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 XI . Proposition 1.1. The evaluation  : XI × I → X defined by (σ, t) = σ(t) is continuous. I Let x0 be a point in X. We denote Ω(X, x0) = {σ ∈ X : σ(0) = σ(1) = x0}. Then I Ω(X, x0) is a closed subspace of X . 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. m+1 The m-dimensional real projective space is the quotient space of R \{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, m ∼ m πn(RP ) = πn(S ). n Here RP is the n-dimensional real projective space. 1 2

m m Proof. Notice that we have a surjective (quotient) map p from S into RP given by m x 7→ [x]. Then p is a 2-sheet covering map. We choose any point x0 on S as our fixed m point and chose y0 = p(x0) as the based point of RP .  1 Corollary 1.3. For any n ≥ 2, πn(S ) = {0}. 1 2πit 1 Proof. Let p : R → S defined by p(t) = e . Then p is a covering over S . Hence ∼ 1 πn(R, 0) = πn(S , 1) for all n ≥ 2. Since R is contractible, by Corollary 1.1, πn(R, 0) = 0 for all n.