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,...,eˆi,...,en]. P Here σ|[e0,...,eˆi,...,en] means the composition of [e0,...,en−1] → [e0,..., eˆi,...,en] given by the canonical linear homeomorphism, followed by σ restricted to [e0,..., eˆi,...,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 coeffi- cients 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.