Loop Space Let I be the closed unit interval [0, 1] and X be a topological space

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 e_x0 on the loop space: let e_x0 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 π_n(X, x₀) = π_n−1(Ω(X, x₀), e_x0).

Proposition 1.3. The homotopic groups π_n(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, π_n(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 R^m+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,

π_n(RP^m) ∼= πn(S^m).

Here RPⁿ is the n-dimensional real projective space.

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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(S¹) = {0}.

Proof. Let p : R → S¹ defined by p(t) = e^2πit. Then p is a covering over S¹. Hence π_n(R, 0) ∼= πn(S¹, 1) for all n ≥ 2. Since R is contractible, by Corollary 1.1, πn(R, 0) = 0 for all n.