(X), which is called the fundamental group of the path connected space X.
48.11. Definition. A path connected space X is called simply connected, if any
two paths α1 : I → X and α2 : I → X such that α1(0) = α2(0) = x0, α1(1) = α2(1) = x1 are homotopic.
48.12. Theorem. A path connected space X is simply connected if and only if
π1(X) = 0.
Proof. Let X be simply connected. Take an arbitrary class [ϕ] ∈ π1(X, x0) and the unit class [constx0
]. These loops can be regarded as paths having coinciding beginning and end: ϕ(0) = ϕ(1) = constx0
(0) = constx0
(1) = x0. The loop ϕ is homotopic to the loop constx0
,
hence [ϕ] = [constx0
], and π1(X) = 0.
Now let π1(X, x) = 0 at the point x ∈ X, which can be considered arbitrary by virtue of
Corollary 48.9. Consider two paths α1 and α2 in X with common beginning x0 and end x1.
Since α1α
−1
1 and α
−1
1 α2 are loops, and π1(X) = 0, then α1α
−1
1 ∼ constx0 and α
−1
1 α2 ∼ constx1
. Hence
α1 ∼ α1constx1 ∼ α1α
−1
1 α2 ∼ constx0 α2 ∼ α2
under fixed-endpoint homotopies. By the transitivity of homotopy equivalence there exists
a homotopy of the paths α1 and α2.
48.13. Examples. 1. R
n, n ∈ N, is simply connected.
2. The open and closed balls Dn, Bn, n ∈ N, are simply connected.
3. The sphere S
2 is simply connected (prove that any loop in S
n is homotopic to a loop not
filling the whole sphere).
§ 49. Induced homomorphism.
49.1. Definition. Let h : (X, x0) → (Y, y0) be a continuous mapping. The equality
h∗([ϕ]) = [h ◦ ϕ] ∈ π(Y, y0)
defines a homomorphism
h∗ : π(X, x0) → π(Y, y0),
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called the homomorphism induced by the mapping h.
49.2. Theorem. The induced homomorphism h∗ is correctly defined (i.e. the class h∗([ϕ]) =
[h ◦ ϕ] does not depend on the choice of the representative ϕ ∈ [ϕ]) and is a homomorphism.
For h : (X, x0) → (Y, y0) and g : (Y, y0) → (Z, z0)
(g ◦ h)∗ = g∗ ◦ h∗.
Proof. The correctness of the definition of the mapping h∗ follows from Lemma 46.6 of Lecture
14. Let us check that h∗ is a homomorphism.
h∗([ϕ] · [ψ]) = h∗([ϕψ]) = [h ◦ (ϕψ)] = [(h ◦ ϕ)(h ◦ ψ)] = [h ◦ ϕ] · [h ◦ ψ] = h∗([ϕ]) · h∗([ψ]).
Let h : (X, x0) → (Y, y0) and g : (Y, y0) → (Z, z0) be continuous mappings. Then
(g ◦ h)∗([ϕ]) = [(g ◦ h) ◦ ϕ] = [g ◦ (h ◦ ϕ)] = g∗([h ◦ ϕ]) = g∗(h∗([ϕ])) = (g∗ ◦ h∗)([ϕ]).
Assignment N 15
1. Prove that homotopy equivalent spaces have the same number of (path)
connectedness components.
2. Find a countable number of pairwise homotopy equivalent spaces, which are not
pairwise homeomorphic.
3. Prove the homotopy equivalence:
(1) of the circle S
1 and the annulus K = {(x, y) ∈ R
2
: 1 ≤ x
2 + y
2 ≤ 4};
(2) of the circle S
1 and the plane with a point removed R
2 \ {O};
(3) of the circle S
1 and the space R
3
with a line removed;
(4) of the circle S
1 and the Mobius band.
4. Prove the homotopy equivalence:
(1) of the plane with two points removed and the wedge of two circles S
1 W
S
1
;
(2) of the torus T
2
with a disk D2
removed (i.e. a handle) and the wedge of two circles S
1 W
S
1
;
(3) of the sphere S
2
with a pair of points identified and the wedge of a sphere and a circle S
2 W
S
1
;
(4) of the torus T
2
with closed disks B2
, glued along the boundary along a meridian and along
a parallel, and the sphere S
2
;
(5) of the spaces of nondegenerate matrices GL(n, R) and orthogonal matrices O(n).
5. Prove that the wedge of two circles S
1 W
S
1
is homotopy equivalent to the union of
a circle and an arbitrary diameter of it.
6. Prove that the relation of homotopy of paths is an equivalence relation.
7. Prove that a contractible space is path connected. Is the converse implication true?
8. Prove that a space is contractible if and only if it has the homotopy type of a point.
9. Prove that a convex subset of R
n is contractible.
Let X be a space. The product X × I of the space X with the segment I is called the cylinder of the space X. If one identifies all the points of the upper base X ×{1}
of the cylinder X × I (contracts X × {1} to a point), one obtains the cone Con(X) over X. Prove that
the cone Con(X) is contractible for any space X.
10. Compute the fundamental group:
(1) of a discrete space;
(2) of R
n, n ∈ N.
Additional problems for Assignment N 15
11. Prove that a connected finite graph is homotopy equivalent to a wedge of circles.
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12. Prove that if the spaces X and Y are homotopy equivalent, then there exists a
one-to-one correspondence between their homotopy classes of mappings into an arbitrary space.
13. Prove that if the spaces X and Y are homotopy equivalent, then there exists a one-to-one correspondence between the homotopy classes of mappings of an arbitrary
space Z into X and Y respectively.
14. Prove that the infinite-dimensional sphere S∞ = {x ∈ `
2
:
P|xi
|
2 = 1} is contractible.
15. Prove that for a path connected space X the following conditions are equivalent:
(1) X is contractible;
(2) π(X, Y ) is trivial (i.e. consists of a single element) for any path connected space Y;
(3) π(Y, X) is trivial for any space Y.
16. Prove that the product X × Y of the spaces X and Y is contractible if and only
if the spaces X and Y are contractible. Is the analogous result true for a countable
(arbitrary) number of factors?
17. Prove that any linear space over the field of real numbers is contractible.
Prove that a retract (see Problem 20 of Assignment 5) of a contractible space is contractible.
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