1. Loop Space
Let I be the closed unit interval [0, 1] and X be a topological space. The space of paths
in X is the set C(I, X) of continuous maps from I to X.
For each compact subset K of I and open subset U of X, we denote
(K : U ) = {σ ∈ C(I, X) : σ(K) ⊂ U }.
The set {(K : U ) : K ⊂ I compact, U ⊂ X open} forms a subbase for a topology on
C(I, X). We call this topology the compact-open topology on C(I, X). For simplicity, we
denote C(I, X) by X I .
Proposition 1.1. The evaluation : X I × I → X defined by (σ, t) = σ(t) is continuous.
Let x0 be a point in X. We denote Ω(X, x0 ) = {σ ∈ X I : σ(0) = σ(1) = x0 }. Then
Ω(X, x0 ) is a closed subspace of X I . We call Ω(X, x0 ) the loop space.
Proposition 1.2. The fundamental group π1 (X, x0 ) of (X, x0 ) is exactly π0 (Ω(X, x0 )).
Given x0 ∈ X, there is a natural based point ex0 on the loop space: let ex0 be the constant
loop at x0 . Then (Ω(X, x0 ), ex0 ) is a pointed topological space. We define
π2 (X, x0 ) = π1 (Ω(X, x0 ), ex0 ).
Inductively, we define the higher homotopy groups of the pointed space (X, x0 ) by
πn (X, x0 ) = πn−1 (Ω(X, x0 ), ex0 ).
Proposition 1.3. The homotopic groups πn (X, x0 ) are all commutative for n ≥ 2.
Let f : (X, x0 ) → (Y, y0 ) be a morphism of pointed spaces. We define
Ω(f ) : (Ω(X, x0 ), x0 ) → Ω(Y, y0 )
by setting Ω(f )(σ) = f ◦ σ for σ ∈ Ω(X, x0 ). Then Ω(f ) is a morphism of pointed spaces.
Hence we can inductively defined a homomorphism of groups:
πn (f ) : πn (X, x0 ) → πn (Y, y0 ).
Proposition 1.4. For each n ≥ 1, πn is a functor from the category of pointed topological
spaces into the category of groups.
Corollary 1.1. If X is contractible, πn (X, x0 ) is trivial for all n.
Proposition 1.5. For all n, there is a canonical isomorphism
πn (X × Y, (x0 , y0 )) ∼
= πn (X, x0 ) × πn (Y, y0 ).
Theorem 1.1. If p : (E, e0 ) → (X, x0 ) is a covering space, then
πn (p) : πn (E, e0 ) → πn (X, x0 )
are isomorphisms for all n ≥ 2.
The m-dimensional real projective space is the quotient space of Rm+1 \ {0} modulo the
relation ∼ defined by
x ∼ y iff x = λy
for some λ 6= 0.
Corollary 1.2. Let m ≥ 2. For all n ≥ 2,
πn (RP m ) ∼
= πn (S m ).
Here RP n is the n-dimensional real projective space.
1
2
Proof. Notice that we have a surjective (quotient) map p from S m into RP m given by
x 7→ [x]. Then p is a 2-sheet covering map. We choose any point x0 on S m as our fixed
point and chose y0 = p(x0 ) as the based point of RP m .
Corollary 1.3. For any n ≥ 2, πn (S 1 ) = {0}.
Proof. Let p : R → S 1 defined by p(t) = e2πit . Then p is a covering over S 1 . Hence
πn (R, 0) ∼
= πn (S 1 , 1) for all n ≥ 2. Since R is contractible, by Corollary 1.1, πn (R, 0) = 0
for all n.