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 x_{0} ∈ 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) = x_{0}. We may identify c with a morphism/map c : (S^{1}, 1) → (X, x_{0}) between
pointed topological spaces. The constant loop at x0 is the constant map ex_{0} : (S^{1}, 1) → (X, x0) so
that ex_{0}(t) = x0 for all t ∈ S^{1}. The space of loops in (X, x0) has a subspace topology induced from
C(S^{1}, X). Together with the constant loop ex_{0}, 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, e_{0}) → (B, b_{0}) be a morphism of pointed spaces such that p : E → B is a
fiber bundle (or a fiber map). Let F = π^{−1}(b_{0}). For each n ≥ 1, there exists a group homomorphism
δ_{n}: π_{n}(B, b_{0}) → π_{n−1}(F, x_{0}) such that the following sequence is exact:

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

Example 1.1. Let p : (R, 0) → (S^{1}, 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 → S^{1}.

Example 1.2. Let S^{n} be the n-dimensional unit sphere in R^{n+1}, i.e.

S^{n+1}= {(x0, · · · , xn) ∈ R^{n+1}: x^{2}_{0}+ · · · + x^{2}_{n}= 1}.

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

Z2→ S^{n}→ RP^{n}.

1

Example 1.3. Let S^{2n+1} be the 2n + 1 dimensional unit sphere in C^{n} i.e.

S^{2n+1} = {z = (z1, · · · , zn) ∈ C^{n} : |z1|^{2}+ · · · + |zn|^{2}= 1}.

The one dimensional compact torus S^{1}= {z ∈ C : |z| = 1} acts on S^{2n+1} by
S^{1}× S^{2n+1}7→ S^{2n+1}, (λ, z) → λz.

The quotient space S^{2n+1}/S^{1}denoted by CP^{n} is called the n-dimensional complex projective space.

Study the long exact sequence of homotopy groups induced from the fiber map
S^{1}→ S^{2n+1} → CP^{n}.

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 R^{1}is 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_{στ}1_{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_{q}ασδσ 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_{q}^{i}: R^{q} → R^{q+1}by

F_{q}^{i}(x0, x1, · · · , xq−1) = (x0, x1, · · · , xi−1, 0, xi, · · · , xq−1)

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

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

∂_{q}σ =

q

X

i=0

(−1)^{i}σ^{(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+1}are 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) ∈ R^{2} : 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 homeomorphism^{3}. 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

S_{1}→ 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 f_{1}^{0}: ∆0→ ∆1is the map f_{1}^{0}(0) = (0, 1) and f_{1}^{1}: ∆0→ ∆1is the map f_{1}^{1}(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 : C_{0}(X, R) → R be the map

X

x∈X

r_{x}x

!

= 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

◦ ∂_{1}β =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} = {x_{1}, · · · , x_{n}} and rxi = r_{i} for 1 ≤ i ≤ n. Then z = Pn

i=1r_{i}x. Choose x_{0} ∈ X. By the path
connectedness of X, for each 1 ≤ i ≤ n, we choose a path β_{i} from x_{0} to x_{n}. Write β =Pn

i=1r_{i}β_{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 C^{0}= (C_{i}^{0}, ∂_{i}^{0}) by setting

C_{i}^{0}=

Ci(X, R) if i ≥ 0

R if i = −1

0 if i < −1.

and ∂_{i}^{0}=

∂i if i > 0

if i = 0 0 if i < 0.

.

The i-th homology of the new complex C^{0}is called the reduced i-th homology of the space X and
denoted by

H_{i}^{#}(X, R) = H_{i}(C^{0}).

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 C^{0} defined by

C_{i}^{0} =

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 C^{0} is called the reduced chain complex associated with (C, ). The complex C^{0}
depends on the choice of . The corresponding homology of C^{0} is called the reduced homology
associated with (C, ) and denoted by

H_{i}^{#}(C) = Hi(C^{0}), 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 C^{0} is acyclic, ker = Im ∂1. In this case,
C_{0}/ Im ∂_{1}∼= R. Note that H0(C) = C_{0}/ Im ∂_{1}, we find H_{0}(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 ^{C}^{q+1}^{(f )}

y ^{C}^{q}^{(f )}

y ^{C}^{q−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)^{i}σ^{(i)}. Thus
Cq−1(f ) ◦ ∂_{q}^{X}(σ) = Cq−1(f )(∂qσ) =

q

X

i=0

(−1)^{i}f ◦ σ^{(i)}=

q

X

i=0

(−1)^{i}(f ◦ σ)^{(i)}.
On the other hand, C_{q}(f )(σ) = f ◦ σ and hence

∂_{q}^{Y}C_{q}(f )(σ) =

q

X

i=0

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

∂_{q}^{Y}Cq(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 ^{f}^{i+1}

y ^{f}^{i}

y ^{f}^{i−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_{A}_{i}: 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 → A^{0} 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, z^{0} ∈ ker ∂i so that z − z^{0} = ∂^{A}_{i+1}w for some w ∈ Ai+1. Since f is a chain map,
fi(z) − fi(z^{0}) = fi(z − z^{0}) = fi∂_{i+1}^{A} w = ∂_{i+1}^{B} fi+1w

Hence fi(z) − fi(z^{0}) ∈ Im ∂_{i+1}^{B} . Thus [fi(z)] = [fi(z^{0})]. 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 H^{i} : 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