1. Loop Space and Higher Homotopy Groups Let K Be a Compact Hausdorﬀ Space and X Be a Topological Space

Home , Loop space

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) = {f ∈ C(K,X): f(A) ⊂ U}. The family {β(A, U)} 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 ∈ 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 ∈ 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 ∈ 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 = {(x0, ··· , xn) ∈ R : x0 + ··· + xn = 1}. 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 {I,A} 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 = {z = (z1, ··· , zn) ∈ C : |z1| + ··· + |zn| = 1}. The one dimensional compact torus S1 = {z ∈ C : |z| = 1} 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 = {(t0, t1, ··· , tq): ti = 1, ti ≥ 0}. 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 = {σ : ∆q → X : σ is continuous}. A singular q-chain over a ring R1 is a function

α : Sq → R so that {σ ∈ Sq : α(σ) 6= 0R} 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 ∈ F, where α, β ∈ Cq(X,R) and a ∈ R. The 0 element of Cq(X,R) is the function 0 : Sq → R defined by

0(σ) = 0R.

When q ∈ 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 {δσ : σ ∈ Sq} forms an R-basis for Cq(X,R) for q ≥ 0. Proof. Show that {δ : σ ∈ S } is linearly independent over R and that α = P α δ for any σ q σ∈Sq σ σ α ∈ 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 ∈ Cq(X,R) can be rewritten as X α = ασσ. σ∈F 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) ∈ 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 |∆q−1 . i i The continuities of the family of functions {fq : 0 ≤ i ≤ q} 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 = σ∈F 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).

σ∈Sq

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 ∈ 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) q∈Z called the singular homology theory of X over R. We also denote M C∗(X,R) = Cq(X,R). q∈Z Example 2.1. Let X = {x} 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 {Cn : n ∈ Z} 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 {Xα : α ∈ Λ} be the set of all path components of X. For all q ≥ 0, ∼ M Hq(X,R) = Hq(Xα,R). α∈Λ Hence to study the singular homology of a space, we only need to ﬁnd out the singular homology of its path components. Assume that X is path connected. The standard 0-simplex is the set ∆0 = {0}. A singular 0-simplex in X is a continuous map σ : ∆0 → X. Then σ is determined by σ(0). The set S0 and X are in one-to-one correspondence:

ι : S0 → X, σ 7→ σ(0).

We identify C0(X,R) with the space of functions α : X → R so that such that {x ∈ X : α(x) 6= 0} P is a ﬁnite set. Thus an element of C0(X,R) is represented as x∈X rxx where rx ∈ R. 2 The standard 1-simplex ∆1 is the subset {(t0, t1) ∈ R : t0, t1 ≥ 0, t0 + t1 = 1}. A singular simplex in X is a continuous map σ : ∆1 → X. Let h : [0, 1] → ∆1 be the map t 7→ (1 − t, t). Then h is a homeomorphism3. We obtain a path σ ◦ h : [0, 1] → X in X. The space of path in X is denoted by P = C([0, 1],X). There is a bijection

S1 → P, σ 7→ σ ◦ h.

We identify C1(X,R) with the space of functions β : P → R so that {c : β(c) 6= 0} is a finite set. P We represent β as a finite sum β = c∈P rcc. Here rc ∈ R and c : [0, 1] → X denotes a path. Now we would like to express the boundary map ∂1 using the new expression. 0 0 1 1 The face map f1 : ∆0 → ∆1 is the map f1 (0) = (0, 1) and f1 : ∆0 → ∆1 is the map f1 (0) = (1, 0). If σ : ∆1 → X is a singular 1-simplex, then σ(0)(0) = σ(0, 1) = σ ◦ h(1), σ(1)(0) = σ(1, 0) = σ ◦ h(0). Recall the boundary of an 1-singular simplex σ in X is defined to be the 0-chain (0) (1) ∂1σ = σ − σ . −1 Thus if c : [0, 1] → X is a path corresponding to the singular 1-simplex c ◦ h : ∆1 → X, then by P the identification described above, we find ∂1c = c(1) − c(0). If β = c∈P rcc, then X ∂1β = rc(c(1) − c(0)) c∈P X X = rcc(1) − rcc(0) c c X X = rcx − rcx. x:c(1)=x c:c(0)=x

Let : C0(X,R) → R be the map ! X X rxx = rx. x∈X x∈X ∼ Then is surjective and thus by the first isomorphism, we find C0(X,R)/ ker = R. Observe that     X X X  rcx = rc =  rcx . x:c(1)=x c x:c(0)=x Since is R-linear, we find X X ◦ ∂1β = rc − rc = 0. c c

3Check as an exercise 6

In other words, ◦ ∂1 = 0 which gives Im ∂1 ⊆ ker . In fact, we can prove

Lemma 3.1. ker = Im ∂1. P P Proof. Let z ∈ ker . Then z = 0. Write z = x∈X rxx. Then x∈X rx = 0. Denote {x ∈ X : rx 6= Pn 0} = {x1, ··· , xn} and rxi = ri for 1 ≤ i ≤ n. Then z = i=1 rix. Choose x0 ∈ X. By the path Pn connectedness of X, for each 1 ≤ i ≤ n, we choose a path βi from x0 to xn. Write β = i=1 riβi. Then n X ∂β = ri(xi − x0) i=1 n n X X = rixi − rix0 i=1 i=1 n n ! X X = rixi − ri x0 i=1 i=1 = z

Here use the fact that r1 + ··· + rn = 0. We ﬁnd z = ∂β for some β ∈ C1(X,R). We prove our assertion.

Since the zeroth singular homology group H0(X,R) is deﬁned to be C0(X,R)/ Im ∂1 and ∂1 = ker , we ﬁnd ∼ H0(X,R) = C0(X,R)/ ker = R when X is path connected. In general by Theorem 3.1, we obtain: Corollary 3.1. Let X be a space and C be the set of all path components of X. Then ∼ M H0(X,R) = R. C

We have set Ci(X,R) = 0 when i < 0 and ∂i : Ci(X,R) → Ci−1(X,R) to be the zero map for all 0 0 0 i < 0. In fact, we can deﬁne a new complex C = (Ci, ∂i) by setting   Ci(X,R) if i ≥ 0 ∂i if i > 0 0  0  Ci = R if i = −1 and ∂i = if i = 0 . 0 if i < −1. 0 if i < 0.

The i-th homology of the new complex C0 is called the reduced i-th homology of the space X and denoted by # 0 Hi (X,R) = Hi(C ). It follows from the deﬁnition that ( H (X,R) if i 6= 0 H#(X,R) = i i 0 if i = 0 and if X is path connected.

When X is an one pointed space, all of its reduced homology modules vanish, i.e.

# Hi (pt,R) = 0, i ∈ Z. This motivates the deﬁnition of acyclic complex.

Deﬁnition 3.1. A complex C = (Ci, ∂i) is acyclic if all of its homological modules vanish, i.e. Hi(C) = 0 for all i ∈ Z. 7

Notice that Hi(C) = ker ∂i/ Im ∂i+1 and hence C is acyclic if and only if ker ∂i = Im ∂i+1, i.e. the complex C forms an exact sequence of R-modules. Let C = (Ci, ∂i) be a complex with the property that Ci = 0 for all i < 0. An augmentation of such a complex C over R is an R-epimorphism : C0 → R such that ∂1 = 0. A complex with an augmentation gives us a new complex C0 deﬁned by   Ci if i ≥ 0 ∂i if i > 0 0  0  Ci = R if i = −1 and ∂i = if i = 0 . 0 if i < −1. 0 if i < 0. The new complex C0 is called the reduced chain complex associated with (C, ). The complex C0 depends on the choice of . The corresponding homology of C0 is called the reduced homology associated with (C, ) and denoted by # 0 Hi (C) = Hi(C ), i ∈ Z. Deﬁnition 3.2. A chain complex C with augmentation is acyclic if its corresponding reduced chain complex is acyclic. Hence the chain complex of an one point space with the above given augmentation is acyclic. Proposition 3.1. A complex C with augmentation is acyclic if ( 0 if i 6= 0 Hi(C) = R if i = 0.

# # Proof. Using the fact that Hi (C) = Hi(C) for i 6= 0, we see that Hi (C) = 0 if and only if Hi(C) = 0 for i 6= 0. ∼ 0 Since : C0 → R is surjective, C0/ ker = R. If C is acyclic, ker = Im ∂1. In this case, ∼ ∼ C0/ Im ∂1 = R. Note that H0(C) = C0/ Im ∂1, we ﬁnd H0(C) = R. This completes the proof. 8

4. Category of complexes and the Homological Functor

Let f : X → Y be a continuous map. Given a singular q-simplex σ : ∆q → X in X, the continuous map f ◦ σ : ∆q → Y determine a q-simplex in Y. We deﬁne an R-linear map X X Cq(f): Cq(X,R) → Cq(Y,R), rσσ 7→ rσ(f ◦ σ). σ σ Then we obtain the following diagrams of R-linear maps

X X ∂q+1 ∂q · · · −−−−→ Cq+1(X,R) −−−−→ Cq(X,R) −−−−→ Cq−1(X,R) −−−−→ · · ·       C (f) C (f) C (f)  y q+1 y q y q−1 y y Y Y ∂q+1 ∂q · · · −−−−→ Cq+1(Y,R) −−−−→ Cq(Y,R) −−−−→ Cq−1(Y,R) −−−−→ · · · . X Y Here ∂q and ∂q are the boundaries maps on Cq(X,R) and on Cq(Y,R) respectively. If σ : ∆q → X Pq i (i) is a singular q-simplex, then ∂qσ = i=0(−1) σ . Thus q q X X i (i) X i (i) Cq−1(f) ◦ ∂q (σ) = Cq−1(f)(∂qσ) = (−1) f ◦ σ = (−1) (f ◦ σ) . i=0 i=0

On the other hand, Cq(f)(σ) = f ◦ σ and hence q Y X i (i) ∂q Cq(f)(σ) = (−1) (f ◦ σ) . i=0 The above two equations imply that Y X ∂q Cq(f) = Cq−1(f)∂q . In other words, the above diagram commutes. This motivates the deﬁnition of chain maps or morphisms between chain complexes. A B Deﬁnition 4.1. Let A = (Ai, ∂i ) and B = (Bi, ∂i ) be two chain complexes over R. A chain map f = {fi} (morphism) from A to B is a sequence of R-linear maps fi : Ai → Bi such that B A ∂i ◦ fi = fi−1 ◦ ∂i , for all i ∈ Z. In other words, the following diagram commutes:

A A ∂i+1 ∂i · · · −−−−→ Ai+1 −−−−→ Ai −−−−→ Ai−1 −−−−→ · · ·       f   f   y i+1y fiy i−1y y B B ∂i+1 ∂i · · · −−−−→ Bi+1 −−−−→ Bi −−−−→ Bi−1 −−−−→ · · · .

The identify morphism idA : A → A is deﬁned by the sequence of identity maps {idAi : Ai → Ai}. If f : A → B and g : B → C are chain maps, we deﬁne their composition g ◦ f to be the sequence of R-linear maps {gi ◦ fi : Ai → Ci}. We leave to the reader to verify that g ◦ f is again a chain map. Proposition 4.1. The collections of all complexes over R together with chain maps forms a category denoted by CompR. The category CompR is called the category of complexes over R. Corollary 4.1. Any continuous map f : X → Y between topological spaces X,Y determines a chain map C∗(f): C∗(X,R) → C∗(Y,R), between the singular chain complexes of X and Y , where C∗(f) = {Ci(f): i ∈ Z} such that the following properties hold.

(1) If idX : X → X is the identity map, then C∗(idX ) = idC∗(X,R) . (2) If f : X → Y and Y → Z are continuous maps, then C∗(g ◦ f) = C∗(g) ◦ C∗(f). 9

This corollary implies that C∗ : Top → CompR deﬁnes a functor from the category of topological spaces into the category of complexes over R.

Lemma 4.1. Let f : A → A0 be a chain map between chain complexes A and B. For each i ∈ Z, we deﬁne Hi(f): Hi(A) → Hi(B), [z] 7→ [fi(z)].

Hi(f) is a well-deﬁned R-linear homomorphism. 0 0 A Proof. Let z, z ∈ ker ∂i so that z − z = ∂i+1w for some w ∈ Ai+1. Since f is a chain map, 0 0 A B fi(z) − fi(z ) = fi(z − z ) = fi∂i+1w = ∂i+1fi+1w 0 B 0 Hence fi(z) − fi(z ) ∈ Im ∂i+1. Thus [fi(z)] = [fi(z )]. The linearity of Hi(f) follows from the linearity of fi. Lemma 4.2. Let f : A → B and g : B → C be chain maps. Then

Hi(g ◦ f) = Hi(g) ◦ Hi(f) for any i ∈ Z. Proof. Routine check.

Corollary 4.2. For each i ∈ Z, the assignment Hi : A 7→ Hi(A) sending a chain complex over R to its i-th homology and Hi :(f : A → B) 7→ Hi(f): Hi(A) → Hi(B) sending a chain map to its corresponding induced map deﬁnes a functor

Hi : CompR → ModR from the category of complexes over R to the category of R-modules ModR. 10

5. Homotopy and Chain Homotopy