Notes for Math 227A: Algebraic Topology

NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY

KO HONDA

1. CATEGORIES AND FUNCTORS 1.1. Categories. A category C consists of: (1) A collection ob(C) of objects. (2) A set Hom(A, B) of morphisms for each ordered pair (A, B) of objects. A morphism f ∈ Hom(A, B) f is usually denoted by f : A → B or A → B. (3) A map Hom(A, B) × Hom(B,C) → Hom(A, C) for each ordered triple (A,B,C) of objects, denoted (f, g) 7→ gf. (4) An identity morphism idA ∈ Hom(A, A) for each object A. The morphisms satisfy the following properties: A. (Associativity) (fg)h = f(gh) if h ∈ Hom(A, B), g ∈ Hom(B,C), and f ∈ Hom(C, D). B. (Unit) idB f = f = f idA if f ∈ Hom(A, B). Examples: (HW: take a couple of examples and verify that all the axioms of a category are satisﬁed.) 1. Top = category of topological spaces and continuous maps. The objects are topological spaces X and the morphisms Hom(X,Y ) are continuous maps from X to Y .

2. Top• = category of pointed topological spaces. The objects are pairs (X,x) consisting of a topological space X and a point x ∈ X. Hom((X,x), (Y,y)) consist of continuous maps from X to Y that take x to y. Similarly deﬁne Top2 = category of pairs (X, A) where X is a topological space and A ⊂ X is a subspace. Hom((X, A), (Y,B)) consists of continuous maps f : X → Y such that f(A) ⊂ B. 3. hTop = homotopy category of topological spaces. The objects are topological spaces X and Hom(X,Y ) is the set of homotopy classes of continuous maps f : X → Y . Recall that f0,f1 : X → Y are homotopic if there is a continuous map F : X × [0, 1] → Y such that F |X×{i} = fi, i = 0, 1. Homotopy induces an equivalence relation ∼ and the equivalence classes are called homotopy classes. 4. Grp = category of groups and group homomorphisms. The objects are groups G and the morphisms Hom(G, H) are group homomorphisms from G to H. 5. R–Mod = category of left R-modules of an associative ring R. The objects are left R-modules and the morphisms Hom(M,N) are left R-module maps. Also let Mod–R be the category of right R-modules. We sometimes write Ab (category of abelian groups) for Z–Mod. 6. Ch(R–Mod) = category of chain complexes of (left) R-modules and chain maps. Recall that a chain complex is a sequence C∗ of R-module maps:

∂n+1 ∂n −−−−→ Cn+1 −−−−→ Cn −−−−→ Cn−1 −−−−→ 1 2 KO HONDA

φn such that ∂n∂n+1 = 0 and a chain map is a collection of R-module maps Cn −→ Dn, often denoted φ : C∗ → D∗, such that the diagram commutes:

∂n+1 ∂n Cn+1 −−−−→ Cn −−−−→ Cn−1

φn+1 φn φn−1     ∂n+1  ∂n  Dny+1 −−−−→ Dyn −−−−→ Dny−1 1.2. Functors. A covariant functor F : C → D from a category C to a category D is a rule which sends: (1) an object A ∈ ob(C) to F (A) ∈ ob(D); and (2) every morphism f ∈ Hom(A, B) in C to a morphism F (f) ∈ Hom(F (A), F (B)) in D, such that: A. F (gf)= F (g)F (f). B. F (idA)=idFA. A contravariant functor F assigns to f ∈ Hom(A, B) an element F (f) ∈ Hom(F (B), F (A)) satisfying F (fg)= F (g)F (f). Examples: (HW: verify the axioms of a functor) 1. The singular chain complex functor Sing : Top → Ch(Z–Mod), which is defined as follows: m Brief review of singular homology. Given v0, . . . , vn ∈ R , let [v0, . . . , vn] be the convex hull of v0, . . . , vn, i.e., n m n [v0, . . . , vn]= { i=0 tivi ∈ R | i=0 ti = 1,ti ≥ 0, i = 0,...,n} . n n+1 The standard n-simplex ∆ is [e0,...,eP n] ⊂ R . P n Let X ∈ ob(Top). A continuous map σ :∆ → X is called a singular n-simplex. We define Cn(X) to the free Z-module generated by singular n-simplices; an element of Cn(X) is called singular n-chain. The boundary map ∂n : Cn(X) → Cn−1(X) is given by: n n ∂nσ = i=0(−1) σ|[e0,...,eî,...,en]. P Here σ|[e0,...,eî,...,en] means the composition of [e0,...,en−1] → [e0,..., eî,...,en] given by the canonical linear homeomorphism, followed by σ restricted to [e0,..., eî,...,en]. ∂n+1 ∂n One can verify that ∂n∂n+1 = 0, i.e., Cn+1 −→ Cn −→ Cn−1 is a chain complex. Given a continuous map f : X → Y , we define the induced map fn : Cn(X) → Cn(Y ) by mapping n n σ :∆ → X to f ◦ σ :∆ → Y and extending linearly. One can verify that ∂n+1fn+1 = fn∂n+1. 1’. Similarly, define the relative singular chain complex functor Sing : Top2 → Ch(Z–Mod) which sends (X, A) to ∂n C∗(X, A) = (→ Cn(X, A) −→ Cn−1(X, A) →), where Cn(X, A) := Cn(X)/Cn(A) and ∂n is the map Cn(X, A) → Cn−1(X, A) induced from ∂n : Cn(X) → Cn−1(X).

2. The homology functor Hi : Ch(R–Mod) → R–Mod which assigns to a chain complex C∗ the homology Hi(C∗) := ker(di)/ im(di+1), NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 3 and to a chain map φ : C∗ → D∗ the induced homomorphism φ∗ : Hi(C∗) → Hi(D∗).

3. The composition Hi ◦ Sing : Top → Z–Mod is the singular homology functor which takes X 7→ sing Hi (X; Z).

4. The fundamental group functor π1 : Top• → Grp sends (X,x) 7→ π1(X,x).

5. Given a ring R and a right R-module N, there is a functor N⊗R : R–Mod → Z–Mod given by M 7→ N ⊗R M. 5’. More generally, given rings R, S, and an S − R bimodule N (this means that S acts on N from the left and R from the right), we have the functor N⊗R : R–Mod → S–Mod given by M 7→ N ⊗R M. We can also extend N⊗R to Ch(R–Mod) → Ch(S–Mod). 6. Given an R-module N, there is a covariant functor Hom(N, −) : R–Mod → R–Mod given by M 7→ HomR(N, M) and a contravariant functor Hom(−,N) : R–Mod → R–Mod given by M 7→ HomR(M,N). We can also extend the Hom functors to Ch(R–Mod) → Ch(R–Mod).

1.3. Adjoint functors. Two functors F : C → D and G : D→C are adjoint if for each A ∈ ob(C) and B ∈ ob(D) there is a bijection

∼ τAB : HomD(FA,B) −→ HomC(A, GB), subject to the following naturality condition: For any f ∈ Hom(A, A′) and g ∈ Hom(B,B′), the diagram commutes:

∗ ′ Ff g∗ ′ HomD(F A ,B) −−−−→ HomD(FA,B) −−−−→ HomD(FA,B )

τA′B τAB τAB′

 ∗    ′ f  Gg∗  ′ HomC(yA , GB) −−−−→ HomC(yA, GB) −−−−→ HomC(A,y GB ).

We say that F is a left adjoint of G and G is a right adjoint of F .

Example: Given an R − S-bimodule N, the functors ⊗RN : Mod–R → Mod–S and HomS(N, −) : Mod–S → Mod–R are adjoint functors.

1.4. Natural transformations. Given two functors F, G : C → D, a natural transformation η : F ⇒ G ′ associates a morphism ηA : F (A) → G(A) for each A ∈ ob(C) such that for each morphism f : A → A in C the following diagram commutes:

Ff F (A) −−−−→ F (A′)

ηA ηA′    Gf  G(yA) −−−−→ G(yA′). 4 KO HONDA

′ ′ Example: A right R-module map φ : N → N gives a natural transformation η : N⊗R ⇒ N ⊗R where ′ ′ N⊗R,N ⊗R : R–Mod → Z–Mod: Given a left R-module map f : M → M ,

(N⊗R)f ′ N ⊗R M −−−−−→ N ⊗R M

ηM ηM′

 ′  ′  (N ⊗R)f ′  ′ N ⊗yR M −−−−−−→ N ⊗yR M . NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 5

2. HOMOLOGY WITH COEFFICIENTS 2.1. Definitions. Let X be a topological space and G be a Z-module (aka abelian group). An n-chain on X with coefficients in G is finite formal sum i niσi, where σi is a singular n-simplex on X and ni ∈ G. The set Cn(X; G) of n-chains on X with coefficientsP in G can be written as

Cn(X; G)= Cn(X) ⊗Z G.

The functor ⊗G : Ch → Ch, where Ch = Ch(Z–Mod), sends C∗(X) to C∗(X; G). (Note that Z is commutative, so left and right modules are the same; we are also tensoring over Z.) In particular this means ∂ ∂⊗idG 2 that Cn(X) → Cn−1(X) passes to Cn(X; G) −→ Cn−1(X; G) and (∂ ⊗ idG) = 0. The homology of C∗(X; G) is written as H∗(X; G) and is called the homology of X with coefﬁcients in G. This is the result of the composition

Hn ◦⊗G ◦ Sing : Top → Z–Mod.

Similarly, we can define Cn(X, A; G) = Cn(X, A) ⊗ G. The relative homology of (X, A) with coefficients in G is its homology group. Remark: If we take homology with coefficients in Z/2Z or Q, then everything involved become vector spaces, and the calculations are often easier.

Question: Is Hn(X; G) ≃ Hn(X; Z) ⊗ G? In order to answer this question, we take a detour in homological algebra.

2.2. Right exactness of ⊗. Let N be a left R-module and

0 → A → B → C → 0 be an exact sequence of right R-modules. Then we have the following:

Fact: The functor ⊗RN : Mod–R → Z–Mod is right exact, i.e.,

A ⊗R N → B ⊗R N → C ⊗R N → 0 is an exact sequence. HW: check this! However, A ⊗ N → B ⊗ N is not always injective. (We omit R from the notation from now on.) Prototypical Example: 0 → Z →n Z → Z/nZ → 0. If we tensor with Z/nZ, then we have

0 Z/nZ → Z/nZ → Z/nZ → 0 exact, but the left arrow is not injective. To continue this exact sequence to the left, we introduce the derived functors of ⊗N. 6 KO HONDA

2.3. Projective resolutions. Definition 2.3.1. A (right) R-module P is projective if it satisfies the following lifting property: given a b map a : P → N and a surjection M −→ N → 0, there exists a lift a : P → M such that a = ba. HW: Prove that P is projective if and only if it is a direct summand of a free R-module. Definition 2.3.2. A projective resolution of an R-module M is an exact sequence

i2 i1 i0 (2.3.1) ···→ P2 → P1 → P0 → M → 0, where Pi are projective R-modules. (We often write P∗ → M.) As a special case, if we take the Pi to be free, we have a free resolution.

Lemma 2.3.3. A Z-module M has a free resolution 0 → F1 → F0 → M → 0.

Proof. Let {aα} be a generating set for M and F0 be the free abelian group generated by {aα}. Then the kernel of the natural map F0 → M is a subgroup of a free abelian group, and hence is free. (Note this is not obvious.) This gives the injection F1 → F0.

Remark: The same proof applies to show that any R-module M admits a free resolution. Example: If A = Z/mZ, then m 0 → Z → Z → Z/mZ → 0 is a free resolution of Z/mZ. We could also have taken i j 0 → Z → Z2 → Z → Z/mZ → 0, where i : 1 7→ (0, 1) and j : (1, 0) 7→ m, (0, 1) 7→ 0. In particular the example shows that projective/free resolutions are not unique! Lemma 2.3.4. ′ (1) Given projective resolutions P∗ of A and P∗ of B and a map φ−1 : A → B, there is a chain map ′ φ : P∗ → P∗ which extends φ−1, i.e., the following diagram commutes: i1 i0 −−−−→ P1 −−−−→ P0 −−−−→ A −−−−→ 0

φ1 φ0 φ−1

 ′  ′  ′ i1 ′ i0  −−−−→ Py1 −−−−→ Py0 −−−−→ By −−−−→ 0 ′ (2) Any two chain maps φ, ψ : P∗ → P∗ extending φ−1 are chain homotopic. ′ ′ Recall that two chain maps φ, ψ : (C∗, ∂) → (C∗, ∂ ) are chain homotopic if there exist maps hi : Ci → ′ ′ Ci+1 such that φi − ψi = ∂i+1hi + hi−1∂i. Moreover, chain homotopic chain maps induce the same maps on homology. ′ Proof. (1) The map φ0 is defined using the lifting property: i0φ0 = φ−1i0. For φ1, we show that φ0i1(P1) ⊂ ′ ′ ′ Im i1 = ker i0. Indeed, i0φ0i1 = φ−1i0i1 = 0. Hence φ1 can be defined using the lifting property. ′ ′ (2) Since i0(φ0 − ψ0) = 0, it follows that (φ0 − ψ0)(P0) ⊂ Im(i1). Hence h0 can be defined using the ′ ′ lifting property. Next observe that i1(φ1 − ψ1 − h0i1) = (φ0 − ψ0 − i1h0)i1 = 0. Hence h1 can also be defined using the lifting property. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 7

2.4. Deﬁnition of Tor. Given a projective resolution P∗ → M, we tensor with a right R-module N to obtain

i2 i1 · · · → P1 ⊗ N → P0 ⊗ N → 0, which is no longer exact but is a chain complex. Its homology is denoted by Tori(M,N).

Theorem 2.4.1. Tori(M,N) only depends on M and N. In particular, it is independent of the projective resolution used for M.

′ Proof. This follows from Lemma 2.3.4: Given projective resolutions P∗, P∗ → M, there exist chain maps ′ ′ ′ φ : P∗ → P∗ and ψ : P∗ → P∗ such that ψφ, id : P∗ → P∗ are chain homotopic. Tensoring with N still preserves this property.

Remark: Given a projective resolution P∗ of M, there is a chain map

i1 −−−−→ P1 −−−−→ P0 −−−−→ 0

0 i0     −−−−→ y0 −−−−→ My −−−−→ 0 which induces an isomorphism on homology. A chain map f : C∗ → D∗ which induces an isomorphism on all homology groups is called a quasi-isomorphism. Hence P∗ is a quasi-isomorphic replacement of M. Properties of Tor:

(1) Tori(M,N) = 0 if i > 1 and R = Z. (This follows from the existence of a free resolution 0 → F1 → F0 → M → 0.) i1 (2) Tor0(M,N)= M ⊗ N. (If P1 → P0 → M → 0 is exact, then P1 ⊗ N → P0 ⊗ N → M ⊗ N → 0 is exact. Hence, Tor0(M,N)= P0 ⊗ N/ Im i1 ≃ M ⊗ N.) (3) Tori(M,N) = Tori(N, M) if R is commutative. (Proof is not obvious.) (4) If M is projective, then Tori(M,N) = 0 for i ≥ 1. (If M is projective, then 0 → M → M → 0 is the projective resolution for M. Hence 0 → M ⊗ N → 0 is the chain complex which computes Tor.) ′ ′ (5) Tori(M ⊕ M ,N) ≃ Tori(M,N) ⊕ Tori(M ,N). m (6) Tor1(Z/mZ, Z/nZ) = ker(Z/nZ → Z/nZ) = Z/kZ, where k = GCD(m,n). In particular, m Tor1(Z/nZ, Z/nZ) = Z/nZ. (Tensor the truncated free resolution 0 → Z → Z → 0 to obtain m 0 → Z/nZ → Z/nZ → 0.)

2.5. More on the Tor functor.

Theorem 2.5.1. Given an exact sequence 0 → A → B → C → 0 of right R-modules, there exist projective i j resolutions F∗ → A, G∗ → B, H∗ → C and exact sequences 0 → F∗ −→ G∗ −→ H∗ → 0 which make 8 KO HONDA the following diagram commutative: 0 0 0

    f1  f0  −−−−→ Fy1 −−−−→ Fy0 −−−−→ Ay −−−−→ 0

i1 i0 i−1     g1  g0  −−−−→ Gy1 −−−−→ Gy0 −−−−→ By −−−−→ 0

j1 j0 j−1     h1  h0  −−−−→ Hy1 −−−−→ Hy0 −−−−→ Cy −−−−→ 0

      y0y 0y 0 Proof. Take projective resolutions F∗ → A and H∗ → C. Then we set Gi = Fi ⊕ Hi and i : Fi → Gi and j : Gi → Hi to be the usual inclusion and projection maps. The goal is to deﬁne the maps g0, g1, etc. For g0, lift h0 to a map h0 : H0 → B. Then let g0 = (i−1f0, h0) : F0 ⊕ H0 → B. We verify that g0 is ′ ′ ′ surjective: Given b ∈ B, there exists b ∈ B such that j−1(b − b) = 0 and b = h0(y) for some y ∈ H0. ′ ′ Now b − b ∈ Im i−1, so there exists x ∈ F0 such that i−1f0(x) = b − b. Thus g0 maps (x,y) 7→ b. The deﬁnitions of g1 etc. are similar and it remains to verify that Im gi+1 = ker gi (HW). We now apply the functor ⊗M where M is a left R-module.

Fact: The sequence 0 → Fi ⊗ M → Gi ⊗ M → Hi ⊗ M → 0 is exact since 0 → Fi → Gi → Hi → 0 splits and Gi ≃ Fi ⊕ Hi. (HW: check this!) Hence we have an exact sequence of chain complexes

0 → F∗ ⊗ M → G∗ ⊗ M → H∗ ⊗ M → 0. The corresponding long exact sequence is:

···→ Tor1(A, M) → Tor1(B, M) → Tor1(C, M) → → A ⊗ M → B ⊗ M → C ⊗ M → 0.

Example: Applying ⊗Z/mZ to m 0 → Z → Z → Z/mZ → 0, we obtain ∼ 0 ∼ 0 → Tor1(Z/mZ, Z/mZ)= Z/mZ → Z/mZ → Z/mZ → Z/mZ → 0.

2.6. Universal Coefﬁcient Theorem. We now restrict to Z-modules. Theorem 2.6.1 (Universal Coefﬁcient Theorem for homology). There is an exact sequence

0 → Hn(X) ⊗ M → Hn(X; M) → Tor1(Hn−1(X), M) → 0, NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 9 and the exact sequence splits, albeit noncanonically. Therefore,

Hn(X; M) ≃ (Hn(X) ⊗ M) ⊕ Tor1(Hn−1(X), M). j HW: Show that if an exact sequence 0 → A →i B → C → 0 splits, i.e., there exists k : C → B such that jk = id, then B ≃ A ⊕ C. Proof. Start with the exact sequence

∂n 0 → Zn → Cn −→ Bn−1 → 0, where Cn = Cn(X), Zn = ker ∂n, and Bn−1 = Im ∂n. The exact sequence can be viewed as a row in an exact sequence of chain complexes 0 → Z∗ → C∗ → B∗ → 0, where the boundary maps for Z∗ and B∗ are zero. Now apply ⊗M: since Bn−1 is a submodule of Cn−1, it is free, and Tor1(Bn−1, M) = 0. Hence the exact sequence remains exact under ⊗M, i.e.,

(2.6.1) 0 → Zn ⊗ M → Cn ⊗ M → Bn−1 ⊗ M → 0 is exact. The corresponding long exact sequence is:

in⊗id in−1⊗id Bn ⊗ M −→ Zn ⊗ M → Hn(X; M) → Bn−1 ⊗ M −→ Zn−1 ⊗ M.

(HW: check that the connecting homomorphism Bn → Zn actually agrees with the inclusion map in.) Hence 0 → (Zn ⊗ M)/ Im(in ⊗ id) → Hn(X; M) → ker(in−1 ⊗ id) → 0. We also have the exact sequence 0 → Bn → Zn → Hn(X) → 0, whose derived exact sequence is:

in⊗id 0 → Tor1(Hn(X), M) → Bn ⊗ M −→ Zn ⊗ M → Hn(X) ⊗ M → 0.

We obtain Hn(X) ⊗ M ≃ (Zn ⊗ M)/ Im(in ⊗ id) and Tor1(Hn−1(X), M) ≃ ker(in−1 ⊗ id); this proves the ﬁrst assertion of the theorem. To prove the noncanonical splitting, notice that there is a splitting j : Bn−1 ⊗ M → Cn ⊗ M of ′ Equation (2.6.1) since Zn, Cn, Bn−1 are all free Z-modules. Restrict j to j on

Tor1(Hn−1(X), M) ≃ ker(in−1 ⊗ id) ⊂ Bn−1 ⊗ M.

Then by the deﬁnition of the connecting homomorphism in−1 ⊗ id for Equation (2.6.1), we see that ′ ′ (∂n ⊗ id)j (x) = 0 ∈ Cn−1 ⊗ M for x ∈ Tor1(Hn−1(X), M). Hence j descends to a map j : Tor1(Hn−1(X), M) → Hn(X; M). The fact that j is a splitting implies that j is a splitting (check this!). 2 Example: X = RP . Recall that H0(X)= Z, H1(X)= Z/2Z, and H2(X) = 0. Then we compute that

H0(X; Z/2Z)= H0(X) ⊗ Z/2Z = Z/2Z,

H1(X; Z/2Z) = (H1(X) ⊗ Z/2Z) ⊕ Tor1(Z, Z/2Z)= Z/2Z, H2(X; Z/2Z) = Tor1(Z/2Z, Z/2Z)= Z/2Z. 10 KO HONDA

3. COHOMOLOGY 3.1. Deﬁnitions. Given an abelian group G, recall the contravariant functor Hom(−, G) : Ab → Ab such that A 7→ Hom(A, G) and f ∈ Hom(A, B) is mapped to f ∗ : Hom(B, G) → Hom(A, G) which sends φ to φ ◦ f. There is also a contravariant functor Hom(−, G) : Ch → Ch which sends a chain complex

∂n ∂1 ···→ Cn −→ · · · −→ C0 → 0, to the cochain complex

δn−1 δ0 ···← Hom(Cn, G) ←− · · · ←− Hom(C0, G) ← 0, ∗ where the cochain map is given by δn−1 = ∂n (note slightly awkward indexing), and f : C∗ → D∗ to Hom(−, G)f : Hom(D∗, G) → Hom(C∗, G). n n Now we compose with the functor Sing : Top → Ch: We denote Hom(Cn, G) by C or C (G) or n ∗ n−1 C (X; G) if Cn = Cn(X). Unwinding the deﬁnition of δn−1 = ∂n, given φ ∈ C and α ∈ Cn, we have δn−1φ(α)= φ(∂nα). Claim: δ2 = 0. Proof. This is ∂2 = 0 dualized: (δ2φ)(α)= δφ(∂α)= φ(∂2α) = 0. The homology of this chain complex Hn(C; G) = ker δ/ Im δ is the nth cohomology group. n 3.2. Interpreting cohomology. The cochain group C (X; G) = Hom(Cn(X), G) is the set of functions φ : {singular n-simplices of X}→ G. In particular, φ ∈ C0(X; R) assigns a real number to each point of X and δφ ∈ C1(X, R) assigns a real number to each arc α :∆1 = [0, 1] → X as follows: δφ(α)= φ(∂α)= φ(α(1)) − φ(α(0)). For example, if a, b ∈ X, α is a path from a to b, and φ(a)= r, φ(b)= s, then δφ(α)= φ(b)−φ(a)= s−r. Note that δφ = 0 means φ assigns the same value to all the points in a connected component of X. Next, if φ ∈ C1(X; R), then δφ ∈ C2(X; R) is given by:

δφ(α)= φ(α|[v1,v2]) − φ(α|[v0,v2])+ φ(α|[v0,v1]).

Observe that δφ = 0 means φ(α|[v0,v2])= φ(α|[v0,v1])+ φ(α|[v1,v2]). n One way of constructing a cochain φ ∈ Csm(X; R), if X is a manifold and all the simplices are smooth, is to integrate an n-form ω, i.e., deﬁne

φ(β)= β ω. R Then δφ(α) = ∂α ω = α dω by Stokes’ Theorem. (Here d is the exterior derivative.) Hence there is a chain map of (co)-chainR complexes:R d . . . −−−−→ Ωn(X; R) −−−−→ Ωn+1(X; R) −−−−→ . . .

  n  δ n+1 . . . −−−−→ Csm(yX; R) −−−−→ Csm y(X; R) −−−−→ . . . Here Ωn(X; R) is the space of smooth n-forms on X. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 11

3.3. Universal coefﬁcient theorem. Just as the functor ⊗G is only right exact, the Hom(−, G) functor is only left exact:

j j∗ ∗ Fact: If A →i B → C → 0 is exact, then 0 → Hom(C, G) −→ Hom(B, G) −→i Hom(A, G) is exact. Proof. j∗ is injective: Since j : B → C is surjective, given f ∈ Hom(C, G), the composition f ◦ j(x) = 0 for all x ∈ B implies that f(y) = 0 for all y ∈ B. Im j∗ = ker i∗. ⊂: Given f ∈ Hom(C, G), i∗j∗f = (ji)∗f = 0. ⊃: If i∗g : A → B → G is zero, then we can take the quotient map C = B/A → G.

Prototypical Example: Given 0 → Z →n Z → Z/nZ → 0, applying Hom(−, Z) gives: n 0 → Hom(Z/nZ, Z) ≃ 0 → Hom(Z, Z) ≃ Z → Hom(Z, Z) ≃ Z.

As in the case of Tor, take a projective resolution P∗ → A and apply Hom(−, G) to obtain:

Hom(P∗, G) := (···← Hom(P1, G) ← Hom(P0, G) ← 0). The (co)-homology of this chain complex is denoted by Exti(A, G). As before, Exti(A, G) does not depend on the choice of projective resolution. Properties of Ext: (1) Ext0(A, G) = Hom(A, G). (Follows from the left exactness of the Hom functor.) i (2) Ext (A, G) = 0 for i> 1 if R = Z. (Use the free resolution 0 → F1 → F0 → 0 for A.) (3) Exti(A, G) = 0 if A is projective. (Use the resolution 0 → A → 0 for A.) (4) Exti(A ⊕ B, G) = Exti(A, G) ⊕ Exti(B, G). (5) Ext1(Z/nZ, Z) = Z/nZ. (Take the resolution 0 → Z →n Z → 0 for Z/nZ, and dualize to obtain n 0 → Z → Z → 0. Then Ext0(Z/nZ, Z) = Hom(Z/nZ, Z) = 0 and Ext1(Z/nZ, Z)= Z/nZ.) Now, given an exact sequence of abelian groups 0 → A → B → C → 0, take the corresponding exact sequence of projective resolutions 0 → P∗ → Q∗ → R∗ → 0 and apply Hom(−, G). Its long exact sequence is: 0 → Hom(C, G) → Hom(B, G) → Hom(A, G) → → Ext1(C, G) → Ext1(B, G) → Ext1(A, G) → . . . We have the following (proof similar to homology case):

Theorem 3.3.1 (Universal Coefﬁcient Theorem for cohomology). If a chain complex C∗ of free abelian n groups have homology groups Hn(C∗), then the cohomology groups H (C∗; G) are given by: 1 n 0 → Ext (Hn−1(C∗), G) → H (C∗; G) → Hom(Hn(C∗), G) → 0. The sequence splits, albeit noncanonically.

2 0 0 Example: X = RP . H (X; Z) ≃ Hom(H0(X; Z), Z) = Z. (In fact, H (X; Z) ≃ H0(X; Z) al- 1 1 ways.) H (X; Z) ≃ Hom(H1(X; Z), Z) always, since H0(X; Z) is always free. Hence H (X; Z) ≃ Hom(Z/2Z, Z) = 0. Finally, 2 1 1 H (X; Z) ≃ Hom(H2(X; Z), Z) ⊕ Ext (H1(X; Z), Z) = Ext (Z/2Z, Z)= Z/2Z. 12 KO HONDA

Remark: The Universal Coefﬁcient Theorem basically states that H∗(X; G) can be computed from the knowledge of H∗(X; G). Then why do we care about cohomology?

3.4. Properties of cohomology. The cohomology groups satisfy all the properties enjoyed by homology, but with the arrows in the other direction. This includes the long exact sequence for relative homology, homotopy invariance, excision, Mayer-Vietoris, etc. For a while we suppress the coefﬁcient module G.

i j Relative groups. Starting with 0 → Cn(A) −→ Cn(X) −→ Cn(X, A) → 0, applying Hom(−, G) yields:

j∗ ∗ 0 → Cn(X, A) −→ Cn(X) −→i Cn(A) → 0,

n where C (X, A) := Hom(Cn(X, A), G). The sequence is exact since Cn(X, A) is free. Note that φ ∈ Cn(X, A) can be viewed as a map φ : {singular n-simplices of X} → G such that φ(σ) = 0 if Im σ ⊂ A. n n+1 We also deﬁne δX,A : C (X, A) → C (X, A) as the dual of the map ∂X,A : Cn+1(X, A) → Cn(X, A). (For the time being also write δX and δA for the duals of ∂X and ∂A.) By dualizing all the arrows for the case of homology, we obtain the commutative diagram:

j∗ ∗ 0 −−−−→ Cn(X, A) −−−−→ Cn(X) −−−−→i Cn(A) −−−−→ 0

δX,A δX δA

 ∗    j  i∗  0 −−−−→ Cn+1y(X, A) −−−−→ Cn+1y(X) −−−−→ Cn+1y(A) −−−−→ 0 and a long exact sequence in relative cohomology:

c → Hn(X, A) → Hn(X) → Hn(A) −→ Hn+1(X, A) → .

n n+1 n n HW: Verify that δX,A agrees with the restriction of δX : C (X) → C (X) to C (X, A) → C (X, A) (this is with respect to the inclusion j∗ : Cn(X, A) → Cn(X)).

HW: Verify that the diagram commutes, where the maps hA and hX,A arise in the Universal Coefﬁcient ∗ Theorem and the map d is the dual of the connecting homomorphism b : Hn+1(X, A) → Hn(A):

Hn(A) −−−−→c Hn+1(X, A)

hA hX,A    b∗  Hom(Hyn(A), G) −−−−→ Hom(Hn+1y(X, A), G).

Homotopy invariance. This follows from observing that if φ, ψ : C∗ → D∗ are chain homotopic chain ∗ ∗ maps, i.e., there exists h : C∗ → D∗+1 such that φ − ψ = h∂C + ∂Dh, then φ , ψ : Hom(D∗, G) → ∗ Hom(C∗, G) are chain homotopic as cochain maps, i.e., there exists h : Hom(D∗+1, G) → Hom(C∗, G) ∗ ∗ ∗ ∗ ∗ ∗ such that φ − ψ = ∂C h + h ∂D. Excision. Suppose Z ⊂ A ⊂ X and the closure of Z is in the interior of A. Recall that the inclusion i : (X − Z, A − Z) → (X, A) induces an isomorphism on homology. The Universal Coefﬁcient Theorem NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 13

is natural in the following way (i.e., the following diagram commutes): 1 n 0 −−−−→ Ext (Hn−1(X, A), G) −−−−→ H (X, A) −−−−→ Hom(Hn(X, A), G) −−−−→ 0

   1  n   0 −−−−→ Ext (Hn−1(X y− Z, A − Z), G) −−−−→ H (X −yZ, A − Z) −−−−→ Hom(Hn(X −yZ, A − Z), G) −−−−→ 0. HW: Check the commutativity of the diagram. The left and right arrows are isomorphisms; hence by the ﬁve lemma the middle one is also. HW: Hatcher, Section 3.1: 3,7,11. 14 KO HONDA

4. CUP PRODUCTS Today we highlight the key difference between homology and cohomology: cohomology has a ring structure.

4.1. Deﬁnition of cup product. Given cochains φ ∈ Ck(X; R) and ψ ∈ Cl(X; R) (here R is a commutative ring), deﬁne the cup product φ ∪ ψ ∈ Ck+l(X; R) by evaluating on a singular (k + l)-simplex σ :∆k+l → X, and extending bilinearly:

P ∞ A graded ring/algebra A over R comes with a decomposition A = ⊕i=0Ai of R-modules and a multiplication Ai × Aj → Ai+j which is R-bilinear. If an element a ∈ Ai we write |a| = i and call it the degree of a. The graded algebra A is a differential graded algebra (dga) if it has a differential d : Ai → Ai+1 such |a| ∗ ∞ i that d(ab) = (da)b + (−1) adb. Hence C (X; R) := ⊕i=0C (X; R) is a dga with multiplication ∪ and differential δ. HW: Verify that the cup product on C∗(X; R) is associative. If δφ = 0 and δψ = 0, i.e, both φ and ψ are closed, then δ(φ ∪ ψ) = 0 by the lemma. Also, if φ = δη and δψ = 0, i.e., φ is exact and ψ is closed, then δ(η ∪ ψ)= δη ∪ ψ = φ ∪ ψ, i.e., φ ∪ ψ is exact. Therefore, the cup product on the chain level induces a map ∪ Hk(X; R) × Hl(X; R) → Hk+l(X; R). Since the cup product is bilinear, by the universal property of tensor product, we have a map ∪ Hk(X; R) ⊗ Hl(X; R) → Hk+l(X; R).

∗ ∞ i Hence H (X; R) := ⊕i=0H (X; R) is an associative graded ring. Lemma 4.1.2. Given f : X → Y , the induced map f ∗ : Cn(X; R) → Cn(Y ; R) satisﬁes f ∗(φ ∪ ψ)= f ∗φ ∪ f ∗ψ. Proof. Just unwind deﬁnitions. Given cocycles φ ∈ Ck(X; R) and ψ ∈ Cl(X; R), ∗ f (φ ∪ ψ)(σ) = (φ ∪ ψ)(f ◦ σ)= φ(f ◦ σ|[e0,...,ek])ψ(f ◦ σ|[ek,...,ek+l]) ∗ ∗ ∗ ∗ = f φ(σ|[e0,...,ek])f ψ(σ|[ek,...,ek+l]) = (f φ ∪ f ψ)(σ). NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 15

4.2. Supercommutativity. Theorem 4.2.1. α ∪ β = (−1)klβ ∪ α, if α ∈ Hk(X; R) and β ∈ Hl(X; R). Such a relation is called graded commutative, skew-commutative, or supercommutative. Note that the cup product is NOT supercommutative on the chain level. The proof (which is rather involved) will be postponed until next time. Comparison with de Rham theory: If we have ω ∈ Ωk(X) and η ∈ Ωl(X), then we have the wedge product ω ∧ η = (−1)klη ∧ ω, already on the chain level! This is because the skew-commutativity was built |σ| into the deﬁnition: ω(x)(v1, . . . , vk) = (−1) ω(x)(σ(v1, . . . , vk)), where |σ| is the sign of a permutation σ. Example: Consider X = T 2 given as in Figure 1. We will do calculations in simplicial (co)-homology. Suppose φ ∈ C1(X) is given by φ(a) = φ(c) = 1 and φ(b) = 0. Then δφ(A) = δφ(B) = 0. Hence

b b c A

FIGURE 1.

δφ = 0. Similarly, ψ ∈ C1(X) is given by ψ(b) = ψ(c) = 1 and ψ(a) = 0, and δψ = 0. Now, (φ∪ψ)(A)= φ(a)ψ(b) = 1 and (φ∪ψ)(B)= φ(b)ψ(a) = 0. Since A−B generates H2(X)= Z, we have (φ∪ψ)(A−B) = 1, in other words, φ∪ψ generates H2(X)= Z. We also verify that (ψ∪φ)(A−B)= −1, although the skew-commutativity does not hold on the chain level! Finally, we discuss a little bit of Poincaréduality. Observe that φ could have been defined as follows: take a closed curve b′ parallel to and oriented in the same direction as b. Then φ(x) is the oriented intersection number of x with b′. Similarly ψ can be defined by taking the oriented intersection number with −a′. The curves b′ and −a′ are said to be the Poincare´ duals of φ and ψ. 4.3. Proof of Theorem 4.2.1. The proof is given in several steps. m n m n Step 1. We denote by [ej0 ,...,ejm ] the restriction to ∆ → ∆ of the linear map R → R that sends ei, n n i = 0,...,m, to eji . A particular case we are interested in is [en,...,e0]:∆ → ∆ which sends ei to en−i. This induces the map ρ : Cn(X) → Cn(X),

σ 7→ εnσ[en,...,e0], n(n+1)/2 where εn is the constant (−1) . 16 KO HONDA

Step 2. ρ is a chain map, i.e., ∂ρ = ρ∂. For σ :∆n → X, i ∂ρ(σ)= ∂(εnσ[en,...,e0]) = εn i(−1) σ[en,..., en−i,...,e0], i P i ρ∂(σ)= ρ( i(−1) σ[e0,..., ei,...,en]) = εn−1 ib(−1) σ[en,..., ei,...,e0] P n−i P = εn−1 i(−1) [en,...,b en−i,...,en], b P n and ∂ρ = ρ∂ holds by observing that εn = (−1)bεn−1.

Step 3. ρ and id are chain homotopic, i.e., there exists an operator P : C∗(X) → C∗+1(X) such that ρ − id = ∂P + P ∂. The chain homotopy operator P is a variation of the prism operator. Consider the prism π :∆n ×[0, 1] → n ∆ ; label its vertices vi = ei × {0} and wi = ei × {1}, i = 0,...,n. We then deﬁne i Pσ = i(−1) εn−iσπ[v0, . . . , vi, wn, . . . , wi], n n where [v0, . . . , vi, wn, . . . , wi] refersP to composition with ∆ → ∆ × [0, 1] sending ej to the jth vertex in i the list. (Recall the prism operator maps σ 7→ i(−1) (σ × id)[v0, . . . , vi, wi, . . . , wn].) Omitting σπ from the notation, we compute:P i ∂P = i(−1) εn−i∂[v0, . . . , vi, wn, . . . , wi] P i+j n−j+1 = i≥j(−1) εn−i[v0,..., vj, . . . , vi, wn . . . , wi]+ i≤j(−1) εn−i[v0, . . . , vi, wn,..., wj, . . . , wi], P j P P ∂ = P ( j(−1) [v0,..., vj, . . .b , vn]) b P j i = ij(−1) (−1) εn−i[v0,..., vj, . . . vbi, wn, . . . , wi] P i+j+1 = i>j(−1) εn−i[v0,..., vj, . . . vib, wn, . . . , wi] P i+j + i

Step 4. Recall that if two chain maps f, g : C∗ → D∗ are chain homotopic, then they induce the same map ∗ ∗ ∗ ∗ on cohomology f , g : H (D∗) → H (C∗). In particular, this holds for id, ρ : C∗(X) → C∗(X). Given cocycles φ ∈ Ck(X), ψ ∈ Cl(X), ρ∗(φ ∪ ψ) is cohomologous to φ ∪ ψ and ∗ ρ (φ ∪ ψ)(σ) = (φ ∪ ψ)(ρσ)= εk+lφ(σ[ek+l,...,el])ψ(σ[el,...,e0])

= εk+lεlεkψ(ρ(σ[e0,...,el]))φ(ρ(σ[el,...,ek+l])) = (−1)klρ∗ψ ∪ ρ∗φ(σ). kl kl by noting that εk+l = (−1) εlεk. This shows that [φ] ∪ [ψ] = (−1) [ψ] ∪ [φ]. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 17

5. COMPUTATIONS HW: Hatcher, Section 3.2: 1–8.

5.1. Preliminaries. Recall the cup product on the level of chains: If φ ∈ Ck(X), ψ ∈ Cl(X), and σ ∈ Ck+l(X), then

(φ ∪ ψ)(σ)= φ(σ[e0,...,ek])ψ(σ[ek,...,ek+l]). The cup product descends to cohomology and gives: Hk(X) × Hl(X) → Hk+l(X), Hk(X, A) × Hl(X) → Hk+l(X, A), Hk(X, A) × Hl(X, A) → Hk+l(X, A).

k k The latter two are straightforward when we recall that φ ∈ C (X, A) is φ ∈ C (X) such that φ(Ck(A)) = 0. Lemma 5.1.1. The cup product descends to a map: Hk(X, A) × Hl(X,B) → Hk+l(X, A ∪ B), if A, B are open subsets of X or subcomplexes of CW complexes of X.

k l Proof. Note that it is not clear whether φ ∪ ψ(σ) = 0 for φ ∈ C (X, A), ψ ∈ C (X,B), σ ∈ Ck+l(A ∪ B). Hence we consider the replacement Ck(X, A) × Cl(X,B) → Ck+l(X, A + B), where elements of Ck+l(X, A + B) vanish on sums of chains in A and chains in B. Now recall the inclusion i : Cn(A + B) → Cn(A ∪ B). It was shown to have a chain homotopy inverse in Math 225C. Hence, dualizing, we obtain a quasi-isomorphism i∗ : Cn(A ∪ B) → Cn(A + B). This in turn implies that the map Cn(X, A ∪ B) → Cn(X, A + B) is a quasi-isomorphism: Apply the ﬁve lemma to: Hn(X, A ∪ B) −−−−→ Hn(X) −−−−→ Hn(A ∪ B)

id ≃    n  n n  H (X,y A + B) −−−−→ H y(X) −−−−→ H (Ay+ B). The lemma then follows.

i We also have a natural pairing h, i : H (X, A) × Hi(X, A) → R induced from i h, i : C (X, A) × Ci(X, A) 7→ R, (φ, σ) 7→ hφ, σi = φ(σ).

∗ i (Check this is well-deﬁned!) Given f : X → Y , there are maps f∗ : Hi(X) → Hi(Y ) and f : H (Y ) → Hi(X) satisfying the adjoint condition: ∗ hφ, f∗σi = hf φ, σi. 18 KO HONDA

5.2. Basic calculation. Given i + j = n, Rn = Ri × Rj, Ri = Ri × {0}, Rj = {0}× Rj, Hi(Rn, Rn − Rj) × Hj(Rn, Rn − Ri) −→∪ Hn(Rn, Rn − {0}) takes generator times generator to generator. Note that Rn − {0} = (Rn − Rj) ∪ (Rn − Ri). Intuition: The basic calculation is the Poincarédual of that fact that two planes Rj and Ri of complemen- tary dimension intersect at a point. Consider the projections n n j i i π1 : (R , R − R ) → (R , R − {0}) n n i j j π2 : (R , R − R ) → (R , R − {0}) j i i i that project out the R -and R -directions. Observe that Hi(R , R − {0}) is generated by an affine linear i i .simplex σi : ∆ ֒→ R which passes through {0} in its interior. Let σi be a lift of σi, i.e., π1σi = σi i i i Let φ ∈ H (R , R − {0}) be a cocycle such that hφ, σii = 1; its existence is guaranteed for example ∗ e e by the Universal Coefficient Theorem. Then hπ1φ, σii = 1. Similarly, there exist σj and ψ such that ∗ n n hπ2ψ, σji = 1. Finally, it is not hard to find an n-simplex σn for Hn(R , R − {0}) whose front i face and e ∗ ∗ back j face agree with some σi and σj. This implies that hπ1φ ∪ π2ψ, σni = 1. e e ∗ n 5.3. Calculation of H (CP e; Z). e e Theorem 5.3.1. H∗(CPn; Z)= Z[α]/(αn+1), and the degree of α is 2. Proof. We suppress Z-coefficients and write Pn for CPn. Recall that Pn has a single i-cell for i = 0, 2, 4,..., 2n. Hence Hi(Pn)= Z for i = 0, 2,..., 2n and 0 otherwise. We argue by induction on n. For n = 1, H∗(P1) = Z[α]/(α2) = 0. Suppose H∗(Pn−1) = Z[α]/(αn). We consider the inclusion i : Pn−1 → Pn and the induced map i∗ : H∗(Pn) → H∗(Pn−1), which is an algebra homomorphism and is an isomorphism for Hi with i ≤ 2n−2. Hence if α is a generator for H2(Pn) then αn−1 generates H2n−2(Pn). It remains to show that αn generates H2n(Pn). Let Pi and Pj, i + j = n, be projective planes in Pn that intersect transversely (i.e., in a point). The fact that they intersect in a point will be translated into αi ∪ αj generating H2n(Pn). We have the diagram ∪ Hi(Pn) × Hj(Pn) −−−−→ Hn(Pn)

a b x x  ∪  Hi(Pn, Pn − Pj) × Hj(Pn, Pn − Pi) −−−−→ Hn(Pn, Pn − {pt})

c d    ∪  Hi(Cn, Cn − Cj) ×y Hj(Cn, Cn − Ci) −−−−→ Hn(Cn, Cyn − {0}). This commutes by the naturality of the cup product. It remains to show that the maps a, b, c, d are isomorphisms. b is an isomorphism by the relative exact sequence for the pair (Pn, Pn − {pt}). d is an isomorphism by excision (excise Z = Pn−1). For one of the components of a, we consider the relative sequence for (Pn, Pn − Pj), observing that Hi(Pn − Pj) = 0 since Pn − Pj deformation retracts onto Pi−1 (HW!). c is similar. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 19

6. EILENBERG-ZILBERTHEOREM This presentation follows Greenberg-Harper, “Algebraic Topology”. ′ 6.1. Statement of theorem. Given chain complexes (C∗, ∂) and (D∗, ∂ ), their tensor product can also be viewed as a chain complex where (C∗ ⊗ D∗)n := ⊕i(Ci ⊗ Dn−i) and the boundary map is (∂ ⊗ ∂′)(x ⊗ y)= ∂x ⊗ y + (−1)ix ⊗ ∂′y, where x ∈ Ci and y ∈ Dn−i. Question: Do you need the (−1)i?

Goal: Given topological spaces X and Y , relate C∗(X) ⊗ C∗(Y ) and C∗(X × Y ). There is a map going one way, called the Eilenberg-Zilber map:

A : Cn(X × Y ) 7→ (C∗(X) ⊗ C∗(Y ))n, n σ = (πX σ, πY σ) 7→ πX σ[e0,...,ei] ⊗ πY σ[ei,...,en], Xi=0 where πX : X × Y → X and πY : X × Y → Y are projections. HW: Verify that A is a chain map. We now view this more categorically: Let Top × Top be the product of Top with itself, i.e., the objects (f,g) are pairs (X,Y ) of spaces and the morphisms are pairs of maps (X,Y ) −→ (X′,Y ′). Then there exist two functors F, F ′ : Top × Top → Ch, ′ F (X,Y )= C∗(X × Y ), F (X,Y )= C∗(X) ⊗ C∗(Y ). HW: Check these are actually functors. Lemma 6.1.1. A : F ⇒ F ′ is a natural transformation.

(f,g) Proof. We unwind deﬁnitions: Given (X,Y ) −→ (X′,Y ′), we verify the commutativity of the diagram

F (f,g) ′ ′ Cn(X × Y ) −−−−→ Cn(X × Y )

A A

 ′   F (f,g) ′  ′ (C∗(X) ⊗yC∗(Y ))n −−−−→ (C∗(X ) ⊗yC∗(Y ))n. One way maps A σ = (πX σ, πY σ) 7→ πX σ[e0,...,ei] ⊗ πY σ[ei,...,en] Xi F ′(f,g) 7→ fπXσ[e0,...,ei] ⊗ gπY σ[ei,...,en]. Xi The other way gives the same result. Theorem 6.1.2 (Eilenberg-Zilber). There exists a natural transformation B : F ′ ⇒ F such that AB : F ′ ⇒ F ′ and BA : F ⇒ F are chain homotopic. 20 KO HONDA

What do we mean by “chain homotopic” natural transformations? Given functors F, F ′ : C → Ch, two natural transformations A, B : G ⇒ G′ are chain homotopic if there exist H(X) : F (X) 7→ F ′(X), where ′ Fi(X) is mapped to Fi+1(X) such that A(X) − B(X) = ∂H(X) − H(X)∂, and such that the following diagram commutes: Ff F (X) −−−−→ F (Y )

H(X) H(Y )    F ′f  F ′y(X) −−−−→ F ′(yY ).

In particular, A : C∗(X ⊗ Y ) → (C∗(X) ⊗ C∗(Y )) is a chain homotopy equivalence.

Remark 6.1.3. The homotopy inverse B : C∗(X) ⊗ C∗(Y ) → C∗(X × Y ) can actually be given explicitly using “shuffle homomorphisms”. Corollary 6.1.4 (Künneth formula). If R is a principal ideal domain (PID), then n n Hn(X × Y ) ≃ (⊕i=0Hi(X) ⊗ Hn−i(Y )) ⊕ (⊕i=0 Tor1(Hi(X),Hn−i−1(Y ))). The proof is very similar to the proof of the Universal Coefficient Theorem and can be found on pp. 253-256 of Greenberg-Harper.

6.2. Acyclic models. A category with models (A, M) is a category A with a set of objects M, called the set of models. If F : A → Ch is a functor, then let Fi : A → R − Mod be the functor which sends X to the degree i part (F (X))i of F (X). (For today assume that Ch = Ch≥0, i.e., chain complexes such that Ci = 0 for i< 0.)

Deﬁnition 6.2.1. A basis for Fi is a collection {dN ∈ Fi(N) | N ∈ Ni}, where Ni ⊂ M, such that, for any X ∈ ob(X), Fi(X) is the free R-module generated by

{Fi(u)(dN ) | N ∈Ni, u ∈ Hom(N, X)}.

We say F is free if all the Fi have bases for i ≥ 0. Examples. The functors F, F ′ : Top × Top → Ch ′ F (X,Y )= C∗(X × Y ), F (X,Y )= C∗(X) ⊗ C∗(Y ). from the previous subsection have models M = {(∆i, ∆j), i, j ≥ 0} and are free. n n n n n n 1. For Fn, choose Nn = {(∆ , ∆ )} and di ∈ Fn(∆ , ∆ ) = Cn(∆ × ∆ ) given by the diagonal map n n n n ∆ → ∆ × ∆ . Given Fn(X,Y ) = Cn(X × Y ), consider σ : ∆ → X × Y . It can be written as a composition

d u=(πX σ,πY σ) ∆n −→n ∆n × ∆n −→ X × Y, so that σ = Fn(πX σ, πY σ)(dn). ′ i j ′ i j i j 2. For Fn, choose Nn = {(∆ , ∆ ) | i + j = n} and δi ⊗ δj ∈ Fn(∆ , ∆ ) = (C∗(∆ ) ⊗ C∗(∆ ))n, such i i j j ′ that δi =id:∆ → ∆ and δj =id:∆ → ∆ . Given σ ⊗ τ ∈ Fn(X,Y ) = (C∗(X) ⊗ C∗(Y ))n where i j ′ σ :∆ → X and τ :∆ → Y , σ ⊗ τ = Fn(σ × τ)(δi ⊗ δj). NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 21

ε An augmented chain complex is a chain complex with Ci = 0 for i < −1, C−1 = R, and C0 → C−1 surjective. Recall that given C∗(X) we can extend it to an augmented chain complex ··· → C1(X) → ε C0(X) → R → 0 by setting ε( i aiσi)= i ai. Its homology is the reduced homology of X. ′ ′ Let Ch ⊂ Ch≥−1 be the (full)P subcategoryP of augmented chain complexes. We can view F and F as ′ functors to Ch instead by augmenting C∗(X × Y ) and C∗(X) ⊗ C∗(Y ).

′ Deﬁnition 6.2.2. A functor F : A→ Ch is acyclic if for every H∗(F (M)) = 0 for every M ∈ M.

Examples. F and F ′ are both acyclic. F is clear, since ∆i × ∆j is contractible and hence its reduced i j ′ homology H∗(∆ × ∆ ) = 0. The case of F is HW.

Theorem 6.2.3.e Let (A, M) be a category with models and F, F ′ : A → Ch′ be functors such that F is free and F ′ is acyclic. Then there is a natural transformation Φ : F ⇒ F ′ which is unique up to chain homotopy.

Corollary 6.2.4. If F and F ′ are both free and acyclic, then F , F ′ are chain homotopy equivalent.

Proof of Theorem 6.2.3. We will prove the existence of the natural transformation Φ. The definition of the chain homotopy H is similar. ′ For each X ∈ ob(A) we want to define Φ(X) : F (X) → F (X). Let {dN ∈ F (N)}N∈Ni be a basis for Fi. Then Fi(X) is freely generated by Fi(u)(dN ) as dN ranges in Ni and u ranged in Hom(N, X). ′ Step 1. Suppose we have defined Φi(N)(dN ) for all N ∈Ni. Then Φi(X) : Fi(X) → Fi (X) is determined by the following commutative diagram:

Φi(N) ′ Fi(N) −−−−→ Fi (N)

′ Fi(u) Fi (u)    Φi(X) ′  Fi(yX) −−−−→ Fi (yX),

′ i.e., Φi(X)(Fi(u)(dN )) = Fi (u)(Φi(N)(dN )). ′ We need to verify the naturality of Φi : Fi ⇒ Fi , i.e., for f : X → Y with Φi(X), Φi(Y ) that we just deﬁned, the diagram is commutative:

Φi(X) ′ Fi(X) −−−−→ Fi (X)

′ Fi(f) Fi (f)    Φi(Y ) ′  Fiy(Y ) −−−−→ Fi (yY ).

This is HW. Note that this step only uses the freeness of F . ′ Step 2. By induction on i we deﬁne Φi(X) for all X ∈ ob(A) so that Φi : Fi ⇒ Fi is a natural transforma- ′ tion. Suppose for all j < i we have deﬁned (the unique) Φj(X) for all X ∈ ob(A) so that Φj : Fj ⇒ Fj is a natural transformation. 22 KO HONDA

We deﬁne Φi(N)(dN ), N ∈ Ni, to be any element such that ∂Φi(N)(dN )=Φi−1(N)∂dN ; such an ′ element exists by the acyclicity of F and the fact that Φi−1(N)∂dN is closed. ∂ Fi(N) −−−−→ Fi−1(N)

Φi(N) Φi−1(N)   ′ ∂ ′  Fi y(N) −−−−→ Fi−1y(N).

Then by Step 1 we can deﬁne Φi(X) for all X ∈ ob(A). ′ When i = 0, we similarly deﬁne Φ0(N)(dN ) to be any element such that ε Φ0(N)(dN )= εdN : ε F0(N) −−−−→ R

Φ0(N) ≃

 ′  ′ ε  F0y(N) −−−−→ R,y and extend to Φ0(X) using Step 1.

Step 3. To combine the natural transformations Φi into Φ, we need to verify that ∂Φi(X)=Φi−1(X)∂: ′ ′ ′ ∂Φi(X)(Fi(u)(dN )) = ∂(Fi (u)Φi(N)dN )= Fi−1(u)∂(Φi(N)dN )= Fi−1(u)Φi−1(N)∂dN =Φi−1(X)Fi−1(u)∂dN =Φi−1(X)∂(Fi(u)dN ). NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 23

7. ORIENTATIONS Today our topological spaces are topological manifolds M of dimension n, i.e., Hausdorff topological spaces covered by open sets homeomorphic to Rn.

7.1. Deﬁnitions. Basic Calculation. Given x ∈ M,

n n n n−1 Hn(M, M − x) ≃ Hn(R , R − x) ≃ Hn−1(R − x) ≃ Hn−1(S ) ≃ Z.

We write Hn(M|A) := Hn(M, M − A). Deﬁnition 7.1.1.

(1) An orientation at x is a choice of generator of Hn(M|x). (2) An orientation of M is a function x 7→ µx ∈ Hn(M|x) with local consistency: for each x ∈ M there exists an open ball B ∋ x and µB ∈ Hn(M|B) such that µy = φB,yµB for all y ∈ B under the natural map φB,y : Hn(M|B) → Hn(M|y). (3) If an orientation exists for M, then M is orientable.

Recall that if M is smooth, then M is orientable if the “determinant line bundle” ΛnT ∗M is trivial (take this as the deﬁnition for M smooth), which is equivalent to the existence of a nonvanishing section (aka volume form) M → ΛnT ∗M. This is equivalent to a smooth (with respect to M) choice of orientation of ∗ Tx M. × Orientation double cover. Denote the units of Hn(M|x) by Hn(M|x) . Then

× M = Hn(M|x) xa∈M f × and there is a map π : M → M which sends Hn(M|x) → x. We topologize M by choosing a basis {U } so that π : M → M is a covering space: Given an open ball B ∋ x and a generator µ of B,µB f f B H (M|B), let U := {φ (µ ) | y ∈ B}. n B,µB f B,y B Claim/HW: M is orientable.

Lemma 7.1.2.f If M is connected, then M is orientable if and only if M has 2 connected components.

Proof. If M has 2 components, then eachf sheet is homeomorphic to Mf, and hence M is orientable. If M is orientable, then it has 2 orientations and each orientation defines a component. f Analogously, we can define R-orientations for R a commutative ring with identity in a similar manner. Note that Hn(M|x; R) ≃ Hn(M|x) ⊗ R ≃ R. Then: × • an R-orientation at x is a choice of unit u ∈ Hn(M|x; R) , × • there is a covering space MR× → M with fibers Hn(M|x; R) , and • an R-orientation of M is a section of MR× → M. 24 KO HONDA

7.2. Fundamental class. Theorem 7.2.1. Let M be a closed (= compact without boundary), connected n-manifold.

(A) If M is R-orientable, then Hn(M; R) → Hn(M|x; R) is an isomorphism for all x ∈ M. (B) If M is not R-orientable, then Hn(M; R) → Hn(M|x; R) ≃ R is an injection with image {r ∈ R | 2r = 0}. (C) Hi(M; R) = 0 for i>n.

A generator of Hn(M; R) ≃ R is called a fundamental class for M with respect to R. Proof. We’ll prove (A) and (C), leaving (B) for HW. If M is R-orientable, then there is a section σ of MR× → M. We inductively prove that for all A ⊂ M compact,

(i) there exists a unique class σA ∈ Hn(M|A, R) such that φA,x(σA)= σ(x), and (ii) Hi(M|A; R) = 0 for i>n. Suppressing R, we show the following: (1) If (i) and (ii) hold for compact sets A, B, A ∩ B, then they hold for A ∪ B. (2) Write A as a union of A1,...,An, where each Ai is contained in an open ball. Since Hn(M|Ai) ≃ n n Hn(R |Ai) by excision, we reduce to the case M ≃ R . (3) If A is a closed ball in M = Rn, (i) and (ii) are immediate. (4) Argue for an arbitrary compact set A ⊂ Rn. (1) follows from the Mayer-Vietoris sequence Φ Ψ 0 −−−−→ Hn(M|A ∪ B) −−−−→ Hn(M|A) ⊕ Hn(M|B) −−−−→ Hn(M|A ∪ B), where Φ(α) = (α, −α) and Ψ(α, β)= α + β. In particular Ψ(σA, −σB)= σA − σB = 0, and comes from some σA∪B under the map Φ. (Check consistency + uniqueness!) n n (4) If A is an arbitrary compact set in R , suppose α ∈ Hi(R |A) is represented by a relative cycle z. Then Supp(∂z) is a compact subset of Rn − A, and hence has positive ﬁnite distance to A. We can then n cover A with closed balls B1,...,Bk disjoint from Supp(∂z) and view it as an element of Hi(R |∪i Bi) n n which maps to α under Hi(R |∪i Bi) → Hi(R |A). If i>n, then (3) implies that α = 0. If i = n, then existence in (i) is clear. (Check uniqueness!) NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 25

8. POINCAREDUALITY´ HW: Hatcher, Section 3.3: 1,2,5,6,16,22,26. 8.1. Cap product. The cap product is deﬁned as follows: l ∩ : Ck(X) × C (X) → Ck−l(X)

(σ, φ) 7→ φ(σ[e0,...,el])σ[el,...,ek]. Lemma 8.1.1. ∂(σ ∩ φ) = (−1)l(∂σ ∩ φ − σ ∩ δφ). This is left for HW; it’s the same kind of veriﬁcation as for the cup product. The lemma implies that ∩ descends to: l ∩ : Hk(X) × H (X) → Hk−l(X). Naturality. Given a map f : X → Y , consider the diagram

l ∩ Hk(X) × H (X) Hk−l(X)

∗ f∗ f f∗

l ∩ Hk(Y ) × H (Y ) Hk−l(Y ).

∗ Lemma 8.1.2. f∗σ ∩ φ = f∗(σ ∩ f φ). ∗ Proof. Follows from φ(fσ[e0,...,el])fσ[el,...,ek]= f φ(σ[e0,...,el])fσ[el,...,ek]. 8.2. Statement of Poincare´ duality. Theorem 8.2.1. Let M be a closed R-orientable n-manifold and [M] an R-fundamental class. Then k [M]∩ : H (M; R) → Hn−k(M; R) is an isomorphism. The most important thing to know about Poincaréduality is that it is really a local result. i Theorem 8.2.2. There exists a cohomology theory Hc(X; R) called compactly supported cohomology and a duality map k ∼ DM : Hc (M; R) −→ Hn−k(M; R). In particular, we may take M = Rn. i 8.3. Compactly supported cohomology. Suppress the coefficient ring R. The chain complex Cc(X) consists of φ ∈ Ci(X) such that there exists a compact set K ⊂ X for which φ(σ) = 0 for all σ with support i+1 ∗ on X − K. Since δφ also has the same property, δφ ∈ Cc (X) and (Cc (X), δ) is a cochain complex. Its i cohomology is Hc(X), called compactly supported cohomology. Comparison with de Rham theory. Recall from Math 225B the de Rham complex: ···→ Ωi(M) −→d Ωi+1(M) → .... In this setting we defined the compactly supported i-forms by i i Ωc(M)= {ω ∈ Ω (M) | Supp(ω) is compact}, 26 KO HONDA

i and its cohomology is Hc,dR(M). Omitting “dR”, the version of Poincar´eduality for de Rham theory is that i n−i n ∼ Hc(M) × H (M) → Hc (M) −→ R

(ω, η) 7→ ω ∧ η 7→ M ω ∧ η i R n−i is a nondegenerate pairing, i.e., for each ω ∈ Hc(M) there exists η ∈ H (M) such that ω ∧ η 6= 0. R Compare this to the situation in singular theory (ignoring signs):

i n−i ([M]∩,id n−i h,i Hc(M) × H (M) −→ Hn−i(M) × H (M) −→ R, (φ, ψ) 7→ ([M] ∩ φ, ψ) 7→ h[M] ∩ φ, ψi = h[M], φ ∪ ψi. Hence [M]∩ corresponds to integration in de Rham theory. i We will now explain a more algebraic definition of Hc(X) in terms of direct limits. The initial observation i i is the following: Each φ ∈ Cc(X) belongs to some C (X|K) for K ⊂ X compact. Also if K ⊂ L there is an inclusion of chain complexes i i φK,L : C (X|K) → C (X|L) and a corresponding map on cohomology i i φK,L : H (X|K) → H (X|L). Definition 8.3.1. A directed system is a partially ordered set (I, ≤) such that given α, β ∈ I there exists γ such that α ≤ γ and β ≤ γ. In our case the directed system I is the set of compact subsets of X and ≤ is given by inclusion. A directed system (I, ≤) can be viewed as a category I such that ob(I) = I and Hom(α, β) is one element if α ≤ β and empty otherwise. Definition 8.3.2. Given a functor F : I→C (so this means we have assignments α 7→ F (α) and morphisms Fαβ : F (α) → F (β)), its direct limit or colimit is an object C in C together with morphisms {φα : F (α) → C} such that if α ≤ β then φα = φβFαβ, and it satisfies the universal property: If A is any other object in C together with morphisms ψα : F (α) → A satisfying ψα = ψβFαβ, then there is a unique morphism f : C → A such that ψα = fφα.

Slightly confusing notation: lim→ F (α) or colim→ F (α). Lemma 8.3.3. Colimits exist in Ab or R − Mod.

Proof. This is an explicit construction: Take ⊔αF (α)/ ∼, where xα ∈ F (α) is identiﬁed with Fαβ(xα) ∈ F (β). Check the universal property!

HW: Show that taking direct limits commutes with taking homology in R − Mod. In particular, the direct limit of exact sequences is exact. i i Claim 8.3.4. lim→ H (X|L)= Hc(X). i Proof. This is a straightforward unwinding of the deﬁnitions. An element of lim→ H (X|L) is represented i i by a cocycle φ ∈ C (X|K) for some K and [φ] = 0 ∈ lim→ H (X|L) if and only if φ is a coboundary in some Ci(X|L) for K ⊂ L. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 27

i 8.4. Definition of DM . We now define DM : Hc(M) → Hn−i(M) for an R-oriented M. For each compact set K ⊂ M, we have the “fundamental class” µK and we define

i µK∩ : H (M|K) → Hn−i(M).

∗ When K ⊂ L compact, we want to show that µK∩ = µL ∩ ◦φK,L, i.e., the following diagram is commutative:

i µK ∩ H (M|K) Hn−i(M) ∗ µL∩ φK,L Hi(M|L)

∗ This follows from µK ∩ φ = φL,K(µL) ∩ φ = µL ∩ φL,K(φ). By the universal property of direct limits, we have a unique map i DM : Hc(M) → Hn−i(M).

8.5. Proof of Poincare´ duality. The proof is given in several steps. n n Step 1. The base case M = R . Let Bj ⊂ R be a closed ball of radius j about the origin. Then i n i n i n Hc(R ) = limj→∞ H (R |Bj). Using excision etc., we ﬁnd that H (R |Bj) ≃ R if i = n and 0 otherwise; i n ∼ i n i n also H (R |Bj) −→ H (R |Bj+1). Hence Hc(R ) ≃ R if i = n and 0 otherwise. Since Hn−i(M) ≃ R if i = n and 0 otherwise, it sufﬁces to check

n n ∼ n µBj ∩ : H (R |Bj) −→ H0(R ).

This is immediate from taking µBj to be an “embedded” n-simplex which contains Bj and taking the n n generator of H (R |Bj) to evaluate to 1 on µBj . Step 2. Assume the following diagram is sign-commutative: For U, V open and M = U ∪ V ,

k k k k k+1 Hc (U ∩ V ) Hc (U) ⊕ Hc (V ) Hc (M) Hc (U ∩ V )

(8.5.1) DU∩V DU ⊕−DV DM DU∩V

Hn−k(U ∩ V ) Hn−k(U) ⊕ Hn−k(V ) Hn−k(M) Hn−k−1(U ∩ V ).

By the ﬁve lemma, if DU , DV , DU∩V are isomorphisms, then so is DM .

Step 3. Suppose there is a ﬁltration U1 ⊂ U2 ⊂ . . . of open sets such that ∪iUi = M and DUi is an isomorphism, then so is DM : Observe that

k k k Hc (Ui) = lim H (Ui|K) ≃ lim H (M|K) K⊂Ui K⊂Ui and hence k k Hc (M) = lim Hc (Ui). i→∞ 28 KO HONDA

Similarly we can prove that Hn−k(M) = limi→∞ Hn−k(Ui) (HW). It is also easy to show (HW) the commutativity of: D k Ui Hc (Ui) Hn−k(Ui)

D k Ui+1 Hc (Ui+1) Hn−k(Ui+1).

Hence DUi isomorphism for each i implies DM isomorphism. n Step 4. For M an open subset of R , cover M by countably many open balls Vi; their ﬁnite intersections are n convex open balls ≃ R . Then by repeated application of Step 2, for Ui := ∪j≤iVj, DUi is an isomorphism. Next Step 3 applied to U1,U2,... implies that DM is an isomorphism. For second countable M, cover n M by countably many open balls Vi; their ﬁnite intersections are open subsets of R for which D∗ is an isomorphism. Hence the same exhaustion procedure shows that DM is an isomorphism for M second countable. In general, need to use Zorn’s lemma.... 8.6. Proof of (8.5.1). We need to verify three things: (a) The exactness of the top row. (b) The commutativity of the left two squares. (c) The commutativity of the right square. (a) follows from the exactness of

Hk(M|K ∩ L) Hk(M|K)

µK∩L∩ µK ∩

Hn−k(U ∩ V ) Hn−k(U). ∗ Since φK,K∩LµK = µK∩L, it sufﬁces to show that iU∩V,U (φK,K∩LµK ∩ φ) = µK ∩ φK,K∩L(φ), where k φ ∈ H (M|K ∩ L). Take a representative a of µK with support in U and a representative α of φ with support on U ∩ V . Then both sides give ha, αi. The three other squares are analogous. (c) is the hardest part:

Hk(M|K ∪ L) r Hk+1(M|K ∩ L) ∼ Hk+1(U ∩ V |K ∩ L)

µK∪L∩ µK∩L∩ s Hn−k(M) Hn−k−1(U ∩ V ) Write A = M − K and B = M − L. We ﬁrst unwind the deﬁnitions of the connecting homomorphisms r, s. For r, consider the short exact sequence: 0 C∗(M, A + B) C∗(M, A) ⊕ C∗(M,B) C∗(M, A ∩ B) 0. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 29 where C∗(M, A + B) means cochains vanishing on chains in A and on chains in B. Starting with φ ∈ i i i C (M, A ∩ B) with δφ = 0, we pick (φA, −φB) ∈ C (M, A) ⊕ C (M,B) so that φ = φA − φB. Then r(φ)= δφA = δφB . Similarly, starting with z ∈ Ci(M) with ∂z = 0, we pick (zU , −zV ) ∈ Ci(U)⊕Ci(V ) such that z = zU − zV . Then s(z)= zU = zV . We also decompose µK∪L = αU−L +αU∩V +αV −K (why can we do this?) with supports on U −L, U ∩ V,V − K, respectively. Draw a picture of this! Note that αU−L + αU∩V represents µK. We then compute

s(µK∪L ∩ φ)= s(αU−L ∩ φ + (αU∩V + αV −K) ∩ φ)

= ∂(αU−L ∩ φ)

= ±∂αU−L ∩ φ = ±∂αU−L ∩ (φA − φB)

= ±∂αU−L ∩ φA

= ±∂αU∩V ∩ φA, where the ﬁrst line to the second follows from observing that αU−L is supported in U and αU∩V + αV −K is supported in V ; the third line to the fourth follows since φB is zero on U − L; the the fourth to ﬁfth follows since φA is zero on V − K and ∂(αU−L + αU∩V ) vanishes on M − K. We also compute

µK∩L ∩ r(φ)= µK∩L ∩ δφA = αU∩V ∩ δφA ∼±∂αU∩V ∩ φB, where ∼ means “cohomologous to”. 8.7. Lefschetz duality. Let M be a compact R-oriented n-manifold with boundary ∂M. Theorem 8.7.1 (Lefschetz duality). The following diagram commutes:

Hk(M,∂M) Hk(M) Hk(∂M) Hk+1(M,∂M)

[M]∩ [M]∩ [∂M]∩ [M]∩

Hn−k(M) Hn−k(M,∂M) Hn−k−1(∂M) Hn−k−1(M), and each of the vertical arrows is an isomorphism. For a slightly more general statement, see Hatcher, Theorem 3.43. We’ll explain the terms in the diagram and indicate the proof: (1) Fact: There is a collared neighborhood of ∂M ⊂ M of the form (−ǫ, 0] × ∂M where ∂M = {0}× ∂M. (2) We deﬁne the fundamental class [M] ∈ Hn(M,∂M) as:

µAǫ ∈ Hn(M − ∂M|Aǫ) ≃ Hn(M,∂M), where Aǫ = M − ((−ǫ, 0] × ∂M). k k k (3) Hc (M − ∂M) = limǫ→0 H (M − ∂M|Aǫ) ≃ H (M,∂M) since the direct limit stabilizes. Hence k k [M]∩ : H (M,∂M) → Hn−k(M) is really µAǫ ∩ : H (M − ∂M|Aǫ) → Hn−k(M). (4) The first, third, and fourth vertical arrows are isomorphisms by Poincaréduality; hence the second one is also by the five lemma. (5) The second and third squares are consequences of ∂([M] ∩ φ)= ±(∂[M] ∩ φ − [M] ∩ δφ). The main thing to check is that ∂[M] = [∂M], which is HW. 30 KO HONDA

8.8. Alexander duality. Refer to Greenberg-Harper, Section 2.7 since we omit the proof. Theorem 8.8.1 (Alexander duality). Let M be a compact R-oriented n-manifold and A ⊂ M a closed subset. Then there is an isomorphism k ∼ DA : lim H (U) −→ Hn−k(M, M − A). U⊃A k The direct limit limU⊃A H (U) is with respect to the directed system which consists of open sets U ⊃ A directed by reverse inclusion: V ⊂ U implies U ≤ V . NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 31

9. HOMOTOPY COEXACTNESS 9.1. Basic constructions. We list some basic constructions. Let X,Y be topological spaces. 1. The cone CX is X × [0, 1]/(x, 1) ∼ (x′, 1) for all x,x′ ∈ X. 2. The suspension SX is X × [−1, 1]/(x, 1) ∼ (x′, 1), (x, −1) ∼ (x′, −1) for all x,x′ ∈ X. For example, SSn = Sn+1. C and S induce functors Top → Top (these are called “endofunctors”). In the case of S, X 7→ SX and f : X → Y is mapped to Sf : SX → SY . 3. The join X ∗ Y is X × Y × [0, 1]/ ∼, (x,y, 0) ∼ (x,y′, 0), (x,y, 1) ∼ (x′, y, 1).

The following are operations in the category Top• of pointed topological spaces. Let (X,x0) and (Y,y0) be pointed topological spaces. red ′ ′ 4. The reduced cone C X = CX/(x0,t) ∼ (x0,t ) for all t,t ∈ [0, 1]. The basepoint is the equivalence class of (x0, 0). ′ ′ 5. The reduced suspension ΣX = SX/(x0,t) ∼ (x0,t ) for all t,t ∈ [−1, 1]. The basepoint is the n n+1 equivalence class of (x0, 0). We also have ΣS = S . red Similarly, C and Σ induce endofunctors Top• → Top•.

6. The smash product of (X,x0) and (Y,y0) is X ∧ Y = X × Y/(X × {y0}) ∪ ({x0}× Y ). The basepoint is the equivalence class of (x0,y0) This is the replacement for X × Y in the pointed category. 9.2. Mapping cones and mapping cylinders. For more details see Spanier, Chapter 7, Section 1. Given f : X → Y , we can form the mapping cylinder Mf = Y ⊔ (X × [0, 1])/(x, 1) ∼ f(x) and the ′ mapping cone Cf = Mf /(x, 0) ∼ (x , 0). In Top• we can analogously form the reduced mapping cylinder and the reduced mapping cone by collapsing {x0}× [0, 1]. Remark 9.2.1. Unfortunately, people usually use the same notation for both the reduced and nonreduced objects; we need to be careful about which category we’re in.

Recall that [X,Y ] is the homotopy class of maps X → Y . In [(X,x0), (Y,y0)] there is a distinguished homotopy class, denoted 0, i.e., the homotopy class of maps that are nullhomotopic to the constant map to y0. For the moment let us work in Top• or hTop•. f g Deﬁnition 9.2.2. The sequence X −→ Y −→ Z is coexact if for all W the sequence

g∗ f ∗ [Z,W ] [Y,W ] [X,W ] is an exact sequence of “pointed sets”, i.e., Im(g∗) = (f ∗)−1(0).

f i Theorem 9.2.3. Given f : X → Y , the sequence X −→ Y −→ Cf is coexact. Here i : Y → Cf is the obvious inclusion. 32 KO HONDA

∗ i∗ f Proof. Let us ﬁrst unwind deﬁnitions: Consider [Cf ,W ] −→ [Y,W ] −→ [X,W ]. We want to show that Im i∗ = (f ∗)−1(0). Note that h : Cf → W is equivalent to the following: • h ◦ i : Y → W , and • a nullhomotopy from h ◦ i : X → W to the constant map which maps to the basepoint w0 ∈ W . This interpretation immediately implies that Im i∗ = (f ∗)−1(0). Corollary 9.2.4. Given f : X → Y , there exists a long coexact sequence: f i j f 1 i1 (9.2.1) X −→ Y −→ Cf −→ Ci −→ Cj −→ Cf 1 . . .

HW: There is also a relative version: Given f = (f0,f1) : (X, A) → (Y,B), its mapping cone is (Cf0 ,Cf1 ) and we have the coexact sequence f i (X, A) −→ (Y,B) −→ (Cf0 ,Cf1 ). 9.3. Puppe sequence. Theorem 9.3.1 (Puppe sequence). The long coexact sequence (9.2.1) can be written as: f i j Σf Σi (9.3.1) X −→ Y −→ Cf −→ ΣX −→ ΣY −→ ΣCf −→ .... 1 Proof. Starting with Equation (9.2.1), we show that (i) Ci ≃ ΣX, (ii) Cj ≃ ΣY , and (iii) f ≃ Σf. For Hatcher’s picture proof see p.397. (i) Note that Ci = Cf ∪Y CY , where we are identifying the “bases” of the two cones, and hence Ci is the union of CX and CY glued with f : X → Y . Let r : Ci → Ci/CY = ΣX be the quotient map. Deﬁne s :ΣX → Ci, where ΣX = X × [−1, 1]/ ∼ and s maps X × (0, 1]/ ∼→ CX by inclusion (x,t) 7→ (x,t) and X × [−1, 0] → Y × [−1, 0] by (x,t) 7→ (f(x),t). HW: Write down precise formulas that show r and s are homotopy inverses. (ii) Now Cj = Ci ∪ C(Cf ) glued along Cf . It is not hard to see that C(Cf ) deformation retracts to CY and we are left with CY ∪ CY glued along Y , which is ΣY . Let t : Cj → ΣY be the corresponding map. (iii) It is not hard to see that tf 1s :ΣX → ΣY is equal to Σf.

9.4. Homological algebra version of Puppe sequence. Consider a chain map φ : (C∗, ∂C ) → (D∗, ∂D). Just as the mapping cone Cf enabled us to put any map f : X → Y into a “coexact” sequence, there is a way to put φ∗ into a long exact sequence in homology. We deﬁne the chain complex (Cone(φ), d), called the mapping cone of f, as follows:

Cone(φ)k = Ck−1 ⊕ Dk and d : Ck−1 ⊕ Dk → Ck−2 ⊕ Dk−1 maps d(x,y) = (∂C x, φ(x) − ∂Dy). We check that 2 2 2 d (x,y) = (∂C x, φ∂C (x) − ∂Dφ(x)+ ∂Dy) = 0. The mapping cone Cone(φ) ﬁts into the following short exact sequence of chain complexes:

0 → D∗ → Cone(φ) → C∗−1 → 0. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 33

HW: (1) Show that its long exact sequence has the form

φ∗ Hi(D∗) → Hi(Cone(φ)) → Hi−1(C∗) → Hi−1(D∗), where the connecting homomorphism is φ∗. (2) If f∗ : C∗(X) → D∗ is the induced map from f : X → Y , then Cone(f∗) is quasi-isomorphic to C∗(Cf ). 34 KO HONDA

10. HIGHER HOMOTOPY GROUPS

Today we introduce the higher homotopy groups πn(X,x0) of a pointed topological spaces (X,x0) and describe their basic properties. 10.1. Deﬁnitions. Let In = [0, 1] ×···× [0, 1] be the n-dimensional unit cube. Denote its boundary ∂In as the set of points in In where at least one coordinate is 0 or 1. As a set, let n n πn(X,x0) = [(I ,∂I ), (X,x0)]. 0 0 When n = 0, we take I to be a point and ∂I = ∅. Hence π0(X,x0) is the set of path components of X. n n n n Given f, g : (I ,∂I ) → (X,x0), deﬁne fg : (I ,∂I ) → (X,x0) via

f(2s1,s2,...,sn), 0 ≤ s1 ≤ 1/2 (s1,s2,...,sn) 7→ g(2s1 − 1,s2 ...,sn), 1/2 ≤ s1 ≤ 1

The product descends to a product on πn(X,x0) — the proof is identical to the case of π1(X,x0).

Remark: There is an apparent asymmetry in the definition, as the first coordinate s1 is preferred. For HW show the independence of fg on the choice of coordinate; this is similar to the proof below of the fact that πn(X) is abelian for n ≥ 2. n The nth homotopy group πn(X,x0) is equivalent to [(S , ∗), (X,x0)] (verify this for HW), by recalling that Sn is the quotient of Dn, where ∂Dn is identified to a point. 10.2. Properties of the homotopy groups.

Lemma 10.2.1. πn is a functor Top• → Grp for n ≥ 1 and Ab for n ≥ 2.

Proof. Given φ : (X,x0) → (Y,y0), the induced homomorphism φ∗ : πn(X,x0) → πn(Y,y0) is given by f 7→ φ ◦ f. It is immediate that:

(1) id∗ : πn(X,x0) → πn(X,x0) is the identity map. (2) (φ ◦ ψ)∗ = φ∗ ◦ ψ∗. n n We give a pictorial proof that fg ≃ gf for n ≥ 2. For HW, make it rigorous. Let rC : I ֒→ I be an ′ ′ embedding with image a small square C in the interior. First homotop f to f so that f (x)= x0 for x 6∈ C ′ −1 ′ ′ ′ ′ ′ and f (x)= frC (x) for s ∈ C. Similarly deﬁne g . Then we homotop f g to g f by precomposing with n an isotopy of I that switches the orders of the squares C1 and C2 for f and g, respectively.

HW: Prove that πn(X × Y, (x0,y0)) ≃ πn(X,x0) × πn(Y,y0).

HW: Prove that πn(X,x0) = 0 for n ≥ 1, if X is contractible, i.e., X has the homotopy type of a point.

HW: Let π : (X,˜ x˜0) → (X,x0) be a path-connected, locally path-connected covering space. Then πn(X,x0)= πn(X,˜ x˜0) for n ≥ 2. m m 1 It follows that, for n ≥ 2, πn(RP )= πn(S ) and πn(S )= πn(R) = 0. m Facts: πn(S ) = 0 if nm. For example 2 π3(S )= Z. (See Hatcher, Section 4.1 for a partial table.) m Open question: Give a general formula for πn(S ). NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 35

2 10.3. Relative homotopy groups. We can generalize πn to Top• → Grp. To deﬁne πn(X,A,x0) of a n−1 n n−1 n n−1 pair (X, A) with x0 ∈ A, view I as part of ∂I with sn = 0, deﬁne J as the closure of ∂I − I and take homotopy classes of maps n n−1 n−1 (I ,I , J ) → (X,A,x0).

When n = 0, let π0(X,A,x0) be the quotient set π0(X,x0)/π0(A, x0). n n The nth relative homotopy group πn(X,A,x0) is equivalent to [(D , ∂D , ∗), (X,A,x0)].

Lemma 10.3.1. πn(X,A,x0) is a group for n ≥ 2 and an abelian group for n ≥ 3.

Note in particular that π1(X,A,x0) is not a group since we cannot compose: given f : [0, 1] → X with f(0),f(1) ∈ A and f(0) = x0, we need f(1) = x0 in order to compose.... Exact sequence for (X, A). Analogous to the relative sequence in homology, we have: Theorem 10.3.2. There is a long exact sequence of homotopy groups

→ πn(A) → πn(X) → πn(X, A) → πn−1(A) → . Proof. This is an application of the Puppe sequence (S0, ∗) → (S0,S0) → (D1, ∂D1) → (S1, ∗) → (S1,S1) → (D2, ∂D2) → . n n n Just apply [·, (X, A)]. It is easy to see [(S , ∗), (X, A)] = πn(X) and [(D , ∂D ), (X, A)] = πn(X, A). n n Finally we also see that [(D , D ), (X, A)] = πn(A). 10.4. Compact-open topology and loop spaces. Given topological spaces X, Y , we can define Y X to be the space of continuous maps f : X → Y . It can be topologized via the compact-open topology, defined by taking a subbasis consisting of sets M(K,U)= {f : X → Y continuous |f(K) ⊂ U}, where K ⊂ X is a compact set and U ⊂ X is open. Hence a basis is given by finite intersections of M(Ki,Ui). For example, if X = I, then two paths f, g : I → Y are “close” if there exists a subdivision 0 = t0 < t1 loop space (ΩX, x˜0) is the set of loops f : (I,∂I) → (X,x0), endowed with the compact-open topology. Its basepoint x˜0 is the constant loop at x0. Corollary 10.4.2. If X is locally compact and Hausdorff, then [ΣX,Y ] ≃ [X, ΩY ]. In particular, if n X = S , then πn+1(Y ) ≃ πn(ΩY ). Proof. Since X is locally compact and Hausdorff, there is a bijection between continuous maps X×[0, 1] → Y and continuous maps X → Y [0,1]. Now observe that a map f :ΣX → Y is equivalent to a map f : X × [0, 1] → Y such that X × {0, 1} and {x0}× [0, 1] map to Y . It can therefore be viewed as a map g : X → Map((I,∂I), (X,x0)) such that g(x0) is the constant map I → X that maps to x0. 36 KO HONDA

In categorical language, Ω is an endofunctor of Top• and Σ and Ω are adjoint functors in the full subcat- egory of hTop• whose objects are locally compact and Hausdorff. m m 10.5. Proof of πn(S ) = 0 for nn. (Check this!) By the method to prove πn(S ) = 0 for m>n, the image of n n f(eα) misses a point on each m-cell of Y and hence can be homotoped into the n-skeleton Y . This works for finite CW complexes; for HW figure out how to deal with infinite CW complexes. 10.7. Whitehead’s theorem.

Deﬁnition 10.7.1. A map f : X → Y is a weak homotopy equivalence if it induces isomorphisms (f∗)n : πn(X,x0) → πn(Y,f(x0)) for all n ≥ 0 and all choices of basepoint x0. We have the following amazing theorem:

Theorem 10.7.2. If f : (X,x0) → (Y,y0) is a weak homotopy equivalence between connected (=path connected) CW complexes, then f is a homotopy equivalence.

Remark 10.7.3. The theorem does not say that two pointed topological spaces (X,x0) and (Y,y0) with isomorphic homotopy groups for all n are homotopy equivalent. You need a map from one to the other. We’ll give a proof for ﬁnite CW complexes. HW: Explain how to extend the proof to inﬁnite CW complexes. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 37

Proof. Suppose that f is an inclusion of a subcomplex. Then by the relative homotopy sequence, πn(Y, X)= n n 0 for all n. Let eα be an n-cell in Y with boundary in X. Then eα can be homotoped into X relative to n ∂eα (note that this needs an argument) and the collection of these homotopies can be combined into a deformation retraction of Y onto X. In the general case, first homotop f : X → Y so that f is a cellular map. Then replace f : X → Y by g : X → Mf , where Mf is the mapping cylinder of f (which glues X ×{1} to Y ) and g is the identification of X with X × {0}. Note that Y ֒→ Mf is a homotopy equivalence and g is an inclusion. It suffices to show that Mf deformation retracts onto X. But this is immediate from the previous paragraph because g is an inclusion of a subcomplex. 10.8. CW approximations. Definition 10.8.1. A CW approximation of a topological space X is a CW complex Z together with a weak homotopy equivalence f : Z → X.

Theorem 10.8.2. CW approximations exist for any pointed topological space (X,x0).

Proof. The point is to model the generators and relations of πn(X,x0) using cellular attaching maps.

Step 1. Without loss of generality π0(X,x0) = 1. Start with one point ∗ and a map f(∗) = x0. Next n n n choose a generating set {fα }α∈In for πn(X,x0). For each fα , take an n-cell eα that is attached to ∗ via the n n n n attaching map φα : ∂eα → ∗. Deﬁne f : eα → X so it agrees with fα . Then n f : Z0 = {∗} ∪ (∪n,αeα) → X is surjective on πn for all n.

Step 2. Next we kill off the kernel of f∗ (aka the relations). Suppose we have constructed up to Zn−1 and n+1 (f∗)i : πi(Zn−1, ∗) → πi(X,x0) is an isomorphism up to i = n − 1. Attach (n + 1)-cells eβ to Zn−1 for n+1 all generators β of ker(f∗)n to obtain Zn = Zn−1 ∪ (∪βeβ ) and extend f to Zn; this is possible since all the β’s are nullhomotopic in (X,x0). n−1 We claim that πn−1(Zn−1, ∗) ≃ πn−1(Zn, ∗), since any homotopy S ×[0, 1] → X can be homotoped n ∼ to a map to the n-skeleton X by the cellular approximation theorem. Also (f∗)n : πn(Zn, ∗) → πn(X,x0): The map πn(Zn−1, ∗) → πn(X,x0) factors into f∗i∗, where i is the inclusion Zn−1 → Zn. Hence (f∗)n is n+1 surjective. The kernel of πn(Zn−1, ∗) → πn(X,x0) is killed by the cells eβ , and we have an isomorphism. n+1 Observe that attaching eβ might make πm(Zn, ∗) larger for m>n, but that is ok. Next we discuss the uniqueness of CW approximations: Theorem 10.8.3 (Uniqueness of CW approximations). Given CW models f : Z → X and f ′ : Z′ → X with Z,Z′, X path-connected, there is a homotopy equivalence h : Z → Z′.

f i Proof. Consider the mapping cylinder Mf ′ as a replacement for X. We view the composition Z → X → ′ ′ ′ Mf ′ as a map (Z, z0) → (Mf ′ ,Z ). Since πn(Mf ′ ) ≃ πn(X) ≃ πn(Z ) for all n, πn(Mf ′ ,Z ) = 0 for all n by the relative exact sequence. This implies that all the cells of Z can be compressed into Z′, ′ ′ giving a homotopy of if to h : Z → Mf ′ with image in Z . This implies that h : Z → Z is a weak homotopy equivalence. Finally, by Whitehead’s theorem, h is a homotopy equivalence since Z,Z′ are CW complexes. Proposition 10.8.4. A weak homotopy equivalence f : Y → Z induces: 38 KO HONDA

(1) bijections f∗ : [X,Y ] → [X,Z] for all CW complexes X, and (2) isomorphisms f∗ : Hn(Y ) → Hn(Z). Proof. We’ll prove (1). See Hatcher, Prop. 4.21 for (2). The idea is similar to the uniqueness theorem. Replace Z by Mf and show that f∗ : [X,Y ] → [X, Mf ] is a bijection: Given g : X → Mf , view it as a map (X,x0) → (Mf ,Y ). Since πn(Mf ,Y ) = 0 for all n, g is homotopic to h : X → Mf with image in Y . This implies the surjectivity of f∗. To prove the injectivity, if we have F : (X × I, X × ∂I) → (Mf ,Y ), the same method shows that F can be homotoped to a map G : (X × I, X × ∂I) → (Mf ,Y ) with image in Y .

in Postnikov towers. Given a CW complex X, there exist spaces Xn with inclusion maps X → Xn such that ∼ (in)∗ : πi(X) → πi(Xn) for i ≤ n and πi(Xn) = 0 for i>n and the Xn can be put in a sequence s3 s2 ···→ X3 → X2 → X1 with snin = in−1. The Xn are “truncations” of X with successively better approximations as i → ∞. Starting with X, we can attach (n + 2)-cells to kill πn+1, then (n + 3)-cells to kill πn+2, etc. to obtain Xn. This gives the desired Xn with inclusion maps in : X → Xn. It remains to deﬁne sn+1 : Xn+1 → Xn as an extension of in : X → Xn. Since Xn+1 is obtained from X by attaching (n + 3)-cells and higher and πn+1(Xn) = πn+2(Xn) = · · · = 0, the attaching map of the cells are nullhomotopic. This implies that X can be extended to (n + 3)-cells and higher.

Deﬁnition 10.8.5. (X,x0) is n-connected if πi(X,x0) = 0 for all 0 ≤ i ≤ n. (X, A) is n-connected if πi(X,A,x0) = 0 for all 0 ≤ i ≤ n and x0 ∈ A. (Note that there is an extra condition that we need to take all x0 ∈ A; this is to take care of the situation where A has multiple path components.) Also recall that a subcomplex of a CW complex A ⊂ X is a union of cells of X such that the closure of each cell of A is contained in A (i.e., for each cell the image of its attaching map is contained in A). Deﬁnition 10.8.6. An n-connected CW model for (X, A) with A a nonempty CW complex is an n-connected CW pair (Z, A) and a map f : Z → X such that f|A = id and f∗ : πi(Z) → πi(X) is an isomorphism for i>n and injective for i = n.

Since πi(Z) agrees with πi(A) for in, Z approximates A up to n and X after n. There is an analogous CW approximation theorem for n-connected models, whose proof we omit. Theorem 10.8.7. For each (X, A) with A a nonempty CW complex, there exists an n-connected CW model f : (Z, A) → (X, A) for all n ≥ 0. We may also assume that Z is obtained from A by attaching cells of dimension >n. The n-connected CW model is unique up to homotopy equivalence. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 39

11. EXCISION HW: Hatcher, Section 4.1: 1,2,8,9,10,11,18,19.

11.1. Excision. Theorem 11.1.1 (Excision). Let X be a CW complex that can be written as X = A ∪ B, where A,B,C = A ∩ B are subcomplexes and C is nonempty and connected. If (A, C) is m-connected and (B,C) is n- connected for m,n ≥ 0, then πi(A, C) → πi(X,B) is an isomorphism for in. Hence we may assume that all the cells of A − C have dimension > m and all the cells of B − C have dimension >n. The idea of the proof can be explained in the following simpliﬁed case where A is obtained from C by attaching a single (m + 1)-cell em+1 and B is obtained from C by attaching a single (n + 1)-cell en+1. To prove the surjectivity, consider a map i i f : (I ,∂I , 0) → (X,B,x0). m+1 n+1 n Pick points p ∈ int(e ) and q ∈ int(e ). By the method of proof of πi(S ) = 0 for i

Corollary 11.2.1 (Freudenthal suspension theorem). The suspension map πi(X) → πi+1(SX) is an isomorphism for i< 2n − 1 and a surjection for i = 2n − 1 if X is (n − 1)-connected. 40 KO HONDA

Proof. Decompose SX into two cones C+X and C−X. Then πi(X) ≃ πi+1(C+X, X) by the relative sequence, πi+1(C+X, X) → πi+1(SX,C−X) is the excision map, and πi+1(SX,C−X) ≃ πi+1(SX) again by the relative sequence. The excision map is an isomorphism for i + 1 < 2n since (C+X, X) is n- connected if X is (n − 1)-connected by the relative sequence for (C+X, X). (HW: check that the sequence of maps agrees with the suspension map.) 2 This implies that πi(X) → πi+1(SX) → πi+2(S X) → . . . eventually stabilize (i.e., are isomorphisms): if X is m-connected, then SX is (m + 1)-connected, and SkX is (m + k)-connected, and even- s 0 tually i + k < 2(n + k) − 1. The direct limit is called the stable homotopy group πi (X). When X = S , s 0 n s πi (S )= πi+n(S ) for n > i + 1. It is also abbreviated πi and called the stable i-stem. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 41

12. HUREWICZISOMORPHISMTHEOREM 12.1. Some calculations. n Example. πn(S )= Z for n ≥ 2. Method 1. Consider the sequence of suspension maps

1 i1 2 i2 3 π1(S ) → π2(S ) → π3(S ) → ....

1 By the Freudenthal suspension theorem, i1 is surjective and i2, i3,... are isomorphisms. Since π1(S )= Z, 2 π2(S ) is a quotient of Z. Now there exists a map, called the Hurewicz map,

n H : πn(X) → Hn(X; Z) ≃ Z, (f : S → X) 7→ f∗α,

n where α is a generator of Hn(S ; Z). (HW: show this is a homomorphism!) n 2 n When X = S , H is surjective since id is mapped to α. Hence π2(S ) = Z and πn(S ) = Z for all n ≥ 2. n Method 2. Given f ∈ πn(S ,x0) for n ≥ 2, we can homotop f so that f restricted to a neighborhood −1 Up of f (p) is a piecewise linear map from a union of n-simplices. Hence by the usual transversality argument we may assume that f −1(p) is ﬁnite and intersects each n-simplex ∆ in its interior. By a further homotopy and subdivision of the simplices, f on the complement of the int(∆)’s map to x0 and f|int(∆) n is a homeomorphism onto S − {x0}; in other words, f is homotopic to the sum of standard deg = ±1 n homeomorphisms. The map H implies that πn(S )= Z. The following two examples can be proved using Method 2. n Example. For n ≥ 2, πn(∨αSα) is free abelian with basis which consists of homotopy classes of inclusions n n .Sα ֒→∨αSα n n n+1 Example. For n ≥ 2, πn+1(X, ∨αSα), where X is obtained from ∨αSα by attaching cells eβ via n+1 n n+1 φβ : ∂eβ →∨αSα, is free abelian with basis in bijection with {eβ }.

12.2. Change of basepoints for higher homotopy groups. We now discuss the change-of-basepoint map βγ : πn(X,x1) → πn(X,x0), n ≥ 2, where γ is a path from x1 to x0. Viewing a representative of n n n n πn(X,x1) as f : (D , ∂D ) → (X,x1), we deﬁne γf : (D , ∂D ) → (X,x0) such that γf(x) = f(2x) 1 1 if |x|≤ 2 and γ(2|x|− 1) if 2 ≤ |x|≤ 1. Then βγ ([f]) = [γf].

Lemma 12.2.1. βγ is a group homomorphism and has inverse βγ−1 . The ﬁrst assertion is not obvious and is left as HW. Hence π1(X,x0) acts on πn(X,x0), i.e., πn(X,x0) is a module over the group ring Z[π1(X,x0)].

1 n 1 n n Example. Consider πn(S ∨ S ) for n ≥ 2. The universal cover M of M = S ∨ S is a line R with S ’s n 1 n attached at all integers, and is homotopy equivalent to ∨k∈ZSk . Hence πn(S ∨ S ) is the free Z-module n 1 n 1 f −1 with basis {Sk }k∈Z. On the other hand, π1(S ∨ S ) ≃ π1(S ) ≃ Z, and its group ring is Z[t,t ]. One 1 n −1 −1 can see that πn(S ∨ S ) is the free Z[t,t ]-module Z[t,t ]. 42 KO HONDA

12.3. The Hurewicz theorem.

Theorem 12.3.1. If X is (n − 1)-connected with n ≥ 2, then the reduced homology groups Hi(X) = 0 for i

n ∂ n πn+1(X, ∨αSα) πn(∨αSα) πn(X) 0

n ∂ n Hn+1(X, ∨αSα) Hn(∨αSα) Hn(X) 0. By the calculations in Section 12.1, the ﬁrst two vertical arrows are isomorphisms. Hence the theorem follows from the ﬁve lemma. ′ We’ll state without proof a slightly more general version of the Hurewicz theorem: Let πn(X,A,x0) be the quotient of πn(X,A,x0) obtained by identifying [γf] = [f] where γ ∈ π1(A, x0). Then the homomorphism h : πn(X,A,x0) → Hn(X, A) ′ ′ descends to h : πn(X,A,x0) → Hn(X, A). Theorem 12.3.2 (Hurewicz, general version). If (X, A) is (n − 1)-connected with n ≥ 2, X, A are path- ′ ′ connected, and A 6= ∅, then h : πn(X,A,x0) → Hn(X, A) is an isomorphism and Hi(X, A) = 0 for i

13. FIBRATIONS 13.1. Deﬁnitions.

Deﬁnition 13.1.1. A map p : E → B satisﬁes the lifting property with respect to the pair (Z, A) if for any commutative diagram

f A E i p g Z B, there exists a map g˜ : Z → E such that pg˜ = g and gi˜ = f.

Definition 13.1.2. A surjective map p : E → B is a fibration (resp. Serre fibration) if it satisfies the lifting property with respect to (A × I, A × {0}) for all topological spaces (resp. CW complexes) A.

−1 Given a fibration p : E → B, if B has a basepoint b0, then p (b0) is the fiber F at b0. Picking x0 ∈ F , i p we often denote the fibration by (F,x0) → (E,x0) → (B, b0). Lemma 13.1.3. A fiber bundle is a Serre fibration.

p −1 ∼ A map E → B is a ﬁber bundle with ﬁber F if there is an open cover U of B and φU : p (U) → U × F for all U ∈U such that p = πU φU . Here πU ,πF : U × F → U are the projections to U and F .

Proof. We may reduce to the case where A = In and Z = In × I since we can do homotopy lifting in stages using the CW structure. We may also subdivide In so that f : Z → B has image inside some U. n −1 n n Given a lift g˜ : I × {0}→ p (U)= U × F , we just need to extend πF g˜ from I × {0} to all of I × I; this is straightforward.

13.2. Fibration sequence.

Theorem 13.2.1. If p : E → B is a Serre ﬁbration, then p∗ : πn(E,F,x0) → πn(B, b0) is an isomorphism for all n ≥ 1. Hence the relative homotopy sequence for (E,F,x0) becomes the ﬁbration sequence

···→ πn(F,x0) → πn(E,x0) → πn(B, b0) → πn−1(F,x0) → . . .

n n Proof. We ﬁrst show that p∗ is surjective: Let f : (I ,∂I ) → (B, b0) be a representative of πn(B, b0). n n−1 n n−1 View it as a map f : (I , J ) → (B, b0). We want to lift it to a map f˜ : (I , J ) → (E,x0). n−1 n n−1 n n−1 The constant map J → {x0} is a lift of f|Jn−1 . Then since (I , J ) and (I ,I × {0}) are ˜ n n−1 ˜ homotopy equivalent, f can be lifted to f on (I , J ). We ﬁnally observe that f|In−1×{1} maps to F ˜ n n n−1 since f|In−1×{1} is the constant map to b0. This gives us f : (I ,∂I , J ) → (E,F,x0). n n n−1 The injectivity of p∗ is similar: Given f˜0, f˜1 : (I ,∂I , J ) → (E,F,x0) and a homotopy ft : n n n+1 n−1 (I ,∂I ) → (B, b0) from f0 = p∗f˜0 to f1 = p∗f˜1, the homotopy can be viewed as f : (I , J ×I) → n n−1 n n−1 (B, b0). f is lifted to f˜ on (I × {0, 1}) ∪ (J × I)= J since J × I is mapped to x0. Again, since n+1 n−1 n+1 n (I , J × I) and (I , J ) are homotopy equivalent, we can lift f to f˜ taking f˜0 to f˜1. 44 KO HONDA

13.3. Examples. Example. The Hopf fibration S1 → S3 → S2 is given as follows: Given the unit sphere S3 ⊂ C2, S1 acts 3 iθ iθ iθ 1 2 freely on S via (e , (z1, z2)) 7→ (e z1, e z2). The quotient is CP = S . It is not hard to see the quotient map is a fiber bundle S3 → S2 with fiber S1. We now consider the fibration sequence: 1 3 2 1 3 πn(S ) → πn(S ) → πn(S ) → πn−1(S ) → πn−1(S ). 1 3 2 2 1 Since πn(S ) = 0 for n > 1 and Z for n = 1, for n ≥ 3 πn(S ) ≃ πn(S ) and π2(S ) ≃ π1(S ). In 2 particular, π3(S ) ≃ Z, generated by the Hopf fibration. NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 45

14. HOMOTOPYANDCOHOMOLOGY 14.1. Eilenberg-MacLane spaces. An Eilenberg-MacLane space K(G, n) with G a group and n> 0 is a connected CW complex such that πi(K(G, n)) = G if i = n and 1 if i 6= n. Theorem 14.1.1. A K(G, n) exists (if n> 1 we assume G is an abelian group) and any two K(G, n)’s are homotopy equivalent. A K(G, n) can be constructed as in the CW approximation theorem using a presentation for G: Let n n+1 {fα}α∈I be the generators of G. Then starting with the wedge ∨α∈I Sα, we attach eβ for the relations. We can then kill off πi for i>n by attaching (n + 2)-cells and higher. Let us call this X. In order to prove the homotopy equivalence of any two K(G, n)’s, it sufﬁces to show that there is a weak homotopy equivalence from X to any other K(G, n). This follows from the uniqueness of CW models up to homotopy equivalence. (Check the details for HW.) Note that Hn(K(G, n); Z) ≃ πn(K(G, n)) ≃ G by the Hurewicz theorem. By taking a wedge of K(G, n)’s we can construct a CW complex with arbitrary homotopy groups. 14.2. Spectra.

Deﬁnition 14.2.1. An Ω-spectrum X is a sequence of based CW complexes {Xn}n∈Z together with weak X homotopy equivalences (the “structure maps”) αn = αn : Xn → ΩXn+1.

Recall that ΩXn is the based loop space of Xn; by a theorem of Milnor, the loop space of a CW complex has the homotopy type of a CW complex. A map of Ω-spectra f : X → Y is a collection of based maps Y X fn : Xn → Yn such that αn fn = fnαn . Denote the category of Ω-spectra by S. It has a zero object given by a sequence ∗ of 1-pointed spaces.

Remark 14.2.2. There are variants of this definition, depending on what the Xn are and what restriction to put on the maps αn, and moreover the definitions are not all the same! The definition in Weibel is that for a spectrum we take Xn to be based topological spaces and αn to be based homeomorphisms and for a prespectrum we take αn to just be based maps.

Example. The Eilenberg-MacLane spectrum is an Ω-spectrum with Xn = K(G, n). Recall the adjunction [Si+1, K(G, n)] = [ΣSi, K(G, n)] = [Si, ΩK(G, n)].

This implies that ΩK(G, n) is a K(G, n − 1). Let αn−1 : K(G, n − 1) → ΩK(G, n) be a homotopy equivalence. ∞ n n Example. The suspension spectrum Σ X of a space X is given by Xn = Σ X and αn :Σ X → n+1 n n+1 ΩΣ X which corresponds to ǫn = id : Σ(Σ )X → Σ X under the adjunction [ΣnX, ΩΣn+1X] = [Σ(ΣnX), Σn+1X]. ∞ Here αn is not a weak homotopy equivalence (and hence Σ X is a prespectrum). Refer to Hatcher, Section 4.J for more information on the topology of ΩΣX. 14.3. From Ω-spectra to cohomology theories. n Theorem 14.3.1. If K is an Ω-spectrum, then the (contravariant) functors X 7→ h (X) := [X, Kn], n ∈ Z, give a reduced cohomology theory on the category CW• of pointed connected CW complexes. 46 KO HONDA

Proof. ∗ Step 0. The functoriality is clear: Given φ : X → Y , we have φ : [Y, Kn] → [X, Kn] given by f 7→ fφ.

Step 1. We ﬁrst give a group structure on [X, Kn]. This is a consequence of

[X, Kn] = [X, ΩKn+1]=[ΣX, Kn+1], where the second equality is the adjointness of Σ and Ω. The key is that there is a map τ :ΣX → ΣX ∨ΣX and that the sum f + g of f, g :ΣX → Kn+1 is given by (f ∨ g)τ. 2 Similarly, the abelian group structure on [X, Kn] follows from [X, Kn]=[Σ X, Kn+2]. Step 2. Verify the reduced cohomology axioms: There are natural coboundary maps δ : hn(A) → hn+1(X/A) such that (1) (Homotopy axiom) If f ≃ g : X → Y , then f ∗ = g∗ : hn(Y ) → hn(X). (2) (Relative sequence) There is a long exact sequence → hn(X/A) → hn(X) → hn(A) → hn+1(X/A) → . n n (3) (Wedge axiom) If X = ∨αXα, then h (X) → Παh (Xα) is an isomorphism. Here “natural” means given f : (X, A) → (Y,B) the following diagram commutes: hn(A) hn(X/A)

hn(B) hn(Y/B). (1) and (3) are clear. For (2), recall the Puppe sequence for (X, A): i A → X → Ci → ΣA → ΣX → ΣCi → ..., where Ci ≃ X/A. Apply [·, Kn] to obtain:

[A, Kn] ← [X, Kn] ← [X/A, Kn] ← [ΣA, Kn] ← [ΣX, Kn] ← [ΣX/A, Kn], where [ΣA, Kn] = [A, ΩKn] = [A, Kn−1], [ΣX, Kn] = [X, Kn−1], [ΣX/A, Kn] = [X/A, Kn−1]. This then becomes: n n n n−1 n−1 n−1 h (A) ← h (X) ← h (X/A) ← h (A) ← h (X) ← h (X/A).

Observe that hn(X) ≃ hn+1(ΣX). This can be proved by using the relative sequence or by checking [X, Kn] ≃ [ΣX, Kn+1].

Corollary 14.3.2. For the Eilenberg-MacLane spectrum Kn = K(G, n), X 7→ [X, K(G, n)] agrees with the (usual) reduced cohomology Hn(X; G) with G-coefficients. Theorem 14.3.1 implies that Xe 7→ [X, K(G, n)] gives a reduced cohomology theory. Note that the Hurewicz map [Sn, K(G, i)] → Hi(Sn; G) is an isomorphism for all i. To prove the corollary it remainse to find natural isomorphisms T : [X, K(G, n)] → Hn(X; G) e NOTES FOR MATH 227A: ALGEBRAIC TOPOLOGY 47 for all CW complexes X; we omit the proof and just remark that the isomorphisms can be constructed cell-by-cell. 14.4. Brown representability. Theorem 14.3.1 has the following amazing converse: Theorem 14.4.1 (Brown representability theorem). Every reduced cohomology theory on CW• has the n form h (X) = [X, Kn] for some Ω-spectrum K = {Kn}n∈Z. We will not give full details of the proof (cf. Hatcher, Section 4.E). A functor F : C → Set is repre- sentable if there exists K and a natural transformation η : Hom(X, K) → F (X) which is a bijection for each X. (So, strictly speaking, we want to view hn as a functor from the homotopy category hCW•.) n i n i If we know h (S ) for all i, then we can try to construct Kn by the requirement πi(Kn)= h (S ). One n case where this works is if h is reduced singular cohomology with G-coefficients. Then Kn must be a K(G, n). This implies Corollary 14.3.2. Theorem 14.4.1 follows from the following slightly more general theorem, together with a step relating the Kn’s (which will not be explained here). Theorem 14.4.2. Let F : CW• → Set• be a contravariant functor such that: (1) (Homotopy axiom) If f, g : X → Y are homotopic, then Ff = F g. (Equivalently view F as a functor from the homotopy category hCW•.) (2) (Sheaf axiom) If X ∈ CW• and X = A ∪ B, where A, B, A ∩ B ∈ CW•, then if a ∈ F (A) and b ∈ F (B) that restrict to the same element in F (A ∩ B), there exists x ∈ F (X) that restricts to a and b. (3) (Wedge axiom) If X = ∨αXα, then F (X) = ΠαF (Xα). Then there exists K ∈ CW• and u ∈ F (K) such that

Tu : [X, K] → F (X), Tu(f)= Ff(u) is a bijection for all X. (Note that Tu is a natural transformation.) Remark 14.4.3. Axiom (3) implies that F (pt) is a one-element set. (HW: Verify this using X ∧ {∗} = X.) Remark 14.4.4. Axioms (1)–(3) for F = hn together with the existence of natural isomorphisms hn(X) ≃ hn+1(ΣX) for all X ∈ CW• is equivalent to the reduced cohomology axioms. (Proof omitted here.)

Example. If T is a pointed connected topological space, then apply Theorem 14.4.2 to the functor [·, T ]. Then there exists C ∈ CW• such that [X,C] = [X, T ] for all X ∈ CW•. In other words, the Brown representability theorem implies the CW approximation theorem. 14.5. Proof of Theorem 14.4.2. Step 1. We construct K inductively cell-by-cell as usual so that the theorem holds for all spheres X = Sn with n ≥ 1. Recall that if we know F (Sn) for all n> 0, then we can try to construct K by the requirement n πn(K) = F (S ) (just like for K(G, i)). The complication in the proof comes from keeping track of the element u ∈ K. Start with K0 = pt. Let u0 be the unique element of F (K0), recalling Remark 14.4.3. Arguing by induction, assume that there exist Kn and un ∈ F (Kn) such that i Tun : πi(Kn) → F (S ), f 7→ Ff(un), is surjective for i ≤ n and has trivial kernel for i

We will construct Kn+1 ⊃ Kn and un+1 that restricts to un: Denote the representatives of the kernel of n n Tun : πn(Kn) → F (S ) by fα : Sα → Kn and set f = ∨αfα. Let Mf and Cf be the mapping cylinder and mapping cone of f. We then set n+1 Kn+1 = Cf ∨ (∨βSβ ), where β ranges over F (Sn+1). Fact/HW: Axioms (1)–(3) in Theorem 14.4.2 imply the exactness of F (X/A) → F (X) → F (A) .for each inclusion A ֒→ X n Apply the fact to F (Cf ) → F (Kn) = F (Mf ) → F (∨αSα). Viewing un ∈ F (Mf ), it is mapped n to 0 ∈ F (∨αSα), and hence comes from w ∈ F (Cf ). We then deﬁne un+1 ∈ F (Kn+1) to restrict to n+1 un ∈ F (Kn) and β ∈ F (Sβ ); this exists by (3) and the fact that un and β restrict to the same point. For

HW, verify that Tun+1 is surjective for i ≤ n + 1 and has trivial kernel for i

This would be a good starting point for the study of vector bundles and K-theory, but alas we are out of time....