07-17-2015 Contents 1. Summary of the Homology of Posets 1 2. Definition of Singular Homology 3 3. The Hurewicz homomorphism 6 4. Some Properties of Singular Homology 7 1. Summary of the Homology of Posets By now, you have realized that what algebraic topologists do is assign invariants to topological spaces which do not depend on the homotopy type. That is, we assign quantities F (X) so that if X ' Y , then F (X) =∼ F (Y ). Further, we have the language of functors to make this more precise. What we've done so far is define functors F : Top ! C where C is some algebraic category (abelian groups, chain complexes, . ), such that if X ' Y then F (X) ' F (Y ) (in the case of chain complexes) or better F (X) =∼ F (Y ) (in the case of homology). Definition 1.1. Let T be some subcategory of topological spaces. A homotopy functor is a functor F : D! C such that if f; g : X ! Y are homotopic, i.e., f ' g, then F (f) = F (g). Exercise 1.2. If F is a homotopy functor and X ' Y , then F (X) =∼ F (Y ). Our first examples were π1 : Top∗ ! Groups; and for n ≥ 2, πn : Top∗ ! AbGroups : Now, we defined the homology of a poset, which gave functors P Hn : A-spaces ! Ab : If we think in terms the category Ab∗ of sequences of abelian groups (define the maps!), we can write this as P H∗ : A-spaces ! Ab∗ : We also proved that Proposition 1.3. Let f; g : X ! Y be maps of A-spaces. If f ≤ g, then f and g induce the same map on homology, that is P P P P H∗ (f) = H∗ (g): H∗ (X) ! H∗ (Y ): However, we only have the following partial result: 1 2 07-17-2015 Theorem 1.4. If X and Y are A-spaces and f; g : X ! Y satisfy f ≤ g, then f ' g. If X and Y are F -spaces, we have a converse. That is, in this case, if f ' g, then there are maps f1; : : : ; fn such that, f ≥ f1 ≤ f2 ≥ f3 ≤ f4 ≥ ::: ≤ fn ≥ g: P Therefore, we have not shown that H∗ is a homotopy functor. However, we have shown that its restriction to F -spaces is such: P Proposition 1.5. H∗ : F -spaces ! Ab∗ is a homotopy functor. Definition 1.6. We also defined, for A ⊂ X, the relative homology of X with respect to A d ker Cn(X)=Cn(A) −! Cn−1(X)=Cn−1(A) HP (X; A) = n d im Cn+1(X)=Cn+1(A) −! Cn(X)=Cn(A) = n'th homology of the chain complex C∗(X)=C∗(A) Therefore, at least on F -spaces, we've proved that H∗ has the following proper- ties. (Property (3) is an exercise) P Proposition 1.7. Let H∗ : F -spaces ! Ab be the poset homology functor. Then P P (1) (Homotopy Invariance) If f ' g, then H∗ (f) = H∗ (g) P P (2) (Dimension Axiom) H0 (∗) = Z and Hn (∗) = 0 if n > 0. ` (3) (Additivity) If X = Xi, where Xi are the different path components of P ∼ L P X, then H∗ (X) = i H∗ (Xi). (4) (Exactness) If A ⊂ X, there is a long exact sequence P P P P ::: ! Hn (A) ! Hn (X) ! Hn (X; A) ! Hn−1(A) ! ::: We also proved that: (a) (Mayer-Vietoris) If A and B are sieves in X and X = A [ B, there is a long exact sequence P P P P P ::: ! Hn (A \ B) ! Hn (A) ⊕ Hn (B) ! Hn (X) ! Hn−1(A \ B) ! ::: (b) (Suspension isomorphism) There is an isomorphism ( P P ∼ H1 (ΣX) ⊕ Z n = 0 Hn (X) = P Hn+1(ΣX) n > 0 (Note that except for (1), these all hold for A-spaces.) Remark 1.8. A little later, we will give an axiomatic description for functors which satisfy the above properties. (1)-(4) will be axioms for such functors. However, the list of axioms is still incomplete and (4) is rather useless since it is not clear now P that H∗ (X; A) has (1) any geometric meaning that makes it interesting to compute or P P (2) is easier to compute than H∗ (X) and/or H∗ (A) so that it could be used as a tool in understanding these terms. P P We need a property that allows us to identify H∗ (X; A) with H∗ (X=A) in favorable cases. We ask ourselves: 07-17-2015 3 Question 1.9. Is there a functor H∗ : Top ! Ab which satisfies all the above P ∼ properties and such that H∗ (X) = H∗(X)? 2. Definition of Singular Homology To define the homology of a space, we must associate to a space X a chain complex C∗(X). Then H∗(X) will simply be its homology. Recall that the n- simplex ∆n is the subspace of Rn+1 given by n n+1 ∆ = f(t0; : : : ; tn) 2 R j 0 ≤ ti ≤ 1; t0 + ::: + tn = 1g Now, note that there are n + 1 copies of ∆n−1 in ∆n. These are the faces of ∆n. We denote them by n n ∆i = f(t0; : : : ; ti−1; 0; ti+1; : : : ; tn) 2 ∆ g Remark 2.1. Let xi = (t0 = 0; : : : ; ti−1 = 0; ti = 1; ti+1 = 0; : : : ; tn = 0). Then n n x0; : : : ; xn are the vertices of ∆ . The face ∆i is the n-simplex with vertices x0;:::; xbi; : : : ; xn. There's a canonical homeomorphism: n−1 n ∆ ! ∆i which sends (t0; : : : ; tn−1) to (t0; : : : ; ti−1; 0; ti; : : : ; tn) Definition 2.2. A singular n-simplex on X is a continuous map σ : ∆n ! X. Let Xn be the set of singular n-simplicies: Xn = fσ : ∆n ! Xg Given a singular n-simplex on X, σ : ∆n ! X, we can obtain n + 1 singular n − 1-simplices by restricting σ to the faces of ∆n. That is, n σj n : ∆ ∼ ∆ ! X ∆i n−1 = i Definition 2.3. The group of singular n-chains on X, denoted Cn(X), is the free abelian group generated by the set of all singular n-simplices, that is ( k ) X Cn(X) = ZfXng = aiσi j σi 2 Xn; ai 2 Z i=0 Further, let @ : Cn(X) ! Cn−1(X) be defined on a generator σ 2 Cn(X) by n X i @ (σ) = (−1) σj n n ∆i i=0 One verifies that @ ◦ @ = 0. The chain complex C∗(X) is called the singular chains on X. Its homology is called the singlular homology of X, denoted ker @n Hn(X) = im @n+1 If f : X ! Y , we get a map C∗(f): C∗(X) ! C∗(Y ). For a singular n-simplex σ : ∆n ! X, we let Cn(f)(σ) = f ◦ σ 2 Cn(Y ): This induces a map H∗(f): H∗(X) ! H∗(Y ): 4 07-17-2015 Hence, singular homology is a functor: H∗ : Top ! Ab∗ Remark 2.4. Note that H0(X) = C0(X)=(im(@1 : C1(X) ! C0(X))): A singular 0-simplex is a function σ : ∆0 = f0g ! X, so it is just a point in X. We let 0 σx : ∆ ! X be the simplex with σx(0) = x: Therefore, ∼ C0(X) = ZfXg where ZfXg is the free abelian group on X. An isomorphism is given by k k k X X X aiσxi 7! aiσxi (0) = aixi: i=0 i=0 i=0 So, we can think of the elements of C0(X) as formal sums k X aixi i=0 where the points xi 2 X and ai 2 Z. Further, suppose that two points x0; x1 2 X are in the same path component, that is, that there is a path σ : I ! X with σ(0) = x0 and σ(1) = x1. Since 1 I = ∆ , we have σ 2 C1(X). Further, @ (σ) = σj 1 − σj 1 1 ∆0 ∆1 = σx1 − σx0 7! x1 − x0: Therefore, in C0(X)= im(@1), we have the relation that x1 = x0 for any two gen- erators which are connected by a path. Therefore, any two generators for C0(X) which are in the same path components are identified when we pass to the quotient H0(X) = C0(X)= im(@1). Further, these are the only relations, so H0(X) is a free abelian group with one generator for each path component of X. This proves: ` Proposition 2.5. If X = α2I Xi where Xi are the distinct path components of X, then ∼ M H0(X) = Z: α2I An explicit isomorphism is induced by the map M C0(X) ! Z α2I given by k ! X X aixi 7! ai i=0 xi2Xα α2I ∼ Proposition 2.6 (Dimension Axiom). H0(∗) = Z and Hn(∗) = 0 if n > 0. 07-17-2015 5 n Proof. We have Cn(∗) = Z since there is only one map σn : ∆ ! ∗, namely, the constant map at ∗. Further, since σ j n = σ n ∆i n−1 we have for n > 0 n X i @ (σ) = (−1) σj n n ∆i i=0 n X i = (−1) σn−1 i=0 ( 0 n is odd = σn−1 n is even The chain complex thus looks like 0 ∼= 0 ∼= 0 ∼= 0 ::: ! Z −! Z −! Z −! Z −! Z ! ::: ! Z −! Z −! Z −! Z ! 0 Remark 2.7. We can define a related functor He∗ : Top∗ ! Ab called the reduced homology as follows. For a based space X, there are maps p ∗ −!i X −!∗ These induce maps on homology i p H∗(∗) / H∗(X) / H∗(∗) Z / H∗(X) / Z This implies that H0(X) has a direct summand isomorphic to Z.