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) = ff 2 C(K; X): f(A) ⊂ Ug: The family fβ(A; U)g 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 x0 2 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 1 such that c(0) = c(1) = x0: We may identify c with a morphism/map c :(S ; 1) ! (X; x0) between 1 pointed topological spaces. The constant loop at x0 is the constant map ex0 :(S ; 1) ! (X; x0) so 1 that ex0 (t) = x0 for all t 2 S : The space of loops in (X; x0) has a subspace topology induced from 1 C(S ;X): Together with the constant loop ex0 ; 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; e0) ! (B; b0) be a morphism of pointed spaces such that p : E ! B is a −1 fiber bundle (or a fiber map). Let F = π (b0): For each n ≥ 1; there exists a group homomorphism δn : πn(B; b0) ! πn−1(F; x0) such that the following sequence is exact: (1.1) πn(i) πn(p) δn · · · −−−−! πn(F; x0) −−−−! πn(E; e0) −−−−! πn(B; b0) −−−−! πn−1(F; x0) −−−−! · · · : Here i : F ! X is the inclusion map. Example 1.1. Let p :(R; 0) ! (S1; 1) be the morphism of pointed spaces corresponding to the map p(t) = e2πit for t 2 R: Then the fiber of p over 1 is Z: Study the long exact sequence of homotopy groups induced from the fiber map 1 Z ! R ! S : Example 1.2. Let Sn be the n-dimensional unit sphere in Rn+1; i.e. n+1 n+1 2 2 S = f(x0; ··· ; xn) 2 R : x0 + ··· + xn = 1g: n n The antipodal map A is a continuous map S ! S sending x to −x: Identify Z2 = Z=2Z with n n n the subgroup fI;Ag of the automorphism group of S : The quotient space S =Z2 denoted by RP called the n-dimensional real projective space. Study the long exact sequence of the homotopy groups induced from the fiber map n n Z2 ! S ! RP : 1 2 Example 1.3. Let S2n+1 be the 2n + 1 dimensional unit sphere in Cn i.e. 2n+1 n 2 2 S = fz = (z1; ··· ; zn) 2 C : jz1j + ··· + jznj = 1g: The one dimensional compact torus S1 = fz 2 C : jzj = 1g acts on S2n+1 by S1 × S2n+1 7! S2n+1; (λ, z) ! λz: The quotient space S2n+1=S1 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 1 2n+1 n S ! S ! CP : 3 2. Singular homology The standard q-simplex is the subset of Rq+1 defined by q X ∆q = f(t0; t1; ··· ; tq): ti = 1; ti ≥ 0g: i=0 It is equipped with the subspace topology induced from the Euclidean topology on Rq+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 = fσ : ∆q ! X : σ is continuousg: A singular q-chain over a ring R1 is a function α : Sq ! R so that fσ 2 Sq : α(σ) 6= 0Rg is a finite set. Here 0R is the additive identity of the ring R: The space of singular q-chains in X is denoted either by Sq(X; R) or by Cq(X; R): We define an R-module structure on Cq(X; R) by (α + β)(σ) = α(σ) + β(σ); (aα)(σ) = aα(σ); σ 2 F; where α; β 2 Cq(X; R) and a 2 R: The 0 element of Cq(X; R) is the function 0 : Sq ! R defined by 0(σ) = 0R: When q 2 Z with q < 0; we set Sq(X; R) = 0 the trivial R-module. For each σ : ∆q ! X; we define a q-chain δσ : Sq ! R by δσ(τ) = δστ 1R: Remark. We denote α(σ) by ασ: Lemma 2.1. The set fδσ : σ 2 Sqg forms an R-basis for Cq(X; R) for q ≥ 0: Proof. Show that fδ : σ 2 S g is linearly independent over R and that α = P α δ for any σ q σ2Sq σ σ α 2 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 2 Cq(X; R) can be rewritten as X α = ασσ: σ2F 2 i q q+1 For each 0 ≤ i ≤ q; we define a map Fq : R ! R by i Fq (x0; x1; ··· ; xq−1) = (x0; x1; ··· ; xi−1; 0; xi; ··· ; xq−1) q+1 i for (x0; x1; ··· ; xq) 2 R : Then Fq are continuous for 0 ≤ i ≤ q: For each 0 ≤ i ≤ q; we define the i i i i i-th face map fq : ∆q−1 ! ∆q to be the restriction of Fq to the standard q −1 simplex fq = Fq j∆q−1 : i i The continuities of the family of functions ffq : 0 ≤ i ≤ qg follow from the fact that Fq are continuous. The i-th face of a singular q-simplex σ on X is a singular q − 1-simplex (i) σ : ∆q−1 ! X 1 R is assumed to be commutative with multiplicative identity 1R: 2 P Thus 0 element in Cq(X; R) can be represented as 0 = σ2F 0Rσ: 4 (i) i defined by σ = σ ◦ fq: The boundary of a singular q-simplex σ on X is a q − 1-chain @qσ on X defined by q X i (i) @qσ = (−1) σ : i=0 In general, the boundary of a singular q-chain α is defined to be the q − 1-chain X @qα = ασ(@qσ): σ2Sq When q < 0; we set @q = 0 to be the zero map. Lemma 2.3. The maps @q : Cq(X; R) ! Cq−1(X; R) is R-linear for all q 2 Z such that @q ◦ @q+1 = 0: Let Zq(X; R) = ker @q and Bq(X; R) = Im @q+1: Elements of Zq(X; R) are called q-cycles and elements of Bq(X; R) are called q-boundaries. Since @q ◦ @q+1 = 0;Bq(X; R) is an R-submodule of Zq(X; R): The quotient R-module Hq(X; R) = Zq(X; R)=Bq(X; R) is called the q-th singular homological module of X with coefficients in R: We set M H∗(X; R) = Hq(X; R) q2Z called the singular homology theory of X over R: We also denote M C∗(X; R) = Cq(X; R): q2Z Example 2.1. Let X = fxg be a topological space with a single point. Then ( R if q = 0 Hq(X; R) = 0 otherwise. Definition 2.1. A complex over a ring R is a sequence of R-modules fCn : n 2 Zg 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 Zn(C∗) = ker @n are called n-cycles while elements of Bn(C∗) = Im @n+1 are called n-boundaries. 5 3. Reduced Homology Theorem 3.1. Let X be a space and fXα : α 2 Λg be the set of all path components of X: For all q ≥ 0; ∼ M Hq(X; R) = Hq(Xα;R): α2Λ Hence to study the singular homology of a space, we only need to find out the singular homology of its path components.