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) = fσ 2 C(I;X): σ(K) ⊂ Ug: The set f(K : U): K ⊂ I compact, U ⊂ X openg 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) = fσ 2 X : σ(0) = σ(1) = x0g: 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 2 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 2 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 n f0g 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 ) = f0g: 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: .