Let β(A, U ) be the family of continuous functions f : K → X such that f (A

1. Loop Space and Higher homotopy groups

Let K be a compact Hausdorff space and X be a topological space. The space of all continuous functions from K to X is denoted by C(K, X). Let A be a closed subset of K (and thus A is compact) and U be an open subset of X. Let β(A, U ) be the family of continuous functions f : K → X such that f (A) ⊂ U, i.e.

β(A, U ) = {f ∈ C(K, X) : f (A) ⊂ U }.

The family {β(A, U )} forms a subbase of a topology on C(K, X); this topology is called the compact- open topology on C(K, X).

A pointed topological space is a topological space X together with a point x₀ ∈ X. A pointed topological space is denoted by (X, x0). A morphism/map from (X, x0) to (Y, y0) is a continuous map f : X → Y such that f (x0) = y0.

A loop in X with based point x0 is a continuous map c : [0, 1] → X

such that c(0) = c(1) = x₀. We may identify c with a morphism/map c : (S¹, 1) → (X, x₀) between pointed topological spaces. The constant loop at x0 is the constant map ex₀ : (S¹, 1) → (X, x0) so that ex₀(t) = x0 for all t ∈ S¹. The space of loops in (X, x0) has a subspace topology induced from C(S¹, X). Together with the constant loop ex₀, the space of loops is denoted by Ω(X, x0) and called the loop space of X with base point x0. The fundamental group of (X, x0) is defined to be

π1(X, x0) = π0(Ω(X, x0)).

Here π0(Y ) denotes the set of connected components of a topological space Y. We call π1(X, x0) the fundamental group of the pointed space (X, x0). Since Ω(X, x0) is again a pointed space, its fundamental group is denoted by π2(X, x0), i.e. π2(X, x0) = π1(Ω(X, x0)). Inductively, we define the higher homotopy groups of (X, x0) by

πn(X, x0) = πn−1(Ω(X, x0)) for n ≥ 1.

One can show that the construction of higher homotopy groups is functorial. In other words, for each n ≥ 1, πn defined a functor from the category of pointed topological spaces into the category of groups.

The following theorem is the most useful tool in homotopy theory.

Theorem 1.1. Let p : (E, e₀) → (B, b₀) be a morphism of pointed spaces such that p : E → B is a fiber bundle (or a fiber map). Let F = π⁻¹(b₀). For each n ≥ 1, there exists a group homomorphism δ_n: π_n(B, b₀) → π_n−1(F, x₀) such that the following sequence is exact:

(1.1) · · · −−−−→ πn(F, x0) −−−−→ π^πn(i) n(E, e0) −−−−→ π^πn(p) n(B, b0) −−−−→ π^δn n−1(F, x0) −−−−→ · · · . Here i : F → X is the inclusion map.

Example 1.1. Let p : (R, 0) → (S¹, 1) be the morphism of pointed spaces corresponding to the map p(t) = e^2πit for t ∈ R. Then the fiber of p over 1 is Z. Study the long exact sequence of homotopy groups induced from the fiber map

Z → R → S¹.

Example 1.2. Let Sⁿ be the n-dimensional unit sphere in Rⁿ⁺¹, i.e.

Sⁿ⁺¹= {(x0, · · · , xn) ∈ Rⁿ⁺¹: x²₀+ · · · + x²_n= 1}.

The antipodal map A is a continuous map Sⁿ → Sⁿ sending x to −x. Identify Z² = Z/2Z with the subgroup {I, A} of the automorphism group of Sⁿ. The quotient space Sⁿ/Z²denoted by RPⁿ called the n-dimensional real projective space. Study the long exact sequence of the homotopy groups induced from the fiber map

Z2→ Sⁿ→ RPⁿ.

1

Example 1.3. Let S²ⁿ⁺¹ be the 2n + 1 dimensional unit sphere in Cⁿ i.e.

S²ⁿ⁺¹ = {z = (z1, · · · , zn) ∈ Cⁿ : |z1|²+ · · · + |zn|²= 1}.

The one dimensional compact torus S¹= {z ∈ C : |z| = 1} acts on S²ⁿ⁺¹ by S¹× S²ⁿ⁺¹7→ S²ⁿ⁺¹, (λ, z) → λz.

The quotient space S²ⁿ⁺¹/S¹denoted by CPⁿ is called the n-dimensional complex projective space.

Study the long exact sequence of homotopy groups induced from the fiber map S¹→ S²ⁿ⁺¹ → CPⁿ.

2. Singular homology The standard q-simplex is the subset of R^q+1 defined by

∆q = {(t0, t1, · · · , tq) :

q

X

i=0

ti= 1, ti≥ 0}.

It is equipped with the subspace topology induced from the Euclidean topology on R^q+1. Let X be a topological space. A singular q-simplex in X is a continuous map

σ : ∆q→ X.

The space of singular q-simplices is denoted by

Sq = {σ : ∆_q → X : σ is continuous}.

A singular q-chain over a ring R¹is a function

α : S_q → R

so that {σ ∈ S_q : α(σ) 6= 0_R} is a finite set. Here 0Ris the additive identity of the ring R. The space of singular q-chains in X is denoted either by S_q(X, R) or by C_q(X, R). We define an R-module structure on C_q(X, R) by

(α + β)(σ) = α(σ) + β(σ), (aα)(σ) = aα(σ), σ ∈ F ,

where α, β ∈ Cq(X, R) and a ∈ R. The 0 element of Cq(X, R) is the function 0 : Sq→ R defined by 0(σ) = 0R.

When q ∈ Z with q < 0, we set S^q(X, R) = 0 the trivial R-module.

For each σ : ∆q → X, we define a q-chain δσ: Sq → R by δ_σ(τ ) = δ_στ1_R. Remark. We denote α(σ) by ασ.

Lemma 2.1. The set {δ_σ : σ ∈ S_q} forms an R-basis for C_q(X, R) for q ≥ 0.

Proof. Show that {δσ : σ ∈ Sq} is linearly independent over R and that α = P

σ∈S_qασδσ for any

α ∈ Cq(X, R). 

Lemma 2.2. Show that the map

δ : Sq → Cq(X, R), σ 7→ δσ

is injective.

We identify σ with its image δσ in Cq(X, R). An element α ∈ Cq(X, R) can be rewritten as α = X

σ∈F

ασσ.

2

For each 0 ≤ i ≤ q, we define a map F_qⁱ: R^q → R^q+1by

F_qⁱ(x0, x1, · · · , xq−1) = (x0, x1, · · · , xi−1, 0, xi, · · · , xq−1)

for (x₀, x₁, · · · , x_q) ∈ R^q+1. Then F_qⁱ are continuous for 0 ≤ i ≤ q. For each 0 ≤ i ≤ q, we define the i-th face map f_qⁱ : ∆q−1→ ∆q to be the restriction of F_qⁱ to the standard q − 1 simplex f_qⁱ = F_qⁱ|∆_q−1. The continuities of the family of functions {f_qⁱ : 0 ≤ i ≤ q} follow from the fact that F_qⁱ are continuous.

The i-th face of a singular q-simplex σ on X is a singular q − 1-simplex σ⁽ⁱ⁾: ∆q−1→ X

1R is assumed to be commutative with multiplicative identity 1R. 2Thus 0 element in Cq(X, R) can be represented as 0 =P

σ∈F0Rσ.

defined by σ⁽ⁱ⁾= σ ◦ f_qⁱ.

The boundary of a singular q-simplex σ on X is a q − 1-chain ∂qσ on X defined by

∂_qσ =

q

X

i=0

(−1)ⁱσ⁽ⁱ⁾.

In general, the boundary of a singular q-chain α is defined to be the q − 1-chain

∂qα = X

σ∈Sq

ασ(∂qσ).

When q < 0, we set ∂_q= 0 to be the zero map.

Lemma 2.3. The maps ∂_q : C_q(X, R) → C_q−1(X, R) is R-linear for all q ∈ Z such that

∂_q◦ ∂q+1= 0.

Let Z_q(X, R) = ker ∂_q and B_q(X, R) = Im ∂_q+1. Elements of Z_q(X, R) are called q-cycles and elements of B_q(X, R) are called q-boundaries. Since ∂_q◦ ∂_q+1= 0, B_q(X, R) is an R-submodule of Z_q(X, R). The quotient R-module

H_q(X, R) = Z_q(X, R)/B_q(X, R)

is called the q-th singular homological module of X with coefficients in R. We set H_∗(X, R) =M

q∈Z

Hq(X, R) called the singular homology theory of X over R. We also denote

C_∗(X, R) =M

q∈Z

Cq(X, R).

Example 2.1. Let X = {x} be a topological space with a single point. Then Hq(X, R) =

(R if q = 0 0 otherwise.

Definition 2.1. A complex over a ring R is a sequence of R-modules {Cn: n ∈ Z} together with a sequence of R-linear maps ∂n : Cn → Cn−1 such that ∂n◦ ∂n+1= 0. A complex over R is denoted by C_∗= (Cn, ∂n). The n-th homology of C_∗is defined to be

Hn(C_∗) = ker ∂n/ Im ∂n+1.

Using the same terminology, elements of Z_n(C_∗) = ker ∂_n are called n-cycles while elements of B_n(C_∗) = Im ∂_n+1are called n-boundaries.

3. Reduced Homology

Theorem 3.1. Let X be a space and {Xα: α ∈ Λ} be the set of all path components of X. For all q ≥ 0,

H_q(X, R) ∼=M

α∈Λ

H_q(X_α, R).

Hence to study the singular homology of a space, we only need to find out the singular homology of its path components. Assume that X is path connected.

The standard 0-simplex is the set ∆₀ = {0}. A singular 0-simplex in X is a continuous map σ : ∆₀→ X. Then σ is determined by σ(0). The set S0 and X are in one-to-one correspondence:

ι : S0→ X, σ 7→ σ(0).

We identify C0(X, R) with the space of functions α : X → R so that such that {x ∈ X : α(x) 6= 0}

is a finite set. Thus an element of C0(X, R) is represented asP

x∈Xrxx where rx∈ R.

The standard 1-simplex ∆1 is the subset {(t0, t1) ∈ R² : t0, t1 ≥ 0, t0+ t1 = 1}. A singular simplex in X is a continuous map σ : ∆1→ X. Let h : [0, 1] → ∆1be the map t 7→ (1 − t, t). Then h is a homeomorphism³. We obtain a path σ ◦ h : [0, 1] → X in X. The space of path in X is denoted by P = C([0, 1], X). There is a bijection

S₁→ P, σ 7→ σ ◦ h.

We identify C1(X, R) with the space of functions β : P → R so that {c : β(c) 6= 0} is a finite set.

We represent β as a finite sum β =P

c∈Prcc. Here rc ∈ R and c : [0, 1] → X denotes a path. Now we would like to express the boundary map ∂1using the new expression.

The face map f₁⁰: ∆0→ ∆1is the map f₁⁰(0) = (0, 1) and f₁¹: ∆0→ ∆1is the map f₁¹(0) = (1, 0).

If σ : ∆1→ X is a singular 1-simplex, then

σ⁽⁰⁾(0) = σ(0, 1) = σ ◦ h(1), σ⁽¹⁾(0) = σ(1, 0) = σ ◦ h(0).

Recall the boundary of an 1-singular simplex σ in X is defined to be the 0-chain

∂1σ = σ⁽⁰⁾− σ⁽¹⁾.

Thus if c : [0, 1] → X is a path corresponding to the singular 1-simplex c ◦ h⁻¹ : ∆1→ X, then by the identification described above, we find ∂1c = c(1) − c(0). If β =P

c∈Prcc, then

∂1β =X

c∈P

rc(c(1) − c(0))

=X

c

rcc(1) −X

c

rcc(0)

= X

x:c(1)=x

rcx − X

c:c(0)=x

rcx.

Let  : C₀(X, R) → R be the map

 X

x∈X

r_xx

!

= X

x∈X

r_x.

Then  is surjective and thus by the first isomorphism, we find C0(X, R)/ ker  ∼= R. Observe that





 X

x:c(1)=x

rcx



=X

c

rc= 



 X

x:c(0)=x

rcx



. Since  is R-linear, we find

 ◦ ∂₁β =X

c

r_c−X

c

r_c= 0.

3Check as an exercise

In other words,  ◦ ∂1= 0 which gives Im ∂1⊆ ker . In fact, we can prove Lemma 3.1. ker  = Im ∂1.

Proof. Let z ∈ ker . Then z = 0. Write z =P

x∈Xrxx. ThenP

x∈Xrx= 0. Denote {x ∈ X : rx6=

0} = {x₁, · · · , x_n} and rxi = r_i for 1 ≤ i ≤ n. Then z = Pn

i=1r_ix. Choose x₀ ∈ X. By the path connectedness of X, for each 1 ≤ i ≤ n, we choose a path β_i from x₀ to x_n. Write β =Pn

i=1r_iβ_i. Then

∂β =

n

X

i=1

ri(xi− x0)

=

n

X

i=1

rixi−

n

X

i=1

rix0

=

n

X

i=1

rixi−

n

X

i=1

ri

! x0

= z

Here use the fact that r1+ · · · + rn = 0. We find z = ∂β for some β ∈ C1(X, R). We prove our

assertion. 

Since the zeroth singular homology group H0(X, R) is defined to be C0(X, R)/ Im ∂1 and ∂1 = ker , we find

H0(X, R) = C0(X, R)/ ker  ∼= R when X is path connected. In general by Theorem 3.1, we obtain:

Corollary 3.1. Let X be a space and C be the set of all path components of X. Then H0(X, R) ∼=M

C

R.

We have set Ci(X, R) = 0 when i < 0 and ∂i: Ci(X, R) → Ci−1(X, R) to be the zero map for all i < 0. In fact, we can define a new complex C⁰= (C_i⁰, ∂_i⁰) by setting

C_i⁰=







Ci(X, R) if i ≥ 0

R if i = −1

0 if i < −1.

and ∂_i⁰=







∂i if i > 0

 if i = 0 0 if i < 0.

.

The i-th homology of the new complex C⁰is called the reduced i-th homology of the space X and denoted by

H_i^#(X, R) = H_i(C⁰).

It follows from the definition that H_i^#(X, R) =

(Hi(X, R) if i 6= 0

0 if i = 0 and if X is path connected.

When X is an one pointed space, all of its reduced homology modules vanish, i.e.

H_i^#(pt, R) = 0, i ∈ Z.

This motivates the definition of acyclic complex.

Definition 3.1. A complex C = (Ci, ∂i) is acyclic if all of its homological modules vanish, i.e.

Hi(C) = 0 for all i ∈ Z.

Notice that Hi(C) = ker ∂i/ Im ∂i+1 and hence C is acyclic if and only if ker ∂i= Im ∂i+1, i.e. the complex C forms an exact sequence of R-modules.

Let C = (Ci, ∂i) be a complex with the property that Ci = 0 for all i < 0. An augmentation of such a complex C over R is an R-epimorphism  : C0→ R such that ∂1 = 0. A complex with an augmentation gives us a new complex C⁰ defined by

C_i⁰ =







C_i if i ≥ 0 R if i = −1 0 if i < −1.

and ∂⁰_i=







∂_i if i > 0

 if i = 0 0 if i < 0.

.

The new complex C⁰ is called the reduced chain complex associated with (C, ). The complex C⁰ depends on the choice of . The corresponding homology of C⁰ is called the reduced homology associated with (C, ) and denoted by

H_i^#(C) = Hi(C⁰), i ∈ Z.

Definition 3.2. A chain complex C with augmentation  is acyclic if its corresponding reduced chain complex is acyclic.

Hence the chain complex of an one point space with the above given augmentation is acyclic.

Proposition 3.1. A complex C with augmentation  is acyclic if Hi(C) =

(0 if i 6= 0 R if i = 0.

Proof. Using the fact that H_i^#(C) = Hi(C) for i 6= 0, we see that H_i^#(C) = 0 if and only if Hi(C) = 0 for i 6= 0.

Since  : C0 → R is surjective, C0/ ker  ∼= R. If C⁰ is acyclic, ker  = Im ∂1. In this case, C₀/ Im ∂₁∼= R. Note that H0(C) = C₀/ Im ∂₁, we find H₀(C) ∼= R. This completes the proof.



4. Category of complexes and the Homological Functor

Let f : X → Y be a continuous map. Given a singular q-simplex σ : ∆q → X in X, the continuous map f ◦ σ : ∆q → Y determine a q-simplex in Y. We define an R-linear map

Cq(f ) : Cq(X, R) → Cq(Y, R), X

σ

rσσ 7→X

σ

rσ(f ◦ σ).

Then we obtain the following diagrams of R-linear maps

· · · −−−−→ C_q+1(X, R) ^∂

X

−−−−→ Cq+1 _q(X, R) ^∂

X

−−−−→ Cq _q−1(X, R) −−−−→ · · ·





y ^{Cq+1(f )}





y ^{Cq(f )}





y ^{Cq−1(f )}



 y



 y

· · · −−−−→ Cq+1(Y, R) ^∂

Y

−−−−→ Cq+1 q(Y, R) ^∂

Y

−−−−→ Cq q−1(Y, R) −−−−→ · · · .

Here ∂_q^X and ∂_q^Y are the boundaries maps on Cq(X, R) and on Cq(Y, R) respectively. If σ : ∆q→ X is a singular q-simplex, then ∂qσ =Pq

i=0(−1)ⁱσ⁽ⁱ⁾. Thus Cq−1(f ) ◦ ∂_q^X(σ) = Cq−1(f )(∂qσ) =

q

X

i=0

(−1)ⁱf ◦ σ⁽ⁱ⁾=

q

X

i=0

(−1)ⁱ(f ◦ σ)⁽ⁱ⁾. On the other hand, C_q(f )(σ) = f ◦ σ and hence

∂_q^YC_q(f )(σ) =

q

X

i=0

(−1)ⁱ(f ◦ σ)⁽ⁱ⁾. The above two equations imply that

∂_q^YCq(f ) = Cq−1(f )∂_q^X.

In other words, the above diagram commutes. This motivates the definition of chain maps or morphisms between chain complexes.

Definition 4.1. Let A = (Ai, ∂_i^A) and B = (Bi, ∂_i^B) be two chain complexes over R. A chain map f = {fi} (morphism) from A to B is a sequence of R-linear maps fi: Ai→ Bi such that

∂_i^B◦ fi= fi−1◦ ∂_i^A, for all i ∈ Z.

In other words, the following diagram commutes:

· · · −−−−→ A_i+1 ^∂

A

−−−−→ Ai+1 _i ^∂ A

−−−−→ Ai _i−1 −−−−→ · · ·





y ^fi+1





y ^fi





y ^fi−1



 y



 y

· · · −−−−→ Bi+1

∂^B_i+1

−−−−→ Bi

∂^B_i

−−−−→ Bi−1 −−−−→ · · · .

The identify morphism id_A: A → A is defined by the sequence of identity maps {id_Ai: A_i→ Ai}.

If f : A → B and g : B → C are chain maps, we define their composition g ◦ f to be the sequence of R-linear maps {g_i◦ f_i: A_i→ C_i}. We leave to the reader to verify that g ◦ f is again a chain map.

Proposition 4.1. The collections of all complexes over R together with chain maps forms a category denoted by Comp_R. The category Comp_R is called the category of complexes over R.

Corollary 4.1. Any continuous map f : X → Y between topological spaces X, Y determines a chain map

C_∗(f ) : C_∗(X, R) → C_∗(Y, R),

between the singular chain complexes of X and Y , where C_∗(f ) = {C_i(f ) : i ∈ Z} such that the following properties hold.

(1) If idX : X → X is the identity map, then C_∗(idX) = id_C∗(X,R).

(2) If f : X → Y and Y → Z are continuous maps, then C_∗(g ◦ f ) = C_∗(g) ◦ C_∗(f ).

This corollary implies that C_∗: Top → Comp_R defines a functor from the category of topological spaces into the category of complexes over R.

Lemma 4.1. Let f : A → A⁰ be a chain map between chain complexes A and B. For each i ∈ Z, we define

Hi(f ) : Hi(A) → Hi(B), [z] 7→ [fi(z)].

Hi(f ) is a well-defined R-linear homomorphism.

Proof. Let z, z⁰ ∈ ker ∂i so that z − z⁰ = ∂^A_i+1w for some w ∈ Ai+1. Since f is a chain map, fi(z) − fi(z⁰) = fi(z − z⁰) = fi∂_i+1^A w = ∂_i+1^B fi+1w

Hence fi(z) − fi(z⁰) ∈ Im ∂_i+1^B . Thus [fi(z)] = [fi(z⁰)]. The linearity of Hi(f ) follows from the

linearity of fi. 

Lemma 4.2. Let f : A → B and g : B → C be chain maps. Then Hi(g ◦ f ) = Hi(g) ◦ Hi(f ) for any i ∈ Z.

Proof. Routine check. 

Corollary 4.2. For each i ∈ Z, the assignment Hⁱ : A 7→ Hi(A) sending a chain complex over R to its i-th homology and Hi : (f : A → B) 7→ Hi(f ) : Hi(A) → Hi(B) sending a chain map to its corresponding induced map defines a functor

Hi: Comp_R→ ModR

from the category of complexes over R to the category of R-modules ModR.

5. Homotopy and Chain Homotopy