Algebraic Topology Math 528, Spring 2011 Professor Donald Stanley Notes by: Pooya Ronagh

Algebraic Topology Math 528, Spring 2011 Professor Donald Stanley Notes by: Pooya Ronagh Department of mathematics, University of British Columbia Room 121 - 1984 Mathematics Road, BC, Canada V6T 1Z2 E-mail address: pooya@math.ubc.ca Chapter 1. Homotopy Theory 1. Cofibrations 2. CW complexes 3. Cellular and CW approximation 4. Hurewicz homomorphism 5. Moore spaces 6. Eilenberg-Maclane spaces/Postnikov approximations 3 3 5 9 14 16 18 Chapter 2. Fibrations 1. Fiber bundles and fibrations 20 20 Chapter 3. Spectral Sequences 1. Construction of spectral sequences 25 27 Chapter 4. H-spaces and algebras 1. H-spaces 2. (co)-Algebras 3. Hopf algebras 34 34 36 37 Chapter 5. Localization (rationalization) 1. Cohomology Serre spectral sequence 42 47 Chapter 6. Loop suspensions 1. Another homotopy computation 50 57 Chapter 7. Classifying spaces 1. Category of G-bundles 2. Fiber bundles 3. Milnor’s construction 60 60 62 64 Contents CHAPTER 1 Homotopy Theory 1. Cofibrations Last time talked about cofibrations in CG, the space of compactly generated, Hausdorff spaces (see appendix of Hatcher for more information). Definition 1. X ∈ CG when this holds: C ⊆ X is closed iff for all K compact, C ∩ K is closed. In particular all CW complexes are in CG. Proposition 1. CG is closed under pushouts, coproducts, direct limits (if A(0) ⊆ A(1) ⊆ A(2) ⊆ ⋯ each A(i) Ð → A(i + 1) is a cofibration, then A(0) Ð → lim A(i) = ∪A(i) is a cofibration). Ð→ Proposition 2. In any category consider A /B /C /E /F D (1) Both squares pushout then rectangle is a pushout. (2) Left square and rectangle pushout then right square is a pushout. (3) Right square and rectangle being pushout then it is not true that the left square is a pushout. Proof. If /C A g f /D B 3 1. COFIBRATIONS is a pushout then 4 A × I ∪ B × {0} / C × I ∪ D × {0} / D×I B×I is a pushout. Really the top right is the pushout of / D × {0} . C × {0} C ×I If C ⊆ D then it is the subspace C × I ∪ D × {0} ⊆ D × I. Hence if f is a cofibration then g is a cofibration. In A /C / C ×I /D / C × I ∪C D B left is a pushout by hypothesis, right is a pushout by defintion. So rectanlge is a pushout. A / A×I / C ×I / A × I ∪A B / C × I ∪C D B left is a pushout be defintion, rectangle is a pushout by last step. So right is a pushout. A×I / A×I ∪B / B×I / C ×I ∪D / D×I C ×I Left is a pushout by last step. Rectangle is a pushout since in CG. So right is a pushout as desired. Next thing to show is that Proposition 3. S n Ð → Dn+1 is a cofibration. Proof. Dn+1 × I ⊆ Dn+1 × R projecting from (0, 2) gives retraction Dn+1 × I Ð → Sn × n+1 I ∪D × {0}. f ig(1). Example 1.1. For all X, ∅ Ð → X is a cofibration. ⌟ 2. CW COMPLEXES 5 2. CW complexes Definition 2. f ∶ A Ð → X has the structure of a relative weak CW complex if X(0) = A and we have pushouts ∐α∈I(i) S n(α) / X(i) / X(i + 1) ∐α∈I(i) Dn(α)+1 with the indexing: n(α) ∈ {−1, 0, ⋯, } and S −1 = ∅ and D0 = ∗. And X = X(∞) = ∪X(i) is a (standard) relative CW complex if for every α ∈ I(i), n(α) = i − 1. And if A = ∅ then X is a (weak) CW complex. In the (standard) CW case then Xi = X(i) is called i-skeleton of X. Remark. In fact we need to define (weak) relative CW complexes up to an isomorphism of maps: f is a relative CW complex if f ≅ g and g can be given the structure of a relative CW complex, where the isomorphism of f and g mean there exists commutative diagrams / X′ X g f /X / Y′ Y f /Y with horizontal arrows isomorphism. ▸ Exercise 1. Show that the (relative) weak CW complexes are the smallest class of maps containing ∅ Ð → ∗ and S n Ð → Dn+1 for all n ≥ 0 and closed under pushouts, coproducts and (countable) direct limits; i.e if /B A g f /D C is a pushout then f is relative weak CW complex then so is g. Proposition 4. If f ∶ A Ð → X is a relative CW complex then f is a cofibraiton. Proof. Given any diagram below we can complete the diagram: A×I ∪X ϕ t X ×I t t t / t: Z 2. CW COMPLEXES 6 Fix X(0) = A and show that X(i) Ð → X(i + 1) is a cofibration. Then X = colimX(i) so A Ð → X is a cofibration by previous proposition. For every α, S n(α) Ð → Dn(α)+1 is a n(α) n(α)+1 cofibration. ∐α∈I(i) S Ð → ∐α∈I(i) D is a cofibration. But then we have a pushout / X(i) ∐α∈I(i) S n(α) i j / X(i + 1) ∐α∈I(i) Dn(α)+1 i cofibation ⇒j cofibration. Then direct limit of cofibrations is a cofibration ⇒A Ð → X is a cofibration. For f ∶ X Ð → Y the mapping cylinder of f is the pushout X x↦(x,0) f /Y i / M (f ) X ×I Y Ð → M (f ) is a cofibration. i has strong deformation retraction. And in particular they are homotopy equivalent. If X is a CW complex then Y Ð → M (f ) is relative (weak) CW complex. If Y is a (weak) CW complex then X Ð → M (f ) via x ↦ (x, 1) is a (weak) relative CW complex. Recall that Dn × I can be given the structure of a CW complex. There is a pushout /Y X / Y ∐X / M (f ) X ∐X X ×I is a pushout so if X ∐ X Ð → X × I is a relative (weak) CW structure so is Y ∐ X Ð → M (f ). Also X Ð → X ∐ Y is relative CW if Y is. Y Ð → X ∐ Y is a relative CW complex if X is. So basically you have to show that X ∐ X Ð → X × I is a relative CW complex. ∐ S i × I ∪ Di+1 ∪ Di+1 / X ∐ X ∪ X(i) × I / X ∐ X ∪ X(i + 1) × I ∐D i+1 ×I And you can more generally write α(i) for any i. 2. CW COMPLEXES 7 Definition 3. f ∶ X Ð → Y is a weak homotopy equivalence if for any x0 ∈ X and any n πn (f ) ∶ πn (X, x0 ) Ð → πn (Y, f (x0 )) is an isomorphism. Theorem 2.1 (Whitehead’s). If X, Y are path-connected (weak) CW complexes and f ∶ XÐ → Y is a weak homotopy equivalence then f is a homotopy equivalence. And in particular if f is a subcomplex then f has a strong deformation retraction. Lemma 1. Let (X, A) be a relative (weak) CW complex and (Y, B) be just any pair and B ≠ ∅. Suppose for evey n such that X − A has n-cells (have the vanishing of obstruction theory given by) πn (Y, B, y0 ) = 0 for any y0 ∈ B. Then for every map f ∶ (X, A) Ð → (Y, B) of pairs, there exists g such that im(g) ⊆ B and f ≅ g rel A. Proof. Recall the compression lemma: πn (Y, B, y0 ) = 0 iff for every f ∶ (Dn , S n−1 , ∗) Ð → (Y, B, y0 ) there exists g such that f ≅ g rel S n−1 and im(g) ⊆ B. Suppose A = X(0) ↪ X(1) ↪ ⋯ ↪ X(n) ↪ ⋯X(∞) = X and we have a sequence of homotopies Hk ∶ fk ≅ fk+1 rel X(k − 1) with fk+1 (X(k)) ⊆ B for all k < n. We will construct the next one: Denote by fn = fn ∣X(n) the restriction. We basically want to complete f ∣X(n−1) X(n − 1) w w w w X(n) fn / w; B /Y But we also have a pushout ∐ S n(α) ∐ h(α) ∐ Dn(α)+1 ∐ h(α) f∣ / X(n − 1)X(n−1) / B w; w w w w / X(n) /Y fn So by the compression lemma for S n(α) fn h(α) Dn(α)+1 fn h′ (α) /B /Y 2. CW COMPLEXES 8 there is S n(α) × I D n(α)+1 constant ×I /B /Y H with H(−, 0) = fn h′ (α) and H(−, 1)(Dn(α)+1 ) ⊆ B. So get by taking coproducts Hn as the left square below is a pushout: h(α) ∐ S n(α) × I ∐D n(α)+1 ×I const. homotopy / X(n − 1) × I /B h(α) / X(n) × I _ _ _ _ _ _ _ _ _ _/ Y Hn X(n) ↪ X is a cofibration so in X ×I H n ∪fn /6 lll Y l l l lll ll∃H l l n lll X(n) × I ∪ X call fn+1 (x) ∶= Hn (x, 1) and Hn (x, 0) = fn (x), Hn ∶ fn ≅ fn+1 rel X(n−1) and fn+1 (X(n)) ⊆ B. Last step is to put all these homotopies together to get H(x, t) = Hk (x, 2k+1 t − (2k+1 − 1)), t ∈ [1 − 1/2k , 1 − 1/2k+1 ]. These glue together to give a homotopy basically since Hk (x, 1) = fk+1 (x) = Hk+1 (x, 0). H is continuous since H∣X(k)×I is for k. Since H in eventually stationary on each X(k) (just composing finitely many homotopies on each X(k)). Proof of Whitehead’s theorem. Suppose i ∶ X ↪ Y is (weak) relative CW complex. From the long exact sequence on πk ∂ ⋯Ð → πn X Ð → πx Y Ð → πn (Y, X) Ð → πn−1 (X) Ð →⋯ implies when πn X ≅ πn Y that πn (Y, X) = 0 for all n. So idY ≅ g rel X and g(Y ) ⊆ X. So g is the homotopy inverse and hence X ≅ Y . More generally for any map f ∶ X Ð → Y , where 3. CELLULAR AND CW APPROXIMATION 9 Y ≅ M (f ) we have that i ∶ X Ð → M (f ) via x ↦ (x, 1) making the following / M (f ) EE EE EE p f E" XE E i Y commute. Since the down arrows are weak homotopy equivalences (f by hypothesis and p since homotopy equivalence implies weak homotopy equivalence), “2 out of 3” implies that i is a weak homotopy equivalence! Also i is relative (weak) CW complex hence homotopy equivalence. So f = i ○ p is homotopy equivalence. 3. Cellular and CW approximation Suppose X, Y are have given CW structures then a map f ∶ X Ð → Y is called cellular if for every n, f (Xn ) ⊆ Yn . Lemma 2. Given diagram /Y Sn f Dn+1 / Y ∪ Dk with k > n + 1 there exists g such that f ≅ g rel S n and g(Dn+1 ) ⊆ Y . Lemma 3. Given diagram Sn / Yn+1 /Y Dn+1 where Y has CW structure, there exists g ∶ D Yn+1 . f n+1 Ð → Y such that f ≅ g rel S n and g(Dn+1 ) ⊆ Proof. By compactness of Dn+1 , f factors through f ′ below; i.e. there exists a commutative diagram / Yn+1 Sn D / Y ′ _ HH f ′ HH HH f HHH # n+1 Y 3. CELLULAR AND CW APPROXIMATION 10 where Y ′ − Yn+1 has finitely many cells all of dimension > n + 1 so apply lemma 2 finitely many times. Lemma 4. Given diagram ; Yn ww w ww ww ww / Yn+1 Xn w; g w w w w /Y Xn+1 f where X, Y have CW structures, there exists a homotopy H ∶ X × I Ð → Y such that H∣Xn+1 ×I ∶ f ≅ g rel Xn and g(Xn+1 ) ≅ Yn+1 . Proof. As in Whitehead’s theorem! First construct H ∶ Xn+1 × I Ð → Y then extend to H using the fact that Xn+1 Ð → Xn is cofibration. Theorem 3.1 (Cellular Approximation). If X, Y are given CW structures and f ∶ X Ð →Y is any map then f ≅ g with g cellular (relative version is also true). Proof. Use lemma 4 and same argument as in Whitehead’s theorem (gluing homotopies). What remains is Proof of lemma 2. Given /X Sn Dn+1 / X ∪ ek f where k > n + 1. Consider ek ≅ Rk . Since f is a map on a compact set, it is uniformly continuous. Hence there exist ε such that 1 ∣x − y∣ < ε ⇒ ∣f (x) − f (y)∣ < . 2 n+1 n+1 Cut D ≅I into cubes with diameters < ε. f igure(2) −1 −1 Let U = f B(0, 1/2), V = f B(0, 1) and let K1 be the union of all cubes intersecting U . Then K1 ⊆ V and f (K1 ) ⊆ B(0, 1) and let now K2 be the union of all cubes which intersect K1 . Note K1 is closed, ∂K2 is closed and hence ∂K2 ∩ K1 = ∅ and also f (K2 ) ⊆ B(0, 1 12 ) ⊆ 3. CELLULAR AND CW APPROXIMATION 11 0,1 Rk . So there exists some ψ ∶ K2 Ð→ and ψ(∂K2 ) = 0 and ψ(K1 ) = 1 by normality of K2 . Triangulate K2 refining the cubes. f ig(3) Then let g ∶ K2 Ð → Rk ≅ ek be the linear map agreeing with f on vertices. Now we just give a homotopy: H(x, t) = (1 − tψ(x))f (x) + tψ(x)g(y) Let h = H(−, 1). Since H is a homotopy relative to ∂K2 we can glue with constant homotopy to get H ∶ Dn+1 Ð → Y ∪ Dk ∶ f ≅ h′ rel S n where h′ ∣K2 = h. Also h′K1 = gK1 . But gK1 is linear on simplices of dimension n + 1 and they are only finitely many mapping into Rk , k > n + 1. So im(g) is in the union of finitely many hyperplanes in Rk so is not surjective. Thus h′ ∶ Dn+1 Ð → Y ∪ Dk factors through Dk with a point removed: / Y ∪ Dk MMM MMM MMM & h′ Dn+1 M Y ∪ Dk − {u} So h′ ≅ f rel S n via /Y \ Sn Dn+1 r / Y ∪ D k − {u} h′ the deformation retraction r. Remark. From the proof it follows that if f is already cellular on Xn then we can assume f ≅ g rel Xn . Corollary 1. If Y = X ∪S n Dn+1 and n > 1, i ∶ X Ð → Y , then π∗ (i) is isomorphism for ∗ < n and is surjective for ∗ = n. Proof. For surjectivity let α = [f ] ∈ πi (Y ) for any i ≤ n. f ∶ S i Ð → Y so f ≅ g, i g∶S Ð → Y that factors though X by lemma 2. Suppose α = [g] ∈ πk (X) for k < n such that i∗ (α) = 0 then we get diagram g Sk Dk+1 h /X / Y = X ∪ D n+1 since k + 1 < n + 1 again by lemma 2, h ≅ f rel S k for f ∶ Dk+1 Ð → X hence [g] = 0 in πk (X). 3. CELLULAR AND CW APPROXIMATION 12 Definition 4. (X, x0 ) is n-connected if πi (X, x0 ) = 0, ∀i ≤ n. Hence 0-connected is the same as path-connected and 1-connected is the same as simply connected. Definition 5. Similary (X, A) is n-connected, if πn (X, A) = 0 for every i ≤ n. We are going to assume (as usual) that X and A are path connected. From the long exact sequence of pair (X, A) ⋯Ð → πi A Ð → πi (X) Ð → πi (X, A) Ð → πi−1 A Ð → πi−1 X Ð →⋯ we get Proposition 5. (X, A) is n-connected if and only if πi (A) Ð → πi (X) is an isomorphism for i < n and is surjective if i = n. Example 3.1. (X ∪S n Dn+1 , X) is n-connected from corollary 1. ⌟ Proposition 6 (CW approximation). Say X is path connected. Then there is a (weak) CW-complex Y and weak homotopy equivalence ηX ∶ Y Ð → X. (It probably works if X is not path-connected?) Remark. CW (−) is a functor and η ∶ CW (−) Ð → id is a natural transformation. In the above statement CW (X) = Y . Before giving a proof we state for the record the following trivial statement: Proposition 7. (Z, z0 ) ↦ (Z ∐ W, z0 ) induces an isomorphism on πi , i ≥ 1 and injection on π0 . Proof of proposition 6. For step one consider Y (1) = ∐ i≥0 ∐ Si. f ∈Hom(S i ,X) Note that this is functorial. And let f (1) ∶ Y (1) Ð → X be the obvious map. Claim: f (1) is surjection on π∗ . Precisely speaking, we mean that for any x0 ∈ X there is y0 ∈ Y (1) such that for any α ∈ π∗ (X, x0 ) there exists β ∈ π∗ (Y (1), y0 ) satisfying f (1)∗ (β) = α. 1 In fact let α = [g ∶ S i Ð → X] then the composition Sgi Ð → Y (1) Ð → X is g and πi (g)([idSgi ]) = [g]. Then apply the previous proposition. 1A way to get around with the technicality here will be considering the wedge product rather than disjoint union and consider only the pointed maps. 3. CELLULAR AND CW APPROXIMATION 13 Assuming we have defined Y (i) and f (i) ∶ Y (i) Ð → X for all i ≤ n, compatible with the inclusions ki,i+1 ∶ Y (i) Ð → Y (i + 1), we will construct Y (n + 1) as the pushout / Y (n) FF FF f (n) kn,n+1FFF FF F" / Y (n + 1) _ _ _/ X ∐i ∐(f,H) S i ∐i ∐(f,H) Di+1 where f ∶ S i Ð → Y (n) and H ∶ Di+1 Ð → X make f Si / Y (n) /X H Di+1 f (n) commute. Define f n+1 in the obvious way. This is compatible with other maps and is functorial. Claim: Suppose α ∈ πi (Y (n)) and f (n)(α) = 0 as element of πi (X) (slightly different for π0 ). Then kn,n+1 (α) = 0. Say [α] = α. f (n)(α) = 0 if and only if there is an extension Si f (n)(α) { { { /X {= Di+1 which gives a commutative diagram Si Di+1 α H / Y (n) f (n) /X and hence a homotopy of α to 0 in Y (n + 1). As the last step of the construction, let Y = lim Y (n), f = lim f (n). Ð→ Ð→ Note that Y and f are both functorial. Claim f is a weak homotopy equivalence. 4. HUREWICZ HOMOMORPHISM In fact 14 /Y z z zz f (1) zzf z }zz Y (1) X commutes so f (1) is surjective. Consequently f is surjective. For injectivity (π0 will be different but,) let α = [g] ∈ πi (Y ) and suppose f∗ (α) = 0. By compactness of S n Hom(S n , lim Y (i)) = lim Hom(S n , Y (i)). Ð→ Ð→ i So g ∶ S Ð → Y must factor through Y (n) for some n. So we get a commutative diagram > Yn DD DDkn,n+1 }} } DD } } DD }} " . g Si A A j Yn+1 AA j ′ yyy A y g AAA yy |yy Y But (kn,n+1 )∗ [g] = 0. So j∗ [g] = 0, thus [g] = 0. So f ∶ Y Ð → X is a weak homotopy equivalence. Remark. Another construction would be to let adding only the n-cells in the n-th step. Using / Y (n) ∨S n / Y (n + 1) ∨D get functorial CW complex (not weak), hence also functorial cellular chains. n+1 4. Hurewicz homomorphism Consider [f ] ∈ πn (X, A, x0 ) = πn (X, A) (or just ∈ πn (X)) where f ∶ (Dn , ∂Dn ) Ð → (X, A) or equivalently f ∶ S n Ð → X. Get f∗ ∶ Hn (Dn , ∂Dn ) Ð → Hn (X, A) ( or f∗ ∶ Hn (S n ) Ð → Hn (X)). Pick an isomorphism Z ≅ Hn (Dn , ∂Dn ) which is really just a choice of ±1. Get the Hurewicz homomorphism h ∶ πn (X, A) Ð → Hn (X, A) via α = [f ] ↦ h(α) ∶= H∗ (f )(1). Remark. (1) This is well defined since f ≃ g implies H∗ f = H∗ g. (2) The isomorphism Hn (Dn , ∂Dn ) ≅ Z is needed in the definition so fix that. 4. HUREWICZ HOMOMORPHISM Proposition 8. 15 (a) h is compatible with exact sequences of pairs (for π∗ and H∗ ). (b) h is a homomorphism for n > 1. Remark. n = 1 is the absolute case is done (abelianization of fundamental group). Proof. (a) Straightforward, check it basically for the connecting homomorphism. (b) In the absolute case recall that concatenation corresponds to addition of the abelian group πn . Alternatively f + g is the composition of the pinch map ψ ∶ S n Ð → S n ∨ S n (mod out the f ∨g fold equator in the first variable) to S n ∨ S n ÐÐ→ X ∨ X Ð→ X. The folding is explicitly ⎧ x x2 = ∗ ⎪ ⎪ ⎪ 1 fold(x1 , x2 ) = ⎨ x2 x1 = ∗ ⎪ ⎪ ⎪ ⎩ ∗ x1 = x2 = ∗ where the wedge product is represented as X ∨ X = {(x1 , x2 ) ∶ x1 = ∗ or x2 = ∗}. Now have h(f + g) = (f + g)∗ (1) = ((fold)(f ∨ g)ψ)∗ (1) 1 2 = fold∗ f∗ (1) + g∗ (1) = f∗ (1) + g∗ (1) = hf + hg. Here the equality (1) follows from ψ(1) = (1, 1) and the identity Hn (S n ∨ S n ) ≅ Hn (S n ) ⊕ Hn (S n ). In fact the following Hn S n ⊕ Hn (S n ) oo7 ooo o ≅ o oo ooo ψ / Hn (S n ∨ S n ) Hn (S) 1↦(1,1) f∗ ⊕g∗ / Hn (X) ⊕ Hn (X) (f ∨g)∗ ≅ / Hn (X ∨ X) commutes since the square commutes on each factor. And (2) follows since (f old)∗ ∶ Hn (X ∨ X) Ð → H∗ (X) is just (f, g) ↦ f + g. For the relative case it is similar. Proposition 9 (Hurewicz’s isomorphism). If (X, A) is (n−1)-connected (n ≥ 2), and X, A are simply connected and A ≠ ∅ then h ∶ πn (X, A) Ð → Hn (X, A) (where the first nontrivial groups happen) is isomorphism and Hi (X, A) = 0 if i < n. We will prove this later using spectral sequences (or see Hatcher). Note 4.1. Note that if n = 1 and we are in the absolute case (A = ∗) then π1 (X) Ð → H1 (X) is the abelianization. So we actually need n ≥ 2 condition. ⌟ 5. MOORE SPACES 16 Example 4.2. In the n = 2 case with A not simply connected, say A = S 1 and S 1 Ð → S ∨ S 2 = X then from the long exact sequence of pairs 1 0 = π2 (S 1 ) Ð → π2 (S 1 ∨ S 2 ) Ð → π2 (S 1 ∨ S 2 , S 1 ) Ð → π1 (S 1 ) 1 ∨ S 2 ), i.e. π of the covering which is homotopy but the second term is isomorphic to π2 (S̃ 2 2 ̃2 (S 2 ) ≅ equivalent to ⋁i∈Z S . Thus this term is infinitely generated but H2 (S 1 ∨S 2 , S 1 ) ≅ H 1 2 1 Z. So we don’t get an isomorphism between π2 and H2 of (S ∨ S , S ). ⌟ Example 4.3. πn (S n ) ≅ Z and πn (S n ∨ S n ) ≅ πn (S n ) ⊕ πn (S n ) ≅ Z ⊕ Z. ⌟ Proposition 10. If X, Y are simply connected CW complexes and f ∶ X Ð → Y , which is homology isomorphism then it is a weak homotopy equivalence (and consequently homotopy equivalence). Proof. We compare long exact sequences: Assume X Ð → Y is a CW-pair (else consider the mapping cylinder and use M (f ) ≃ Y ). Consider the long exact sequence on π∗ and H∗ , πn X / πn Y / πn (Y, X) / πn−1 (X) / πn−1 (Y ) / Hn (Y ) / Hn (Y, X) / Hn−1 (X) / Hn−1 (Y ) Hn (X) We know that Hn (Y, X) = 0 for all n, so by induction on n and from Hurewicz’s homomorphism it follows that πn (Y, X) = 0 for all n (note that we have assumed simply connectedness). Now finish using Whitehead’s theorem. 5. Moore spaces Given an abelian group G, and n ≥ 2, there exists (uniquely up to homotopy equivalence) a CW-complex M (G, n), called a Moore space, such that ̃k M (G, n) = { G k = n H 0 else Notation: P n (p) = M (Z/p, n − 1). A quick example: M (Z, n) = S n . and πk M (G, n) = { G k=n . 0 i<n 5. MOORE SPACES 17 We now show the existence of the Moore space for G = Z/`: Either consider the mapping ×` → Sn cone of S n Ð ×` / Sn S n KK KKK KKK ≃ KKK % M ∶= M (×`) / C(×`) / M (×`)/S n or the pushout ×` Sn / Sn / M (G, n) Dn+1 as definition. Check that these two definitions are the same. Get long exact sequence for the pair (M, S n ) from the diagram above: ̃i (M /S n ) Ð Hi (S n ) Ð → Hi (M ) Ð → Hi (M, S n ) = H → Hi−1 (S n ) For i ≠ 0, n, n+1 the vanishing of Hi (M ) is immediate. For i = 0 and i = 1 derive Hi (M, S n ) = 0 from the long exact sequence again, and for i = n ×` 0Ð →ZÐ →ZÐ →? Ð →0Ð →0 is exact and get ? = Z/`Z. For uniqueness suppose there is another simply connected CW-complex M ′ with ̃k (M ′ ) = {Z/` k = n H 0 else By induction and Hurewicz M ′ is (n-1)-connected. By Hurewicz homomorphism again f πk (M ′ ) ≅ Z/` so there is S n Ð → M ′ for which the induced πn (S n ) Ð → πn (M ′ ) via 1 ↦ [1] is surjection on πn . Hence by Hurewicz it is also sujrective on Hn . On the other hand 0 = `[f ] ∈ πk (M ′ ), meaning we have a homotopy H ∶ `f ≃ 0 completing the diagram Sn ×` / Sn . Dn+1 f / M′ H But M (Z/`, n) is a pushout so there is ϕ ∶ M (Z, `) Ð → M ′ arising from the pushout. From the triangle S n KK KK f KK KK KK % / M′ M (Z/`, n) ϕ 6. EILENBERG-MACLANE SPACES/POSTNIKOV APPROXIMATIONS 18 we get Z DD DD f∗ DD DD! ! Z/` ϕ∗ / Z/` on the n-th homology level. So ϕ∗ is a surjection and thus an isomorphism. This completes the proof that ϕ∗ is an isomorphism on H∗ . Now apply the previous proposition. 6. Eilenberg-Maclane spaces/Postnikov approximations Proposition 11. For all X (path-connected as always!) and for all n, there is Y =∶ Pn (X) with f ∶ X Ð → Y such that π (X) i ≤ n πi (Y ) = { i 0 i>n and πi (f ) is an isomorphism. Note again that this construction can be made functorial. Sketch of the proof. Let Yn = X. Assuming Yk is defined, then Yk+1 is the pushout ⋁g∈Hom(S k ,Yk ) S k / Yk / Yk+1 ⋁ g∈Hom(S k ,Y k) Dk+1 observe that πi (Yk ) = πi X for i ≤ n and 0 if k > i > n. And let Y = Y∞ = lim Yk and Ð→ f ∶X Ð → Y is the inclusion. Proposition 12. X Ð → Pn X is relative CW complex and for all X Ð → W such that πi (W ) = 0 if i > n then there is a unique (up to homotopy) dashed extension / Pn X DD DD DD " XD D W so we can think of the functor Pn as a localization. Proof. Exercise. Proposition 13 (Eilenberg-Mac Lane spaces). For all n ≥ 1, and abelian group G there exists a unique (up to homotopy) CW complex K(G, n) such that πi K(G, n) = { G i=n 0 else 6. EILENBERG-MACLANE SPACES/POSTNIKOV APPROXIMATIONS 19 Proof. n = 1 is handled with K(π, 1) = Bπ spaces (and G need not be abelian for that matter). So let n > 1 the existence is easy: let K(G, n) = Pn+1 M (G, n) and this is OK since πn M (G, n) ≅ Hn M (G, n) ≅ G. For uniqueness the idea is to use the last proposition: Suppose G i=n πi K ′ = { 0 else and construct the mapping M (G, n) Ð → K ′ as in proof of existence of M (G, n). Then ′ there is ϕ ∶ Pn+1 (M (G, n)) Ð → K through which the former map factors and which is π∗ equivalence and thus by Whitehead’s theorem a homotopy equivalence. Terminology: A product of EM spaces is called a GEM (generalized Eilenberg- Mac Lane) space. Example 6.1. πi (∏i (K(G, n))) = ⊕πi (K(Gi , i)) = Gi so we can construct spaces with any homotopy groups we want. ⌟ Example 6.2. K(Z, 1) = S 1 and K(Z, 2) = CP ∞ . ⌟ CHAPTER 2 Fibrations 1. Fiber bundles and fibrations Definition 6. Recall p ∶ E Ð → B has the homotopy lifting property with respect to X if for any solid diagram below there is a dashed extension ϕ completing the commutative diagram: /E X x< x x x x ϕ X ×I /B Definition 7. p ∶ E Ð → B is a (Hurewicz) fibration if p has HLP with respect to all X. Definition 8. p ∶ E Ð → B is called a Serre fibration if p has HLP with respect to all disks Dn , n ≥ 0. Remark. (Dn × I, Dn × {0} ∪ S n−1 × I) is homeomorphic to (Dn × I, Dn × {0}). Therefore p is a Serre fibration if and only if for every solid arrow diagram below there exists a map completing the commutative diagram: / o7 E o o ooo ooϕo o o ooo /B Dn × I Dn × {0} ∪ S n−1 × I Remark. Covering space ⇒Hurewicz fibration ⇒Serre fibration. Note that covering spaces are Hurewicz fibrations where homotopies lift uniquely. Idea of fibration: Consider the pullback (B path-connected) /E p−1(b) p {b} / B. If p is fibration, all fibers (i.e. all p−1(b)’s) should be homotopy equivalent and “vary continuously”. This comes historically from the notion of fiber bundles: 20 1. FIBER BUNDLES AND FIBRATIONS 21 Definition 9. A fiber bundle structure on E with fiber F is a map p ∶ E Ð → B such that for all b ∈ B there is a neighborhood U of b and homeomorphism h ∶ p−1(U ) Ð → U × F over B. h is called a local trivialization (and there are not unique). B is the base space of the bundle and E is the total space of the bundle and F is the fiber. We will sometimes denote p a fiber bundle with these data as F Ð →EÐ →B Example 1.1. p1 ∶ X × Y Ð → X gives a fiber bundle structure. ⌟ Example 1.2. E = I × [−1, 1]/ ≅ with the relation (0, v) ≅ (1, −v). The p ∶ E Ð → S 1 is a fiber bundle induced by I × [−1, 1] Ð → (I Ð → S 1 ) and fibers are [−1, 1]. ⌟ Example 1.3 (Projective spaces). p C∗ Ð → Cn+1 − {0} Ð → Cn+1 − {0}/C∗ with the standard trivialization Ui = (xi ≠ 0). One can create analogous spaces using R or H (the field of quaternions). The unit vectors versions will respectively be S1 Ð → S 2n+1 Ð → CP(n) S3 Ð → S 4n+3 Ð → HP(n) S0 Ð → Sn Ð → RP(n). ⌟ Theorem 1.1. Suppose p ∶ E Ð → B is a Serre fibration. Choosing b0 ∈ B, x0 ∈ F ∶= p−1(b0 ) then p∗ ∶ πn (E, F, x0 ) Ð → πn (B, b0 ) p coming from the map of triples (E, F, x0 ) Ð → (B, b0 , b0 ) is an isomorphism for all n ≥ 1 and so there is a long exact sequence p∗ ∂ → πn (B, b0 ) Ð → πn−1 (F, x0 ) Ð →⋯ ⋯Ð → πn (F, x0 ) Ð → πn (E, x0 ) Ð Note: ∂ has to be calculated via the isomorphism p∗ / πn−1 (F, x0 ) . n7 nnn n n nnn nnn πn (B, b0 ) O ≅ p∗ πn (E, F, x0 ) Proof. We prove the surjectivity only: let α = [f ] ∈ πn (B) for f ∶ (I n , ∂I n ) Ð → (B, b0 ). Let J = ∂I n − {sn = 0} as we had denoted before. We want f̃ ∶ (I n , ∂I n , J) Ð → (E, F, x0 ). Define f̃∣J ∶ J Ð → E via β ↦ x0 . 1. FIBER BUNDLES AND FIBRATIONS 22 Note that pf̃∣J = f ∣J . E begin a Serre fibration, the following diagram completes f̃∣J g3/ E g g g g g g p g g g g g g g / B. Dn = I n f Dn−1 × {1} ∪ S n−2 × I = J So pf̃(∂I n ) = b0 and f̃(∂I n ) ⊆ p−1(b0 ) = F and so [f̃] ∈ πn (E, F, x0 ) since f̃(J) = x0 and also p∗ [f̃] = [f ]. Proposition 14. p ∶ E Ð → B is a Serre fibration if and only if p has the HLP with respect to all weak CW pairs. Proof. Sufficiency is obvious. For necessity given a Serre fibration p ∶ E Ð → B and a relative weak CW complex X ↪ Y we want to complete X × I ∪ Y × {0} Y ×I q q q q q / q8 E /B to a commutative diagram. Look at the rectangle below with the left square being a pushout / X(i) × I ∪ X(i + 1) × {0} e e/2 E e e e e e e e e e e e e e e e e e e e e e / X(i + 1) × I /B ∐D × I ∐α∈I(i) S n(α) × I ∪ ∐ Dn(α)+1 So we get lifts ϕ(i) ∶ X(i + 1) × I Ð → E and then take the direct limit of these lifts. Proposition 15. A fiber bundle p ∶ E Ð → B is a Serre fibration. Remark. If B is paracomplact then p is actually a fibration. Proof. We want to complete the following In x In × I x x H x /E x; /B to a commutative diagram. Break I n ×I into smaller boxes Iα,j ∶= cα ×[tj , tj+1 ] and suppose α ∈ J where J is some nicely ordered indexing set. Let this partition be fine enough so that H(Iα,j ) ⊆ Uα,j ⊆ B so that p∣Uα,j is trivial and say hα,j ∶ p−1(Uα,j ) Ð → Uα,j × F is 1. FIBER BUNDLES AND FIBRATIONS 23 trivialization. Inductively assume that the lift has been defined (compatibly) on each cβ for β < α: ⊆ / In /5 cβ kkk E k kkk kkk k k kkk kkk⊆ / In × I cβ × I H /B To extend to cα consider the diagram ρ′ + /2 p−1(Uα,j ) ≅ / Uα,j × F / (∪β<α cβ ∩ cα ) × [tj , tj+1 ] ∪ cα × {tj } d d h d d d rr d d d d π rrr d d d r d d r d r d d d d xrrr d d d d ≅ n / / cα × [tj , tj+1 ] D × Iρ 4 Uα,j Dn × {0} ≅ also assume ϕα ∶ cα ×I Ð → E has been defined for t ≤ tj so constructing ψ will define ϕ∣α (−, t) for t ≤ tj+1 putting them together gives ϕα gluing together for all α gives lift ϕ ∶ I n × I Ð → E. One can check that ψ(d, t) = h−1(ρ(d, t), π2 ρ′ (d)) works. Note that , hψ((d, 0)) = (ρ(d, 0), π2 ρ′ (d)) = (π1 ρ′ (d), π2 ρ′ (d)) where , equality is because the solid arrow diagram commutes. η Example 1.4. Compute π3 (S 2 ) ≅ Z using the fibration S 1 Ð → S3 Ð → CP (1) = S 2 . So η is a fiber bundle hence is a Serre fibration. So we look at the long exact sequence on π∗ πe (S 1 ) Ð → π3 (S 3 ) Ð → π3 (S 2 ) Ð → π2 (S 1 ) Ð → π2 (S 3 ) Since the unviersal cover of S 1 is R, the R Ð → S 1 induces an isomorphism on πi for i ≥ 2 π3 (since it is a convering map) so πi (S 1 ) = 0 if i ≥ 2. Hence π3 (S 3 ) Ð → π3 (S 2 ) is isomorphism. 3 2 3 2 By Hurewicz π3 (S ) ≅ Z so π3 (S ) ≅ Z. In fact πi (S ) ≅ πi (S ) for i≥ 3. ⌟ Example 1.5. Similarly consider the fiber bundle S 3 Ð → S7 Ð → S 4 . Gives exact sequence πn (S 3 ) Ð → πn (S 7 ) Ð → πn (S 4 ) Ð → πn−1 (S 3 ) but the first map factors through zero since S 3 Ð → S 7 ≃ ∗ by cellular approximation so we get short exact sequences 0Ð → πn (S 7 ) Ð → πn (S 4 ) Ð → πn−1 (S 3 ) Ð → 0. In fact all of these short exact sequences are split. So πn (S 4 ) ≅ πn (S67) ⊕ πn−1 (S 3 ). In fact ΩS 4 ≅ ΩS 7 × S 3 which we will turn back to later! ⌟ 1. FIBER BUNDLES AND FIBRATIONS 24 Let X be a CW complex. Let f ∈ πn X. Then we can get Σf ∈ πn (ΣX) where ΣX = X × I/(x, 1) ≃ (x, 0) ≃ (∗, t) is the reduced suspension. Σ is a homotopy preserving functor (i.e. f ≃ g ⇒ Σf ≃ Σg Σ ∶ T op∗ Ð → T op∗ . So α = [g] ∈ πn (S) induces Σα ∶= [Σf ] ∈ πn+1 (ΣX). Theorem 1.2 (Freudenthal suspension). Suppose the connectivity of X is at least n, conn X ≥ n.Then Σ ∶ πn (X) Ð → πn+1 (ΣX) is an isomorphism for i ≥ 2n and surjective for i = 2n + 1. The functor Ω ∶ CG∗ Ð → CG∗ is the functor: ΩX = Map∗ (S 1 , X) with compact open topology. Then Σ is the left adjoint to Ω: Θ ∶ HomCG∗ (ΣX, Y ) ≅ HomCG∗ (X, ΩY ) in fact this correspondence is a hemeomorphism. Let us denote Θf by f a . Then idaΣX = Θ(idΣX ) ∈ Hom(X, ΩΣX) where [f ] ∈ πn (X) = [ΣS n−1 , X] and [f a ] ∈ πn−1 (ΩX). Theta in fact induces an isomorphism πn X Ð → πn−1 (ΩX) a via [f ] ↦ [idΣX .f ] ∈ πn (ΩΣX) ≅ πn+1 (ΣX) which is just Σf . Recall that f ∶ X Ð → Y can be turned into a cofibration / M (f ) EE EE EE E" XE E Y f where the injection X ↪ M (f ) is a cofibraiton and M (f ) ↠ Y is a fibration. Given X Ð →Y. Let Ef = {(x, p) ∶ f (x) = p(0)} ⊆ X × Y I be the pullback / YI X ×YI ev(0) /Y X Now p(1) ∶ Ef Ð → Y via (x, p) ↦ p(1) is a fibration. So we get a commutative diagram E > f AA } AAp(1) ∗ }} } AA }} A } } /Y X f where ∗ is a strong deformation retract. Then p(1)−1(y) is a homotopy fiber of f . If Y is path connected the homotopy type of p(1)−1(y) is independent of y. CHAPTER 3 Spectral Sequences r Consider a sequence of bigraded modules (vector spaces) Epq r ≥ n ≥ 0 for some starting r point n (usually 0, 1 or 2), together with differentials d with degree (−r, r − 1), r r dr ∶ Epq Ð → Ep−r,q+r−1 r+1 satisfying dr ○ dr = 0 and isomorphisms Epq ≅ Hpq (E r , dr ) = ker drpq / im drp+r,q−r+1 . The r + r sequence E = {Epq , dpq }r is called a spectral sequence. These form a category where r′ r compatible with the differentials. Ð → Epr arrows are sequences of bigraded maps Epq r ≠ 0 ⇒ p ≥ 0 and q ≥ 0. E + is a first quadrant spectral sequence if Epq qO ● gOOO ● ● ● OOO d OO2O OOO d1OO ●o ● d0 ● ● ● ● ● ● / p r is p + q. Thus the total degree of dr is always −1. Suppose E + Total degree of α ∈ Epq r is first quadrant. Then for all p, q, Epq stabilizes, i.e. given p, q there is k such that, ∞ k r r ≥ k ⇒ Epq ≅ Epq . We call these terms Epq . We will consider the first quadrant case from now on. If there exists a finite filtration 0 = F−1 Hn ⊆ F0 Hn ⊆ ⋯ ⊆ Ft Hn = Hn then ∞ Epq = Fp Hp+q /Fp−1 Hp+q r is a spectral sequence for which Epq ⇒ Hpq . i π Example 0.6 (Leray-Serre spectral sequence). Let F Ð →EÐ → B be a Serre fibration. We want B to be simply connected. Then there exists a first quadrant spectral sequence [Serre’s PhD] with 2 Epq = Hp (B, Hq (F )) ⇒ Hp+q (E) 25 3. SPECTRAL SEQUENCES 26 ∞ i.e. ⊕p+q=n Epq = Hn (E) up to extensions (with is equality if we work over a field). Also note that Hp (B, Hq (F )) ≅ Hp B ⊗ Hq (F ) ⊕ TorZ1 (Hp−1 B, Hq F ) where the Tor term is zero if we are over a field. Consider turning ∗ Ð → X into a fibration, i.e. P X = ∗ ×X X I = {ϕ ∶ I Ð → X ∶ ϕ(0) = ∗} where π = ev1 ∶ P X Ð → X is the evaluation map at 1, ϕ ∶↦ ϕ(1) which is a fibration. And π −1(∗) = ΩX. So we get the path-loop fibration ΩX Ð → PX Ð → X. Suppose X is path connected in which case P X is contractible, P X ≃ {∗}. Now we apply the Leray-Serre spectral sequence when X = S n : 2 Epq ⇒ Hp+q (E) = 0 unless n = q = 0. 2 Here Epq = Hp B ⊗ Hq F since Hp B is free and H0 F ≅ Z from the long exact sequence of homotopy. So the second page looks like this: q O 2 ? 0 0 ? 1 ? 0 0 ? 0 Z 0 0 Z 0 n−1 1 / p n ∞ = 0 so there is r such that drn0 (1) ≠ 0 for some r and is fact is an We know that En0 isomorphism. So 2 ≅Z E0,n−1 2 ≅ Z so Hn−1 F ≅ Z. Also derive But that means that Z ⊗ Hn−1 F ≅ H0 B ⊗ Hn−1 F ≅ E0,n−1 2 that E∗,i = 0 if 0 < i < n − 1. By same argument H2n−2 F ≅ Z and in general Hi F ≅ { Z n − 1 ∣ i, i ≥ 0 . 0 else Similarly if ΩX ≃ S 1 get path loop fibration S 1 Ð → PX Ð → X. A spectral sequence argument shows that Z 2 ∣ i, i ≥ 0 Hi (X) ≅ { . 0 else 1. CONSTRUCTION OF SPECTRAL SEQUENCES 27 ⌟ 1. Construction of spectral sequences There are generally two ways of getting spectral sequences: filtrations and exzact couples. 1.1. From exact couples. We will talk about exact couples here: An exact couple is a (non-commuting) diagram of modules E ∶ D `A A k i AA A E /D } } } ~}} j which consists of two modules and three maps and is exact at each vertex, e.g. ker j = im i. Let d = jk and we get d2 = 0. Then we get the derived couple i′ =i / iD E 1 ∶ iD eJ JJ tt JJ t t J yttj ′ =ji−1 k=k′ H(E, d) that is the maps are determined by, i′ = i∣D , j ′ (i(d)) = [j(d)], k ′ [e] = ke. Check that these are well-defined and E 1 is an exact couple. One can iterate this procedure and get the r-th derive couple E r ∶ Dr = ir (D) i hPPP PPPk PPP PPP / Dr r r j (r) =ji−r rrr rrr xrrr E r = H(E r−1 ) i.e j (r) ir (d) = [j(d)]. Suppose now that D, E are bigraded Dpq , Epq , such that i has bidegree (1, −1); i ∶ Dpq Ð → Dp+1,q−1 , j has bidegree (0, 0) and k has bidegree (−1, 0). Then r−1 r j (r) has bidegree (−r, r), dr has bidegree (−r − 1, r). We have a spectral sequence (Epq ,d ) as above. Example 1.1 (Bockstein Spectral Sequence (BSS), due to Browder). Consider the short ×` π exact sequence 0 Ð →ZÐ →ZÐ → Z/` Ð → 0 where ` is a prime. Tensor this with C∗ X (or any free chain complex) and get short exact seqeunce of complexes 0Ð → C∗ (X, Z) Ð → C∗ (X, Z) Ð → C∗ (X, Z/`) Ð → 0. 1. CONSTRUCTION OF SPECTRAL SEQUENCES 28 The long exact sequence on H∗ gives an exact couple ×` H∗ (X, Z) gOOO OOOj OOO OO / H∗ (X, Z) . pp ppp p p p π pw pp ∗ H∗ (X, Z/`) The associated spectral sequence is the BSS. ⌟ ▸ Exercise 2. Calculate BSS replacing C∗ X with the chain complex `k → M0 = Z Ð →0Ð →⋯ ⋯Ð →0Ð → M1 = Z Ð given any integer k ≥ 1. We do the first cases here for illustration: when k = 1 the exact couple is / H∗ C H∗ C fM MMM q qq MMM qqq M q M q k M xqq j H∗ (C ⊗ Z/`) i where k is the connecting homomorphism. In degree 0 we get Z/` / Z/` 0 aCC CC C k C ≅ {{{ { }{{ j Z/` and in degree 1 0 `AA AA AA /0. } } }} ~}} Z/` Thus the BSS is given by E01 = Z/` ≅ E11 = Z/`, d = kj ∶ E11 ≅ Z/` Ð → E01 ≅ Z/` E∗2 = 0. In k = 2 case, degree 1 0 _? ?? ?? ?? ? Z/` degree 0 ×` /0 / Z/`2 Z/`2 7 o a D D z DD o o zz DD o o zzπ D z δ δ D o o }zz Z/` 1. CONSTRUCTION OF SPECTRAL SEQUENCES 29 The differential is d1 = jk = πδ ∶ E11 ≅ Z/` Ð → Z/` ≅ E01 [1] ↦ πδ[1] = π[`] = 0. Now passing to the derived couple is given by 0 _? ?? ?? ?? ? Z/` ×` / Z/` . /0 7 Z/` aC o C { o C { o CC {{ CC o o o[δ] {{ ji−1 C { }{ o Z/` with differential of the spectral sequence being 1 d2 = j ′ k ′ = (π )[δ] ∶ E12 ≅ Z/` Ð → Z/` ≅ E02 ` 1 [1] ↦ π [`] = π[1] = [1] ` which is an isomorphism and therefore E∗3 = E∗∞ = 0. 1.2. From filtrations on chain complexes. Let C be a chain complex. A filtration on C is a sequence of sub-chain-complexes Fi = Fi (C) 0 = F−1 ⊆ F0 ⊆ ⋯ ⊆ Fp ⊆ ⋯ ⊆ C. The filtration is exhaustive (or co-omplete ) if C = ∪Fp C (complete corresponds to the limit being 0). Also assume C is non-negatively graded so that we get a first quadrant spectral sequence. The zero page is 0 = (Fp C)p+q /(Fp−1 C)p+q . Epq and d0 is induced by the differentials on quotient. q O ● ● ● 2 ● 1 ● ● ● ● 0 ● ● ● ● / F0 F1 /F0 F2 /F1 F3 /F2 1 0 So in first page Ep,q = Hp+q (Ep,∗ , d0 ). Now let Arp = {c ∈ Fp C ∶ d(c) ∈ Fp−r C} p 1. CONSTRUCTION OF SPECTRAL SEQUENCES 30 i.e. c ∈ Fp is a cycle mod Fp−r C; you may call is an approximate cycle. As a piece of notation also define η p ∶ Fp C Ð → Fp C/Fp−1 C = Ep0 (and we are dropping q’s from our notations). Then Zpr = ηp (Arp ) ⊆ Ep0 r+1 0 Bp−r = ηp−r d(Arp ) ⊆ Ep−r r Zp∞ = ∩∞ r=1 Zp r Bp∞ = ∪∞ r=1 Bp Observe that we have filtration 0 ⊆ Bp0 ⊆ Bp1 ⊆ ⋯ ⊆ Bp∞ ⊆ Zp∞ ⊆ ⋯ ⊆ Zp0 = Ep0 The r-th page would be Zpr /Bpr : Arp ∩ Fp−1 C = Ar−1 p−1 Zpr ≅ Arp /Ar−1 p−1 Arp + Fp−1 C Arp ≅ Epr = Zpr /Bpr ≅ r−1 dAr−1 d(Ar−1 p+r−1 + Fp−1 (C) p+1 ) + Ap−1 and the most important part defining the differential: r drp ∶ Epr _ _ _/ Ep−r OO OO defined since d2 = 0 r Zpr _ _d _/ Zp−r OO Arp OO / Arp−r d d lands in here since d2 = 0 drp is unique since the vertical maps are surjections. To see that drp exists (making the diagram commute) it suffices that d′ (Bpr ) = 0. But since d im(dAr−1 p+r−1 ) = 0 this is the case. r 2 r+1 r Clearly (d ) = 0. Finally check that E ≅ H∗ (E ) to complete the construction of our spectral sequence. Example 1.2. Let our chain complex be concentrated only in degrees 3 and 4, where C4 ≅ Ze31 ⊕ Ze22 , C3 = Ze21 ⊕ Ze12 and we have chosen the indices this way since if the differential d ∶ C4 Ð → C3 is given via (a, b) ↦ (a + kb, b) then F1 C = Ze21 , F2 C = Ze12 ⊕ Ze22 ⊕ Ze21 , F3 C = C 1. CONSTRUCTION OF SPECTRAL SEQUENCES qO E ∶ 0 qO 0 Z Z 0 0 0 Z Z 0 0 0 0 E ∶ 1 / 0 Z 0 0 0 0 Z/k Z 0 0 0 0 / p qO E2 ∶ 31 p qO E3 ∶ 0 Z gOOO 0 Z 0 0 0 0 0 0 Z 0 0 0 0 Z/k 0 0 0 / OOO OOO OOO 0 0 kZ 0 0 / p p ⌟ ▸ Exercise 3. Compute the differentials in the previous example. In a filtration of positively graded chain complex C, that is co-complete we have the convergence E r ⇒ Fp Hp+q /Fp+1 Hp+q and one can check that this is the same as the spectral sequence associated to the exact couple H∗ Fi gN shift NNN NNN N / ⊕i H∗ Fi+1 . m mm mmm m m vm H∗ (Fi+1 /Fi ) Let π ∶ E Ð → B be a fibration and assume that B has a CW structure. Let f ∶ B ′ Ð → B be a weak homotopy equivalence. The by long exact sequence on π∗ , f ′ ∶ π −1B ′ Ð → B ′ is also a weak homotopy equivalence. Let Bn be the n-skeleton of B. Consider the filtration Fn = im C∗ (π −1Bn ) where π −1Bn Ð → E is the pullback. Then F−1 = 0 and filtration is co-complete. Since /E Dr H O H H H H$ ? π −1(Bn ) 1. CONSTRUCTION OF SPECTRAL SEQUENCES 32 factors for some n through π −1Bn since Dn is compact. So the associated spectral sequence is a Serre spectral sequence. As always Ep0 = Fp /Fp−1 Ep1 = H∗ (Fp /Fp−1 ). Note 1.3 (Recall Cellular homology). Let Cncell = ⊕α∈I(n) Zα be the freely generated Z-module generated by n-cells in a (not weak) CW structure. To define a map cell cell Cn X Ð → Cn−1 . We have a mapping that is the composition f (α) q → Xn−1 /Xn−2 ≅ Sαn−1 ÐÐ→ Xn−1 Ð pβ Sβn−1 Ð→ Sβn01 . ⋁ β∈I(n−1) Let pβ ⊕ Z Ð → Z be the projection. And consider deg(pβ qf (α)1β ). (Notice S n−1 is compact cell so this will be a finite sum.) This induces a differential dn ∶ Cncell Ð → Cn−1 . ⌟ The following squares are pullbacks: π −1(Bn−1 ) Bn−1 / π −1B /E n π /B / Bn and we have short exact sequence 0 Ð → Fp−1 Ð → Fp Ð → Fp /Fp−1 Ð → 0 in the form 0Ð → C∗ (π −1Bn−1 ) Ð → C∗ (π −1Bn ) Ð → C∗ (π −1Bm , π −1Bn−1 ) Ð → 0. So H∗ (Fp /Fp−1 ) ≅ H∗ (π −1Bn , π −1Bn−1 ). But by excision H∗ (Bn , Bn−1 ) ≅ Hn ( ∐ Dn , ∐ S n−1 ) α∈I(n) α∈I(n) H∗ (π Bn , π Bn−1 ) ≅ Hn ( ∐ π Dn , ∐ π −1S n−1 ). −1 −1 −1 α∈I(n) α∈I(n) Consider i∗ Dn ∶ π −1Dn /E π /B Dn i ≃ ∗ =∶ 0 so by proposition 4.62 of Hatcher, i∗ Dn is fiber homotopy equivalent to 0∗ Dn , i.e. there are maps f, g making the following diagram commute. u i∗ DnG g i 5 0∗ D f GG ww GG w w π GG# {www π Dn n 1. CONSTRUCTION OF SPECTRAL SEQUENCES 33 H where by homotopies over Dn , gf ≃ idi∗ Dn (for some homotopy H) and f g ≃ id0∗ Dn . Thus this induces an equivalence of pairs (i∗ Dn , i∗ S n−1 ) ≅ (0∗ Dn , 0∗ S n−1 ) ≅ (Dn × F, S n−1 × F ) where F = π −1(∗). Now H∗ (Fn , Fn−1 ) ≅ H∗ (∐ π −1Dn , ∏ π −1S n−1 ) ≅ ⊕α∈I(n) H∗ (π −1Dn , π −1S n−1 ) ≅ ⊕H∗ (Dn × F, S n−1 × F ) ≅ Cncell (B) ⊗ H∗ F. The last equality above holds because, Zα ⊗ H∗ (F ) in degree n ⋯ / H∗ (S n−1 × F) / H∗ (D × F ) n 0 / H∗ (D × F, S n−1 × F ) n Kunneth 0 /H ̃∗ (S n−1 ) ⊗ H∗ (F ) / H∗ (S n−1 ) ⊗ H∗ (F ) / H∗ (F ) /0 Note 1.4. If the base is not simply connected the above argument does not work. (Then say at n = 0, in S n−1 × F we can have disjoint copies of F and the kernel is not the above one??) ⌟ 1 = Cpcell (B) ⊗ Hq (F ) and d1 = dcell ⊗ id so So Epq E 2 ≅ H∗cell B ⊗ H∗ F ⊕ Tor(H∗ B, H∗ F ) ignoring the degrees. ▸ Exercise 4. Work out the “fibration” S 1 ∨ S 2 Ð → S 1 : You may turn this into a fibration −1 1 1 1 FÐ → π (∗) Ð → PS ,π ∶ PS Ð → S . Here F is homotopy equivalent to the covering space of S1 ∨ S2. /⋯ CHAPTER 4 H-spaces and algebras 1. H-spaces Definition 10. An H-space (H, ϕ) is a topological space H with base point e and a product map ϕ∶H ×H Ð →H (x, y) ↦ x.y making H ∨ HG GG GGfold GG GG # H ×H ϕ / H commute relative to {e} and only up to homotopy. Definition 11. H is homotopy associative if ϕ×idH H ×H ×H idH ×ϕ / H ×H ϕ H ×H ϕ /H commutes relative to {(e, e, e)} and up to homotopy. Definition 12. H has homotopy inverse ̂. ∶ H Ð → H if ̂ e = e and H: × H ∆ vvv H vv vv id×̂. / {e} commutes relative to {e} and up to homotopy. 34 H × HH HH ϕ HH HH $ 4H 1. H-SPACES 35 Example 1.1. Topological groups are homotopy associative H spaces with homotopy inverse (take all homotopies to be trivial). So are loop spaces ΩX: ΩX = Map∗ (S 1 , X) with compact open topology. The box product is ϕ ∶ ΩX × ΩX Ð → ΩX (α, β) ↦ α traversing the first and then the second path with twice speed and ̂. ∶ ΩX Ð → ΩX α ↦ α ∶ t ↦ α(t − 1). ⌟ Definition 13. A homotopy associative H-space with homotopy inverse is called an Hgroup. One can dualize all the above notions: Definition 14. A co-H-space is a pair (C, ψ) with C a topological space with base point {e}, and coproduct ψ ∶ C Ð → C ∨ C making C; ∨ C x xx x x xx xx / C ×C C ∆ ψ commute up to homotopy. Homotopy co-associativity and homotopy co-inverses are defined likewise. A co-H-group is a homotopy co-associative co-H-space with homotopy co-inverse. Example 1.2. ΣX (the reduced suspension) is a co-H-group, where the coproduct is the pinching morphism along the equator: // // / // /// // // / / / ψ ∶ // Ð→ /// // // / // // / ⌟ Proposition 16. (1) If Y is an H-group and X is any space, then [X, Y ]∗ is a group. (2) If X is a co-H-group and Y is any space, then [X, Y ]∗ is a group. (3) If Y is an H-group and X is a co-H-group then [X, Y ]∗ is abelian (and both multiplications in the above cases give the same result). 2. (CO)-ALGEBRAS 36 Proof. In (1) the group operation is the composition f ×g ∆ ϕ f.g ∶ X Ð→ X × X ÐÐ→ Y × Y Ð →Y and in (2) f ∨g ψ fold f.g ∶ X Ð → X ∨ X ÐÐ→ Y ∨ Y ÐÐ→ Y. For (3) look at [Bredon, p. 442]. Remark. The question of studying pairs of spaces X and Y for which ΩX ≃ ΩY (deev looping!) is related to the above ideas. In fact, recall the fibration F = ΩX Ð → P X Ð→ X and observe that P X ≃ ∗. The long exact sequence on π∗ looks in part like ≅ ⋯Ð → πn (ΩX) Ð → πn (P X) = 0 Ð → πn (X) Ð → πn (ΩX) Ð →⋯ thus πn X ≅ πn−1 (ΩX). So it is hopeless to try to recover X and Y from the loop spaces. One need more structure on the spaces to tackle the problem. 2. (co)-Algebras Fix R (or k), your favorite underlying commutative ring or field. An algebra is a triple → A and unit µ ∶ R Ð → A. (A, ϕ, µ) with multiplication ϕ ∶ A ⊗ A Ð / A ⊗ A o id⊗µ A ⊗ R t JJJJ tttt JJJ∼J ttttt ϕ JJJJ t t JJJJ ttttt ∼ J ttt µ⊗id R ⊗ AJJJ A We may want to have a grading on A and want ϕ to respect the grading, ϕ(Ak ⊗As ) ⊆ Ak+s , im(µ) ⊆ A0 . ϕ is said to be associative: A⊗A⊗A id⊗ϕ ϕ⊗id ↺ A⊗A ϕ / A⊗A ϕ /A (without the position of parentheses matter) and is (graded-)commutative if A ⊗ AF T FF FF FF FF " commutes such that T (α ⊗ β) = (−1) ∣α∣∣β∣ A / A⊗A x xx xxϕ x x x| x β ⊗ α, where ∣.∣ denotes the degree. A coalgebra is the dual object: a triple (C, ∆, η) where C is an R-module, ∆ ∶ C Ð →C ⊗C and η ∶ C Ð → R and everything is defined dually likewise! A (co)algebra is connected if A0 = R and it is non-negatively graded. 3. HOPF ALGEBRAS 37 Remark. When the coalgebra C is connected then for α ∈ C n ∆(α) = ∑ αi′ ⊗ αj′′ ∈ (C ⊗ C)n = ⊕i+j=n Ci ⊗ Cj i+j=n idC ⊗ η(∆(α)) = ∑ αi′ ⊗ η(αj′′ ) = αn′ ⊗ α0′′ = α ⊗ 1 i+j=n ´¹¹ ¹¸ ¹ ¹ ¶ =0if j≠0 So we can change αn′ , α0′′ so that αn′ = α and α0′′ = 1. Similarly for the other factor. So we can write ′′ ∆(α) = α ⊗ 1 + 1 ⊗ α + ∑ αi′ ⊗ αn−i 0<i<n and therefore we have a reduced diagonal ∆ given via ∆(α) = ∆(α) − (α ⊗ 1 + 1 ⊗ α). If ∆(α) = 0 then α is called primitive, and the R-module of such elements is denoted by P (C). Example 2.1. H ∗ (X, R) is an associative commutative algebra and H∗ (X, R) is a co-associative co-commutative coalgebra and are connected if X is path-connected. ⌟ 3. Hopf algebras Definition 15 (Hopf algebra). A Hopf algebra is a quintuple (A, ϕ, ∆, µ, η) where (A, ϕ, µ) is an algebra and (A, ∆, η) is a coalgebra and the structures are compatible: ϕ is a coalgebra map (or equivalently ∆ is an algebra map), i.e. ∆⊗∆ / A ⊗ AO A⊗A⊗A⊗A OOO ∆(A⊗A) OOO ϕ OOO id⊗T ⊗id OO' A⊗A⊗A⊗A A OOO OOO OOO ϕ⊗ϕ OOO O' t ϕA⊗A A⊗A commutes (but be aware that T introduces signs). Remark. In a Hopft algebra A, P (A) is a Lie algebra with product [α, β] = αβ − (−1)∣α∣∣β∣ βα 3. HOPF ALGEBRAS 38 where the products on the right hand side are multiplication in A. For if α, β ∈ P (A) ∆[α, β] = ∆(α)∆(β) − (−1)∆βδα since ∆ is linear and an algebra map = (α ⊗ 1 + 1 ⊗ α)(β ⊗ 1 + 1 ⊗ β) − (−1)∣α∣∣β∣ (β ⊗ 1 + 1 ⊗ β)(α ⊗ 1 + 1 ⊗ α) = (αβ − (−1)∣α∣∣β∣ βα) ⊗ 1 + 1 ⊗ (αβ − (−1)∣α∣∣β∣ βα) = [α, β] ⊗ 1 + 1 ⊗ [α, β]. Example 3.1. To be safe (?) let k be an underlying field and (X, ϕ) be a pathconnected H-space. So we know that H 0 (X, k) ≅ k. Then H ∗ (X, k) is a connected, commutative, Hopf algebra. If X = ΩY or is a topological group then H ∗ (X, k) will be co-associative: ∆ ∶ X Ð → X × X induces the isomorphism fitting into the following diagram: ≅ / H 8 (X × X) RRR RRR RR H ∗ (∆) ϕ RRRRR R( H ∗ (X) ⊗ H ∗ (X) H ∗X ϕ Since X is an H-space we also have X × X Ð → X which induces H ∗ (X) s H ∗ (ϕ) ∗ / H (X × X) ≅ / H ∗ (X) ⊗ H ∗ (X) 6 ∆ completing the construction of our Hopf algebra structure. Similarly H∗ (X, k) is a Hopf algebra. If X = ΩY then this is associative, but typically not commutative. However if X = Ω2 Y then H∗ (X, k) will be commutative. Here Ω2 Y ∶= ΩΩY fitting in the adjunction [Σ2 X, Y ] ≅ [ΣX, ΩY ] ≅ [X, Ω2 Y ]. ⌟ Note 3.2. Note that Hom(A × B, Y ) ≅ Hom(B, Hom(A, Y )) true as sets or as mapping spaces if spaces satify basic topological properties! Similarly Map∗ (S 1 ∧ X, Y ) ≅ Map∗ (X, Map∗ (S 1 , Y )) where X ∧ Y = X × Y /X ∨ Y is the smash product. Since ΣX ≅ S 1 ∧ X we have Map∗ (ΣX, Y ) ≅ Map∗ (X, ΩY ) which is compatible with homotopies and therefore [ΣX, Y ] ≅ [X, ΩY ] i.e. Σ and Ω are adjoint functors (actually π0 Map∗ (ΣX, Y ) = [ΣX, Y ]). ⌟ 3. HOPF ALGEBRAS 39 Let R[α] be the polynomial algebra over R on one generator α with positive even degree ∣α∣. Let α be primitive so that ∆α = 1 ⊗ α + α ⊗ 1 and in particular ∆(αn ) = (∆α)n = (α ⊗ 1 + 1 ⊗ α)n n n = ∑ ( )αi ⊗ αn−i i=0 i Note that the coproduct on generators always determine the coproduct. And here since this algebra is free the above formula gives a well-defined coproduct. Let E(α) = ∧R [α] be the exterior algebra on generator α with the degree of α being odd. And assume α is primitive. Then E(α) is a free graded commutative algebra on α. This will imply that the formula for ∆ is well-defined. Example 3.3. Say char(k) ≠ 2 and ∣α∣ = 2 contrary to the above assumption. We want to have ∆(α) = 1 ⊗ α + α ⊗ 1. Then 0 = ∆(α2 ) = (∆α)(∆α) = (1 ⊗ α + α ⊗ 1)(1 ⊗ α + α ⊗ 1) = 1 ⊗ α2 + α ⊗ α + (−1)∣α∣∣α∣ α ⊗ α + α2 ⊗ 1 =0 =0 = 2α ⊗ α ≠ 0. k Therefore we do not get a Hopf algebra unless in characteristic 2. More generally k[α]/(αp ) with α primitive, determines a Hopf algebra if and only if char(k) = p. ⌟ If A and B are Hopf algebras then so is A ⊗ B in the obvious way. Theorem 3.1. If char(k) = 0, A is a (graded-)commutative associative connected Hopf algebra. Then as algebras (and not Hopf algebras) we have an isomorphism A ≅ E(a1 , ⋯, an , ⋯) ⊗ k[b1 , ⋯, b` , ⋯] with ∣ai ∣’s odd and ∣bj ∣’s even. Notation: We denote the above by ⋀(a1 , ⋯, an , ⋯, b1 , ⋯, b` , ⋯) the free graded commutative algebra on the ai and bj . Example 3.4. We have seen using Serre’s spectral sequence that Hi (ΩS n , Z) ≅ { Thus H i (ΩS n , Q) ≅ { Z n − 1∣i . 0 otherwise Q r − 1∣i 0 otherwise. The latter is a Hopf algebra so if n is odd we have H i (ΩS n , Q) ≅ Q[a] with ∣a∣ = n − 1. 3. HOPF ALGEBRAS 40 Working overR = Z let ∣a∣ be even and ∣cn ∣ = n∣a∣ then the divided powers algebra Γ(a) = ⊕n∈Z≥0 cn Z is defined via c0 = 1, c1 = a and ci cj = (nj)cn . Then ∆(cn ) = ∑i+j=n ci ⊗ cj . And note that Γ(a) ≅ k[a] if char k = 0 (in fact a Hopf algebra isomorphism if the isomorphism is defined in a convenient way). Finally Γ(a)∗ = HomZ (Γ(a), Z) ≅ R[a] as Hopf algebras. ⌟ We work out the proof only when An is finite dimensional for all n, i.e. A is of finite type. A CW-complex X is called finite type if H∗ X is finite type so this is a concrete case where the following prove goes through. Proof. Say A has a generating set a1 , ⋯, an , ⋯ with ∣ai ∣ ≤ ∣ai+1 ∣. Set A(0) = k, and A(n) the subalgebra (not sub-Hopf algebra) generated by a1 , ⋯, an . It turns out that A(n) is also a sub-Hopf algebra: ∆(ai ) = ai ⊗ 1 + 1 ⊗ ai + something in ⊕ p+q=∣ai ∣,p>0 Ap ⊗ Aq so the q’s above have to be ∣q∣ < ∣ai ∣ therefore the extra terms are in ⊕Ap ⊗ Aq ⊆ A(n − 1) ⊗ A(n − 1). We conclude that ∆A(n) ⊆ A(n) ⊗ A(n). Assume A(n − 1) ≅ γ(a1 , ⋯, an−1 ) if ∣an ∣ even. f There is a surjection A(n − 1) ⊗ k[an ] Ð → A(n) (uses commutativity). It is injective as well: say f (α) = ∑0≤i≤t αi ain , αi ∈ A(n − 1). ∆f α = ∑ ∆(αi ain ) 0≤i≤t = ∑ (∆αi )(∆an )i since Delta is an algebra map. 0≤i≤t But observe that (∆αi )(∆an )i = (αi ⊗ 1 + A(n − 1) ⊗ A(n − 1) + 1 ⊗ αi )(an ⊗ 1 + A(n − 1) ⊗ A(n − 1) + 1 ⊗ an )i . so ∆f (α) = ∑0≤i≤t (αi ⊗ 1)(an ⊗ 1 + 1 ⊗ an )i . Let I ⊆ A(n) be the ideal generated by A(n − 1) and a2n . Consider the composition ∆ A(n) Ð→ A(n) ⊗ A(n) Ð → A(n) ⊗ A(n)/I The image of f α under composition is therefore i−1 i ∑ (αi ⊗ 1)(ian ⊗ an ) + ∑ (α ⊗ 1)(an ⊗ 1) 0≤i≤t 0≤i≤t 3. HOPF ALGEBRAS 41 so if f (α) = 0 then ∆f (α) = 0 thus we should have ∈ A(n) ∑ iαi ai−1 n = 0. 1≤i≤t If we take α ∈ ker f of lowest non-zero degree the we have found an element of lower degree in the kernel which is a contradiction. So α has to be of degree zero. But f is injective on degree zero and we are done with proving the injectivity. We conclude that f is an isomorphism. The case of ∣an ∣ being odd is similar, so taking unions (direct limits) we get that A ≅ γ(a1 , c . . . , an ). Corollary 2. Hi∗ (ΩX; Q) ≅ ⋀(a1 , ⋯, an , ⋯). Remark. The quesiton of realizability of an algebra as the cohomology of a space or more specifically the loop space of some topological space in rational coefficient will be done later in this course. ∃X, H ∗ (ΩX, Q) ≅ ⋀(a1 , ⋯, an , ⋯). The question is difficult integrally though. It is known however that any polynomial algebra Z[x1 , ⋯, xk ] can be realized with a space H ∗ (X, Z) ≅ Z[x1 , ⋯, xk ]. CHAPTER 5 Localization (rationalization) There are two approaches: Serre classes (C classes) of 50’s and localization of 60’s. We will study the latter in what follows. All spaces will be simply connected, CW complexes. Definition 16. For a topological space X, suppose there is a space X(0) and a map ̃∗ (X(0) , Z) is a vector η∶X Ð → X(0) such that η induces an isomorphism on H∗ (−, Q), H ̃ space over Q and for all f ∶ X Ð → Y such that H∗ (Y, Z) is a Q-vector space, there is a unique map ϕ ∶ X(0) Ð → Y (up to homotopy), making η / X(0) CC CC ϕ C f CC ! XC C Y commute up to homotopy. Then X Ð → X(0) is called the rationalization of X. ̃∗ (S n , Z) is a Q-vector space. This is a Aside: There’s a construction to show that H construction by Sullivan to make a rational sphere by a telescope construction from maps ×p S n Ð→ S n for all primes appearing infinitely many times. Proposition 17. There is a functorial localization ηX ∶ X Ð → X(0) on the category of simply connected CW complexes. Denote ηX (f ) by f(0) , then f(0) is homotopy equivalence if and only if H∗ (f ) ⊗ Q is an isomorphism if and only if (using simply connectedness) π∗ (f ) ⊗ Q is an isomorphism (for certain tensor product). Localizations preserves fibrations (everything is simply connected again). Recall K(Π, n) for abelian group Π: πi K(Π, n) = { Π i=n 0 otherwise. By this we mean a more general construction than what we presented above which covers additive abelian groups of infinite generation; In gist, take a presentation of any abelian group G, 0 Ð → ⊕Z Ð → ⊕Z Ð →GÐ → 0. Consider a map ∨S n Ð → ∨S n such that passing to 42 ? 5. LOCALIZATION (RATIONALIZATION) 43 H∗ gives the injective homomorphism of the sequence. Construct a space K(G, n) by the gluing map. If we assume K(Π, n) is a CW complex then it is determined up to homotopy. We want to calculate the localizations of these spaces. Note that (0.5) ΩK(Π, n) ∼ K(Π, n − 1). In fact this is a weak homotopy equivalence from what we know. But a result of Milnor implies that loop spaces are CW-complexes and therefore by Whitehead’s theorem, the computation of π∗ as follows, results the existence of the homotopy equivalence 0.5 of CW-complexes. Using the long exact sequence on π∗ for the fibration ΩK(Π, n) Ð → P K(Π, n) Ð → K(Π, n) so π∗−1 ΩK(Π, n) ≅ π∗ K(Π, n).1 Example 0.6 (Localization of CP ∞ ). The fibrations S 1 Ð → S 2n+1 Ð → CP n form commutative diagrams with compatible actions: S1 / S 2n+1 / CP n S1 / S 2n+3 / CP n+1 . Taking the direct limit we get a fibration S1 Ð → S∞ Ð → CP ∞ . We know that π∗ (lim(Xi )) ≅ lim π∗ (Xi ) since S n is compact, and since π∗ (S n ) Ð → π∗ (S n+1 ) Ð→ Ð→ ∞ is the zero map we get that π∗ (S ) = 0 and therefore S ∞ ∼ ∗ since S ∞ has the a natural CW-structure (or directly see that S ∞ ∼ ∗). S 1 ∼ K(Z, 1) as we know.2 We conclude that ΩCP ∞ ∼ S 1 and Z i=2 πi (CP ∞ ) = { 0 otherwise thus CP ∞ = K(Z, 2). ∞ Note also that π∗ (X(0) ) ≅ π∗ X ⊗ Q so CP(0) = K(Q, 2). So ? ∞ CP(0) ∼ K(Q, 2) ∼ ΩK(Q, 3) ∞ and in particular CP(0) is a loop space. Before doing another example we state a general fact that will be lectured on later: 1It is more generally the case that π ∗−1 (ΩX) ≅ π∗ (X) if X is simply connected. 2And in fact S 1 ∼ K(Q, 1). (0) ⌟ 5. LOCALIZATION (RATIONALIZATION) 44 Theorem 0.2. K(Π, n) represents H ∗ (−, Π): One can think of [−, K(Π, n)] as a functor from the category of pointed homotopy classes of maps to groups. If Π is cyclic [−, K(Π, n)] ≅ H ∗ (−, Π) via f ↦ f ∗ (ιn ), where ιn ∈ H n K(Π, n) ≅ πn K(Π, n) ≅ Π is the generator. Example 0.7 (Localization of ΩS 3 ). We start with the observation that H ∗ (CP ∞ , Z) ≅ Z[a], ∣a∣ = 2. In fact H ∗ (CP n , Z) ≅ Z[a]/(an+1 ) and H ∗ (CP n , Z) Ð → H ∗ (CP n−1 , Z) maps a to a under the isomorphism above. We want to find H ∗ (CP ∞ , Q) ≅ ⋀(a1 , ⋯, an , ⋯). Observe that H 0 (CP ∞ , Q) = Q, H 1 (CP ∞ , Q) = 0, H 2 (CP ∞ , Q) ≅ Q.a, ⋯. Therefore there is an injection3 Q[a] Ð → H ∗ CP ∞ . Checking dimensions implies that the map is also surjective: dim(Q[a])k = { 1 2∣k, k ≥ 0 1 2∣k, k ≥ 0 , dim(H ∗ CP ∞ )k = { . 0 otherwise 0 otherwise So any injective (graded) map Q[a] Ð → H ∗ CP ∞ is sujrective. On the other hand H ∗ (ΩS 3 ; Q) ≅ Q[a′ ], ∣a′ ∣ = 2. (Compute H ∗ (ΩS 3 , Q) as a ring with the Serre spectral sequence; Then take the map of algebras Q[a′ ] Ð → H ∗ (ΩS 3 , Q), ∣a′ ∣ = 2 via a′ ↦ ι2 . This is injective, and by dimension count of the Q-vector spaces in all degrees it is surjective.) ≅ By 0.2 there is a map f ∶ ΩS 3 Ð → CP ∞ , with f (a) = a′ . Since f ∗ is an algebra map it 3 must be an isomorphism, after tensoring with Q. Thus ΩS(0) ∼ K(Q, 2). Really H ∗ (f, Q) is an isomorphism, hence H ∗ (f(0) , Z) is isomorphism: This follows from the commutative diagram ̃ ∗ (X(0) , Z) ≅ Q ̃ ∗ (X, Q) ≅ / H H ̃ ∗ (Y, Q) H f(0) ≅ /H ̃ ∗ (Y(0) , Z) ≅ Q 3 f(0) is a homotopy equivalence ΩS(0) ÐÐ→ K(Q, 2). ⌟ We will end this section with a generalization of the above example, that is the following 3When we omit the coefficient in cohomologies we are assuming rational coefficients unless stated otherwise. 5. LOCALIZATION (RATIONALIZATION) 45 2n+1 2n+1 Theorem 0.3. ΩS(0) ∼ K(Q, 2n) and S(0) ∼ K(Q, 2n + 1) for all n ≥ 1. In fact, the computation in previous example motivates H ∗ (K(Q, 2n)) ≅ Q[a], ∣a∣ = 2n, H ∗ (K(Q, 2n + 1)) = E(a), ∣a∣ = 2n + 1, where E(a) is the exterior algebra in one generator. We will be proving these in what follows. 2n+1 Lemma 5. There exists g ∶ S(0) Ð → K(Q, 2n + 1) that is an isomorphism on π2n+1 and hence on H2n+1 . Proof. π2n+1 (K(Z, 2n + 1)) ≅ Z with generator ι2n+1 . Let g ′ ∶ S 2n+1 Ð → K(Z, 2n + 1) ′ be representing ι. Thus π2n+1 (g ′ ) is an isomorphism. By π∗ (f(0) ) = π∗ f ⊗ Q, g = g(0) is an isomorphism on π2n+1 ≅ H2n+1 . Proposition 18. Assume H ∗ (K(Q, 2n)) = Q[a], ∣a∣ = 2n. Then (1) f = Ωg is a homotopy equivalence. (2) g is a homotopy equivalence. Proof. We have a map between fibration sequences (ΩS 2n+1 )(0) / (P S 2n+1 )(0) / S 2n+1 (0) ≅ ≅ / P S 2n+1 (0) / S 2n+1 (0) / P K(Q, 2n + 1) / K(Q, 2n + 1) 2n+1 Ω(S(0) ) ΩK(Q, 2n + 1) ≅K(Q,2n) Deriving the long exact sequence of π∗ on the fibration sequence we get that f is an isomorphism on π2n and consequently on H2n and also on H 2n . H ∗ (ΩS 2n+1 ) ≅ Q[a′ ], ∣a′ ∣ = 2n, hence H ∗ (f )(a′ ) = αa for some α ≠ 0. Hence H ∗ (f ) is an isomorphism. Thus H∗ (f ) (which is an algebra morphism) is an isomorphism so f is a homotopy equivalence by Whitehead’s theorem. For g use the same long exact sequence on π∗ and note that π∗ (f ) is an isomorphism in all dimensions. Proposition 19. Assuming H ∗ (K(Q, 2n + 1)) ≅ E(a), ∣a∣ = 2n + 1, we have H ∗ (K(Q, 2n + 2)) ≅ Q[b], ∣b∣ = 2n + 2. 5. LOCALIZATION (RATIONALIZATION) 46 Proof. By our assumption Hi (K(Q, 2n + 1)) = { Q i = 0, 2n + 1 . 0 otherwise Consider Serre spectral sequence for homotopy fibration sequence / K(Q, 2n + 2) K(Q, 2n + 1) /∗ ∼ΩK(Q,2n+2) ∼P K(Q,2n+2) in which 2 Epq ≅ Hp (K(Q, 2n + 2)) ⊗ Hq (K(Q, 2n + 1)) ⇒ H∗ (∗) ≅ Q. So we know that the spectral sequence looks like O q E2 ∶ 2n+1 Q 2n 0 ⋯ 0 ⋯ ⋯ 0 ⋯ ⋮ 1 0 0 Q / 2n+2 0 p The only page at which E0,2n+1 may degenerate at 0 is r = 2n + 1. We also want E2n+2,0 to degenerate at 0 on the same page (otherwise it will never stabilize at 0). Therefore the dr -differential has to be the isomorphism shown below O q E2 ∶ 2n+1 2n Q fMM 0 ⋮ MMM MMM 0 MMM ⋯ MMM≅ MMM MMM MMM ⋯ MMM 0 MM 1 0 0 Q Q 0 2n+2 . ⋯ ⋯ / p 2 Likewise analysis shows that all Ek,0 , 1 ≤ k ≤ 2n + 1 should already be zero in the second page. 1. COHOMOLOGY SERRE SPECTRAL SEQUENCE 47 2 2 Now from E2n+2,0 = Q we conclude that E2n+2,2n+1 ≅ Q also. We can now use the same 2 argument on the terms E k, 0, 2n + 3 ≤ k ≤ 4n + 2. Inductively we conclude that H i (K(Q, 2n + 2)) = { Q 2n + 2∣i, i ≥ 0 0 otherwise. Hence as a commutative, connected, Hopf-algebra over a field of characteristic zero H i (K(Q, 2n + 2)) ≅ Q[b], ∣b∣ = 2n + 2 as desired. Proof of theorem 0.3. This is the result of the induction starting with ∞ H i (CP(0) )≅{ Q i even , i ≥ 0 0 i odd and induction steps of propositions 18 and 19. So we have computed π∗ (S 2n+1 ) ⊗ Q. What about the even dimensional spheres? π∗ (S 2n ) ⊗ Q =? 2n Ð → K(Q, 2n) such that π2n (f ) is an isomorphism. This map exists since Consider f ∶ S(0) K(Q, 2n) represents H 2n . Form the Serre spectral sequence associated to the fibration sequence 2n FÐ → S(0) Ð → K(Q, 2n). 1. Cohomology Serre spectral sequence Given a fibration sequence F Ð →E Ð → B where B is simply connected and all spaces are path-connected, there exists a spectral sequence E2pq ⇒ H ∗ (E) with the E2 -terms E2pq ≅ H p (B) ⊗ H q (F ). This is also a spectral sequence of algebras. In particular the differentials satisfy the Leibnitz rule (1.1) dr (ab) = (dr a)b + (−1)∣a∣ adr b, where the product structure is induced by the product structure on the tensor product. Note that since dr satisfies the Leibnitz rule ??, E r+1 has an induced algebra structure. 1. COHOMOLOGY SERRE SPECTRAL SEQUENCE 1.1. Edge homomorphism. In the Serre spectral sequence of the fibration p i FÐ →EÐ →B as above (B simply-connected and F connected) we have the mappings Hn (p) ∞ / En,0 ≅ / E2 n,0 ⊂ Hn (E) Fn H Fn H/Fn−1 H. ) / Hn (B) ∞ 2 The composition En,0 Ð → En,0 Ð → Hn (B) is called the edge homomorphism. We wanted to find πi (S 2n ) ⊗ Q. Consider a fibration sequence f 2n FÐ → S(0) Ð → K(Q, 2n) so that f is π>n -isomorphism. Take the associated cohomology Serre spectral sequence: E2pq ≅ H p K(Q, 2n) ⊗ H q F. We know that H 0 F ≅ Q so E2p,0 ≅ H p K(Q, 2n) ≅ Q[a], ∣a∣ = 2n. Also the edge homomorphism ≅ p,0 2n H pB Ð → E2p,0 ↠ E∞ Ð → H p (S(0) ) tells us 2n pq ={ ≅ H p S(0) E∞ Q p = 0, 2n . 0 otherwise So in the second page of the spectral sequence, since im dr ∩ (Er2n,0 ) = 0 H q F = 0, 0 < q < 4n − 1. 0,4n−1 So there is α ∈ E4n such that d4n α ≠ 0 and hence E20,4n−1 ≅ Q. Therefore E2p,4n ≅ { Q 2n∣p, p ≥ 0 0 otherwise. So since this is a spectral sequence of algebras we can rewrite this computation as E2∗,4n−1 ≅ Q[a] ⊗ Q⟨b⟩, ∣a∣ = 2n, ∣b∣ = 4n − 1. So since dr (a) = 0 = dr (b) then dr (Er∗,4n−1 ) = 0 if r < 4n. Also since d4r (1 ⊗ b) = a2 ⊗ 1 d4n (ak ⊗ b) = (d4r ak ).b + (−1)∣a ∣ ak .d4r b = ak+2 ⊗ 1. k 48 1. COHOMOLOGY SERRE SPECTRAL SEQUENCE So we get that pq E∞ ≅{ 49 Q p = 0, 2n, q ≤ 4n − 1 0 otherwise. Consequently E20,q = 0, q ≥ 4n so H iF = { Q i = 0, 4n − 1 0 otherwise. If α ∈ H q F , the dr (α) = 0 for all r: The only interesting cases are d2n and d4n . If d2n α = c(a ⊗ b), c ≠ 0 then we should have d4n (a ⊗ b) = 0 but this is a contradiction by the computation we did: d4n (a ⊗ b) = a3 ⊗ 1. So we should have d2n (α) = 0. For the 4n-page just use the fact that d4n is a differential, so d4n (α) = 0. For all other r dr (α) ∈ Err,q−r = 0 0,q but E∞ = 0 if q ≠ 0 (by edge homomorphism). We conclude that Hi (F ) ≅ H i (F ) ≅ { Q i = 0, 4n − 1 0 otherwise. 2n Ð → B(0) we get Claim F ∼ F(0) : Because in the long exact sequence on π∗ for F Ð → S(0) that π∗ (F ) ≅ π∗ F ⊗ Q. Then ⎧ Z i=0 ⎪ ⎪ ⎪ H (F, Z) ≅ ⎨ Q i = 4n − 1 ⎪ ⎪ ⎪ ⎩ 0 otherwise. 4n−1 So F ∼ M (Q, 4n − 1) = S(0) = K(Q, 4n − 1). i 2n Now use long exact sequence on π∗ for F Ð → S(0) Ð → K(Q, 2n) so 2n πi (S(0) )={ This completes our comuptation. Q i = 2n, 4n − 1 0 otherwise. CHAPTER 6 Loop suspensions Recall we have adjoint functors Σ and Ω and unit of adjunction X Ð → ΩΣX. Let X be any pointed space (with base point ∗). Let JX be the free monoid on X with identity ∗, i.e. JX = {x1 ⋯xk ∶ k ∈ N, xi ∈ X}/ ≃ where the equivalence relation is x1 ⋯xi ⋯xn = x1 ⋯x̂i ⋯xn if xi = ∗. In fact let Jn X ⊂ JX be the set of all k-tuples for all k ≤ n of JX: Jn X = X n / ≃ with the quotient topology. Then JX = lim Jn X, Ð→ and it is a topological monoid by concatenation. This is called James’ construction. It is an associative H-space, with a strict identity; i.e. ϕ(e, x) = x (not just upto homotopy). There is an obvious inclusion η ∶ X Ð → JX via x ↦ x. JX has the universal property that given f ∶ X Ð → H there is f ∶ JX Ð → H making ηX / JX CC CC f f CC! XC C H commutes exactly. f is well-defined since e is strict (we define it by (x1 ⋯xn ) ↦ (x1 (x2 (⋯(xn )⋯))) or any other choice of parenthesis). And if H is strictly associative then f is strictly multiplicative. / JX JX × JX f ×f H ×H ϕ f / H. In fact if H is strictly associative the choice of parenthesis does not matter since H is strictly associative. We will show that JX is a model for ΩΣX (i.e. it is homotopy equivalent to ΩΣX; in fact more it true, it is multiplicative homotopy equivalent to this space). 50 6. LOOP SUSPENSIONS 51 Definition 17 (Moore loops). Let (X, ∗) be a based space. The More loops of X is Ωm X = {(η, s) ∶ η ∶ R≥0 Ð → X, s ∈ R≥0 , η(t) = ∗ if t ≥ s or t = 0} with subspace topology of Map(R≥0 , X) × R≥0 . ΩM X is associative H-space with strict identity. The structure is given as follows: ΩM X × ΩM X Ð → ΩM X (η, s) × (η ′ , s′ ) ↦ (η ◻ η ′ , s + s′ ) where η ◻ η ′ (t) = { η(t) 0≤t≤s ′ η (t − s) s ≤ t and e = (e∗ , 0), i.e. e∗ (t) = ∗ is the identity map. Consider the continuous mapping f ∶ ΩX Ð → ΩM X η ↦ (η, 1) and the rescaling map g ∶ ΩM X Ð → ΩX (η, s) ↦ η ′ ∶ η ′ (t) = η(st). Proposition 20. g and f are homotopy multiplicative inverse homotopy equivalences. Proof. gf = idΩX and f g ≃ idΩM X by scaling to length 1. Rescaling also gives a homotopy ΩX × ΩX f ×f ΩX f / ΩM X × ΩM X / ΩM X by the top right composition one is mapping a path to some (∗, 2) and by the bottom left composition to (∗, 1). For the other direction let r be the rescaling (η, s) ↦ (η, 1/2). Then the outer pentagon commutes exactly in ΩM5 X × ΩMSX S r×rkkkkk kk kkkk ΩM X × ΩM X g×g ΩX × ΩX SSSSid SSSS S) / ΩM X × ΩM X g / ΩX. 6. LOOP SUSPENSIONS 52 Recall the map XÐ → ΩΣX x ↦ ϕ, ϕ(t) = (x, t) ∈ ΣX = X × I/ ≅ . This factors through X Ð → ΩM ΣX via x ↦ (ϕ, 1). We need an extra hypothesis ∗ Ð → X is a cofibration (one says X is well-pointed ). Proposition 21. Suppose (X, ∗) is well-pointed and f ∶ X Ð → ΩM ΣX is the map above. Then f ≃ g such that g(∗) = e. Proof. Consider H∪f / ΩM ΣX 7 n n n n ̃ n n H ∗ × I ∪ X × {0} X × I. where H is a path in ΩM ΣX from f (∗) to e: say f (∗) = (ϕ, 1) where ϕ(t) = (∗, t). Then H(s) = (ϕ, 1 − s). ̃ making the diagram commute is our homotopy. Now H Notation: Let g ∶ X Ð → ΩM ΣX be the mapping ̃ g(x) = H(x, 1) and θ ∶ JX Ð → ΩM ΣX its multiplicative extension. We will show that H∗ (θ) is an isomorphism. ̃∗ (X; k) as algebras. Theorem 0.1. Let k be any field. Then H∗ (ΩΣX; k) ≅ T H Recall that if V is a (graded) vector space, then T V = ⊕0≤n<∞ V ⊗n , called the tensor algebra and has the obvious multiplicative structure. In fact in case of X = S n and char k = 0, we know that H ∗ (ΩS n+1 , k) = k[a] where a is primitive for degree reasons. So we know H ∗ (ΩS n+1 , l) as Hopf algebra and (H ∗ (ΩS n+1 , k))∗ ≅ H∗ (ΩS n+1 , k) as Hopf algebras (universal coefficients theorem). Recall that X ∧ X = X × X/X ∨ X and from that ̃∗ (X ∧ X) ≅ H ̃∗ (X) ⊗ H ̃∗ (X). H Proposition 22. If X is connected, Σ(X × X) ≅ Σ(X ∨ X) ∨ Σ(X ∧ X). 6. LOOP SUSPENSIONS 53 Proof. Consider the cofibration sequence X ∨X Ð →X ×X Ð → X ∧ X. (∗) By this we mean a sequence A ↪ B Ð → C where C is a quotient of a cofibration A ↪ B. We have maps pi ∶ X × X Ð →X (x1 , x2 ) ↦ xi . Then (∗) gives a sequence / Σ(X × X) Σ(X ∨ X) / Σ(X ∧ X) pinch twice Σ(X × X) ∨ Σ(X × X) ∨ Σ(X × X) id ΣX ∨ ΣX Σp1 ∨Σp2 ∨Σπ / ΣX ∨ ΣX ∨ Σ(X ∧ X) id / Σ(X ∧ X) The diagram commutes up to homotopy and the bottom row is cofibration sequence. So using long exact sequence on H∗ of bottom and top cofibration sequences and the five lemma we get that Σ(X × X) Ð → ΣX ∨ ΣX ∨ Σ(X ∧ X) is homology isomorphism and hence homotopy isomorphism since the suspension of connected spaces are simply connected. Note that we also using the fact that the (weak) product of CW complexes are CW complexes and Σ of CW complexes are CW complexes. Now we apply Whitehead’s theorem. Remark. The above is a space level version of the homology decomposition H∗ (X × X) ≅ H∗ (X) ⊗ H∗ (X) the last one being only linearly equivalent to ̃∗ (X) ⊕ H ̃∗ (X) ⊕ H ̃∗ (X ∧ X). H∗ (X) ⊕ H The idea is to start from a decomposition of algebraic invariants and prove a decomposition of objects with more structure. Recall T n X = {(x1 , ⋯, xn ) ∈ X n ∶ xi = ∗ for some i } is the flat wedge of X. Proposition 23. (1) Let X be a CW-complex, then the diagram T nX / Jn−1 X / Jn X Xn 6. LOOP SUSPENSIONS 54 is a pushout. (2) X n /T n X ≅ Jn X/Jn−1 X. (3) Also T n X Ð → X n is a cofibration so Jn−1 X Ð → Jn X is a cofibration. Proof. (1) is obvious. (2) follows since if A /B /C C is a pushout, then C/A f Z / D/B ϕ is a homeomorphism. If ϕ exists such that the diagram commutes, ϕ induced by map DÐ → C/A determind by projection C Ð → C/A and A/A B |> || | || || EE EE EE EE " / C/A clearly agree on A so we get a map D/B Ð → B/A. Check f and ϕ are inverses. We skip the proof of (3). Proposition 24. T n X Ð → Xn Ð → X ∧n is a cofibration sequence i.e. X n /T n X ≅ X ∧n . Proof. Look at X ∨ Y Ð →X ×Y Ð → X ∧ Y = X × Y /X ∨ Y . So X ∧n = X n / ≅ where the equivalence is generated by (x1 , ⋯, xn ) ≅ ∗ if some xi = ∗. This proves the assertion. Corollary 3. Jn−1 X Ð → Jn X Ð → X ∧n is a cofibration sequence. Proof. Follows from two previous propositions. Since J1 X = X we get / H∗ JX H∗ X t9 u u t uu t u t uu t ϕ uz u / TH ̃∗ X ̃∗ X. H So take ϕ to be the unique algebra extension. 6. LOOP SUSPENSIONS 55 Proposition 25. If H∗ X = H∗ (X, R) is a free R-module then the map ϕ above is an isomorphism. Proof. Let ̃∗ X) = ⊕0≤i≤n (H ̃∗ X)⊗i Tn (H be the tensors of lengths ≤ n. Claims: ̃∗ X) ⊆ H∗ Jn X. (1) ϕ(Tn H ϕ ̃∗ (X)⊗n ↪ Tn H ̃∗ (X) Ð (2) H → H∗ Jn X Ð → H∗ (X ∧n ) is an isomorphism. Considering how ϕ is defined we have ̃∗ X)⊗n (H / H∗ (JX)⊗n / H∗ (J1 X)⊗n / (H∗ X)⊗n >>NNN o7 >> NNN ooo o o >> NNN multiply ooo NN' >> ooo >> >> H∗ (X)⊗n ○ H∗ (JX) >> 7 O o o o >> oo o >> o oo > ooo / H∗ (Jn X) H∗ (X n ) which proves claim (1). For claim (2) look at † ̃∗ (X)⊗n H / H∗ (X n ) ' /H ̃∗ (X ∧n ) MMM MMM MMM MM& H∗ (Jn (X) ‡ /H ̃∗ (X ∧n ) † is an isomorphism by Kunneth formula and induction. ‡ is also an isomorphism since cofibers are heomoemorphic. So we get maps between short exact sequences 0 0 /T H ̃ n−1 ∗ (X) /T H ̃ n ∗X /H ̃∗ (X)⊗n / H∗ (Jn−1 X) / H∗ (Jn X) /0 ≅by claim (1) /H ̃∗ (X ∧n ) /0 ̃∗ X Ð where the vertical maps are induced by ϕ. So by induction on n we get that Tn H → H∗ (Jn X) is an isomorphism. 6. LOOP SUSPENSIONS 56 So we have a sequence of isomorphisms ̃∗ X ≅ lim Tn H ̃∗ X TH Ð→ ≅ lim H∗ (Jn X) Ð→ ≅ H∗ lim Jn X Ð→ ≅ H∗ (JX) since lim commutes with H∗ Ð→ since JX ∶= lim Jn X Ð→ Therefore ϕ is an isomorphism. Proposition 26. The map JX Ð → ΩΣX from the last class is a homotopy equivalence. Proof. We are assuming everything is of finite type. We have a homotopy commuting diagram / JX XF FF FF FF Θ F" ΩΣX where Θ is homotopy multiplcative. So taking H∗ we get multiplcative map H∗ (Θ). We have a commutative diagram ̃∗ (X)) T (H 9 ss s s ≅ sss sss / H∗ (JX) H∗ X9 L 99 LLL 99 LLL H∗ Θ 99 LLL % 99 99 H∗ (ΩΣX) 99 O 99 99 ≅ by Bott-Samuelson theorem 9 ̃∗ (X)) T (H so H∗ (T heta, k) is an isomorphism for any field k. So H∗ (Θ, Z) is an isomorphism. So Θ is a homotopy equivalentce. Since X is a CW complex T X is a CW complex and ΩΣX is homotopy equivalent to a CW-complex. Proposition 27. ΣJX ≅ Σ(⋁1≤i<∞ X ∧i ). Proof. Consider the commutative diagram 0 / Tn X / Xn / X ∧n /0 0 / Jn−1 X / Jn X / X ∧n / 0. 1. ANOTHER HOMOTOPY COMPUTATION 57 The top row splits after suspension. Therefore so does the bottom row. So we have ΣJn X ≃ ΣJn−1 X ∪ ΣX n . Now induction and passing to lim gives the proposition. Ð→ Theorem 0.2 (Freudenthal suspension theorem). Suppose the connectivity of X is ≥ n − 1. Then πi X Ð → πi (ΩΣX) ≅ πi+1 (ΣX) is an isomorphism for i ≥ 2n − 1 and is a surjection for i ≤ 2n − 1. Proof. Look at Serre spectral sequence for fibration FÐ →X Ð → ΩΣX. 2 ̃∗ (X)). Lowest degree homology is at least Hn (X) so the lowest degree Then Ep,0 ≅ T (H homology in the cokernel will at least be ̃n X ⊗ H ̃n X. H So connectivity of the fiber, F , is at least 2n − 2. Then we use the long exact sequence onπ∗ to complete the proof. 1. Another homotopy computation Recall that for the fibration g f FÐ → ΩS 3 Ð → CP ∞ = K(Z, 2) By Serre’s method, g induces an isomorphism on πi for i > 2 and πi (F ) = 0 for i ≤ 2, and f induces isomorphism on π2 and is zero on πi for i > 2. As a result of applying Hurewitcz’s theorem we had π3 F = H3 (F, Z) = π3 (ΩS 3 ) ≅ π4 S 3 . We now want to compute H3 (F, Z). As an algebra H∗ (ΩS 3 , Z) ≅ Z[a], ∣a∣ = 2. This is a consequence of Bott-Samuelson theorem: we have ΩS 3 = ΩΣS 2 and hence ̃∗ (S 2 , R)) = T (R.a) ≅ R[a] H∗ (ΩS 3 ) = T (H ̃∗ (S 2 ) is a free R-module. since H Moreover as Hopf algebras we have an isomorphism H ∗ (CP ∞ , Z) ≅ Z[a′ ] with a′ primitive (just work out the diagonal ∆(a′ ) to see this). Since i+j ∆(a′ )n = ∑ ( )(a′ )i ⊗ (a′ )j i 1. ANOTHER HOMOTOPY COMPUTATION 58 the dual algebra is the divided powers algebra, Γ(α) ∶= ⊕i≥0 Zαi )αi+j : where αi = (ai )∗ and αi αj = (i+j u H∗ (CP ∞ , Z) ≅ (H∗ (CP ∞ , Z))∗ ≅ Γ(α) both as Hopf algebras. In Z/p coefficients we have H∗ (f, Z/p) is an isomorphism for ∗ < 2p and is zero if ∗ = 2p. H∗ (ΩS 3 , Z/p) ≅ Z/p[a] Ð → Γ(α) ⊗ Z/p ≅ H ∗ (CP ∞ , Z/p) ap ↦ (α1 )p = p!(αp ) = 0 Note 1.1. In cases of both coefficients H∗ (f ) is algebra map, since f = Ωg. So f is a Hopf algebra map. ⌟ In the next step we look at Serre’s spectral sequence with coefficients in Z/p: in the second page we have E`,0 ≅ H` B ≅ H` CP ∞ . so q E2 ∶ O 2p Z/p 2p-1 Z/p 0 Z/p Z/p Z/p 0 2 2p / p The edge homomorphism is ∞ 2 H` (ΩS 3 ) Ð → E`,0 ≅ E`,0 Ð → H2 (CP ∞ ) which is an isomorphism. Hence ∞ E2p,0 = 0. So on second page we need to have 2 ≅ Z/p. E0,2p−1 2 ∞ Also E0,2p ≅ Z/p since ⊕i+j=2p Eij = Z/p ≅ H2p (ΩS 3 ) but we don’t need this. So the spectral sequence yields H2p−1 (F, Z/p) = Z/p, Hi (F, Z(p) ) = 0 if i < 2p − 1. 1. ANOTHER HOMOTOPY COMPUTATION 59 From the short exact sequence 0Ð → Z(p) Ð → Z(p) Ð → Z/p Ð →0 we get H2p−1 (F, Z(p) ) = Z(p) or Zpn but we can exclude Z/p from knowing H∗ (f ) ⊗ Q is isomorphism so H∗ (F ) ⊗ Q = 0. On the other hand assuming finite type, and localization techniques we have n Hi (F, Z(p) ) = 0, i < 2p − 1 (so F(p) is 2p − 2 connected). By Hurewicz’s theorem π2p−1 (F ) ⊗ Z(p) ≠ 0. Again from H∗ (f ) ⊗ Q being an isomorphism, we have pk H2p−1 (F, Z(p) ) ≅ π2p−1 (F ) ⊗ Z(p) = 0 so π2p−1 (F ) ⊗ Z(p) ≅ Z/pn . Now we look at the Serre spectral sequence of 3 ∞ F(p) Ð → ΩS(p) Ð → CP(p) with Z-coefficients. O q E2 ∶ 2p-1 Z(p) 0 Z(p) Z(p) Z(p) 0 2 2p / p By the edge homomorphism 2 ∞ Ð → E2p,0 ≅ H2p (CP ∞ ) H2p (ΩS 3 ) Ð → E2p,0 ∞ ∞ 2 is multiplication by p, Z(p) Ð → Z(p) . So αp ∈/ Z2p and pαp ∈ Z2p aso there is β ∈ E0,2p−1 and 2 pβ = 0. And β mut generate E0,2p−1 . So 2 = Z/p ≅ H2p−1 (F(p) , Z(p) ) ≅ π2p−1 (F ) ⊗ Z(p) E0,2p−1 ̃` (F(p) , Z) = 0, ` < 2p − 1. Since πi (F ) ≅ πi (ΩS 3 ) ≅ πi+1 (S 3 ) for i > 2 Also H πi (S 3 ) ⊗ Z(p) = 0, 3 < i < 2p and get π2p (S 3 ) ⊕ Z(p) = Z/p. CHAPTER 7 Classifying spaces 1. Category of G-bundles Let G be a topological group, a space with a continuous right G-action is called a right G-space. We will usually be working with right actions and shortening the above term by G-space. Morphisms of G-spaces are G-equivariant maps. We know that if X is a G-space then the is an open map π ∶ X Ð → X/G to the quotient space with quotient topology. p Definition 18. A map ξ ∶ X Ð → B is a G-bundle if it is isomorphic to X ′ Ð → X ′ /G for some G-space X. Example 1.1. The action of GL(n, R) on Rn gives a quotient space Rn / GL(n, R) consisting of two points with one of them being open and the other one not open. It is obvious from this example that G-bundles may not be fibrations. ⌟ Definition 19. a G-space X is free is xs = x implies s = 1. If X is free then we make a notation X ∗ ∶= {(x, xs) ∶ x ∈ X, s ∈ G} ⊂ X × X. The map τ ∶ X ∗ Ð → G via (x, xs) ↦ s is well-defined since G is free and is called the translation function. Definition 20. A G-space X is principal if X is free and τ ∶ X ∗ Ð → G is continuous. A p G-bundle X Ð → B is principal if X is a principal G-space. p Proposition 28. A principal G-bundle X Ð → B has fiber G (i.e. for all b ∈ B, p−1(b) is homeomorphic to G). Proof. Let b ∈ B pick x ∈ p−1(b) define u ∶ G Ð → p−1(b) via x ↦ u(s) ∶= xs. The inverse is y ↦ τ (x, y). Example 1.2. Let G = Z/2 act on S n . Then S n Ð → S n /G is principal. Clearly the action is free and observe that τ from the space {(x, ±x) ∶ x ∈ S} ⊂ S n × S n to {1, −1} is continuous. ⌟ 60 1. CATEGORY OF G-BUNDLES p 61 f If X is a G-space with associated G-bundle X Ð → B then any map B ′ Ð → B can pullback p ′ to a G-bundle on B : X′ p′ B′ f ◻ f /X p / B. The G-action on X ′ is given via (x, b′ ).s = (xs, b′ ). It is clear that f is a map of G-spaces. Also if X is principal then X ′ is one as well. Definition 21. For any map f ∶ Y Ð → Y ′ of G-spaces we get a commutative diagram / Y′ Y / Y ′ /G Y /G a map of G-bundles is something isomorphic to this (here meaning only consistent homeomorphisms). Let Bun [G] be the category of G-bundles and G − Bun B is the groupoid of G-bundles over B. Any map f ∶ B ′ Ð → B gives a functor f ∗ ∶ Bun B [G] Ð → Bun B ′ [G]. The fact that this is a groupoid will be left unproved, Proposition 29. If f ∈ HomBun B [G] (X ′ , X) then f is an isomorphism. Proof. Let f be the equivariant map over B in the diagram X@ f @@ @@ p @@@ B / X′ } } }} }} p′ } }~ Injectivity: If f (x1 ) = f (x2 ) then p(x1 ) = p′ (f (x)) = p′ (f (x2 )) = p(x2 ). So x1 s = x2 for some s ∈ G. Thus f (x1 )s = f (x1 s) = f (x2 ) = f (x1 ). ′ Therefore x = 1 since ξ is free. So x1 = x2 . Sujrectivity: Given x′ ∈ X ′ , let x ∈ X be such that p(x) = p′ (x′ ) (by surjectivity of p). So p′ (x′ ) = p(x) = p′ (f (x)). Thus x′ = f (x)s for some x ∈ G. Therefore x′ = f (xs) so f is surjective. Now check that f −1 is continuous. 2. FIBER BUNDLES 62 2. Fiber bundles p Let ξ ∶ (X Ð → B) be a principal G-bundle. Let also F be a left G-space and we have a diagonal right action of G on X × F via (x, f ).s = (xs, s−1f ) and the orbit space will be denoted by XF ∶= X × F /G = X ×G F. The map pF ∶ XF Ð → B is the unique factorization of X ×F Ð →X Ð →B pF through X ×G F . Then ξF ∶ (XF Ð→ B) is called a fiber bundle over B with fiber F (as a G-space). One Session missing here Definition 22. A covering {Ui } of X is a numerable overing if there is a locally finite partition of unity {Ui }i∈S . In particular Ui ((0, 1]) ⊆ Ui , ∀i ∈ S. For Hausdorff spaces and paracompact spaces all coverings are numerable. A G-bundle ξ is numerable if it is trivial over a numerable covering. Proposition 30. For a map of bases f ∶ B ′ Ð → B, numerable G-bundle ξ pulls back to ∗ numerable f (ξ). Proof. Use u′i = ui ○ f and Ui′ = f −1(Ui ). For convenience we write E(ξ) for the total space of a G-bundle ξ. Proposition 31. Given r ∶ B × I Ð → B × I via (b, t) ↦ (b, 1) and ξ a numerable G-bundle over B ×I then there is a G-morphism (g, r ∶ ξ Ð → ξ such that there is a commutative diagram E(ξ) g B×I r / E(ξ) / B × I. Corollary 4. ξ and r∗ ξ are isomorphic over B × I. 2. FIBER BUNDLES 63 Proof. There is a unique map making the diagram E(ξ) I I g I I I$ E(r∗ ξ) $ B×I $ / E(ξ) / B×I commute. So check that it is a G-equivariant map and by last session it must be an isomorphism. Theorem 2.1. Given f, g ∶ B ′ Ð → B and ξ a bundle over B, if f ≅ g that f ∗ (ξ) ≅ g ∗ (ξ). Proof. Let H be a homotopy from f to g. Let E0 ∶ B ′ Ð → B; ×I be b ↦ (b, 0). We have HE0 = f , and HrE0 = g. Now we have f ∗ (ξ) = (HE0 )∗ (ξ) = E0∗ H ∗ (ξ) ≅ E0∗ r∗ (H ∗ (ξ)) ≅ (HrE0 )∗ (ξ) = g ∗ (ξ). Notation: kG (B) is the set of isomorphism classes of numerable principal G-bundles over B. And by isomorphism we mean isomorphisms over the identity map. Corollary 5. For f ∈ [X, Y ] we get kG (f ) ∶ kG (Y ) Ð → kG (X). And kG (f ) is a bijection if f is a homotopy equivalence. p0 Definition 23. A principal G-bundle ω ∶ E0 Ð→ B0 is a universal (numerable) G-bundle if ω is numerable and [−, B0 ] Ð → kG (−) is an isomorphism of contravariant functors, i.e. it’s a natural transformation such that for all X, [X, B0 ] Ð → kG (X) is a bijection. Note that [−, B0 ] is clearly a functor on the homotopy category Top ≅ so is kG (0) morphisms given by pullback. Proposition 32. A numerable principal G-bundle ω is universal if and only if (1) ∀ξ principal numerable G-bundle over X, there is f ∶ X Ð → B0 , ξ ≅ f ∗ (ω) over X. (2) For any f, g ∶ X Ð → B0 . f ∗ (ω) ≅ g ∗ (ω) implies f ≃ g. 3. MILNOR’S CONSTRUCTION 64 3. Milnor’s construction Definition 24 (The join). A ∗ B is a topological space A × B × I/ ≅ with the relations (a, b, 1) = (a, b′ , 1), (a, b, 0) = (a′ , b, 0). We can view it as having points (t0 a, t1 b) with t0 + t1 = 1 with relations (0a, t1 b) = (0a′ , t1 b). A ∗ B fits into a pushout (I) A∗B A × CB CA × B A∗B where CX is the notation for cone on X. Consider the homotopy pushout (II) A∨B A ∨ CB CA ∨ B CA ∨ CB the inclusion II Ð → I is a cofibration on each space in part. colim(I/II) ≅ colim I/ colim II ≅ A ∗ B since colim II ≃ ∗. But I/II ∶ A∧B / A ∧ CB(≃ ∗) (∗ ≃)CA ∧ B and this diagram is “equivalent” to A∧B / C(A ∧ B) / Σ(A ∧ B) ∗ and we conclude that colim I/II ≃ Σ(A ∧ B). And finally that A ∗ B ≃ ΣA ∧ B. 3. MILNOR’S CONSTRUCTION 65 We make the notation En G = ∗n G = (∗n−1 G) ∗ G. And as a set we should view it as {(t0 x0 , t1 x1 , ⋯, tn−1 xn−1 ) ∶ ∑ ti = 1}/ ≅ with the relations that if ti = 0 then (⋯, t2 xi , ⋯) ≅ (⋯, ti x′i , ⋯). Clearly En G Ð → En+1 G via (t0 x0 , ⋯, tn−1 xn−1 ) ↦ (t0 x0 , ⋯, tn−1 xn−1 , 0). As an aside note that one can show that En G Ð → En+1 G is null-homotopic. We now make the final definition EG ∶= ∗∞ G ∶= colimn (En G). Proposition 33. EG ≃ ∗. Proof. En G ≃ En−1 G ∗ G ≃ Σ(En−1 G ∧ G). So it’s connected to all degrees. En G with diagonal action is a free G-space. In fact, since for any (t0 g0 , ⋯, tn−1 gn−1 ) we have ∑ ti = 1, some ti ≠ 0 so the action is free. The action is continuous as well (check it!). So we can define Bn G ∶= En G/G, BG ∶= EG/G. pn p Proposition 34. En G Ð→ Bn G and ωG ∶ EG Ð → BG are locally trivial, numerable, principal G-bundles. Proof. We will only work out the proofs of local triviality and numerability for the EG case. Let ti ∶ EG Ð → I be given via (t0 x0 , ⋯, ti xi , ⋯) ↦ ti . 0,1 There is a unique ui ∶ BG Ð→ such that ui p = ti . Let −1 vi = u−1 i ((0, 1]) = p(ti ((0, 1])) ⊆ B. To show that ωG is trivial over vi , we show that there is a section over vi , s i ∶ vi Ð → t−1 i (0, 1] ⊆ EG . So define s′i ∶ t−1 → t−1 i (0, 1] Ð i (0, 1] that is constant on G-orbits s′i ∶ t−1 → t−1 i (0, 1] Ð i (0, 1] to be constant on G-orbits, via s′i ∶ t−1 → t−1 i (0, 1] Ð i (0, 1] (t0 x0 , ⋯, ti xi , ⋯) ↦ a(xi )−1. s′i is well-defined since ti ≠ 0. And si (ag) = s′i (a) for all g ∈ G, hence s′i induces si ∶ t−1 → t−1 i (0, 1]/G Ð i (0, 1]. 3. MILNOR’S CONSTRUCTION 66 It is easy to check that psi = idvi . So ωG is locally trivial, also numerable since the ui ’s give a partition of unity. p Corollary 6. EG Ð → BF is a fibration with fiber ΩBG and so B is an inverse to Ω on the homotopy category. p Proof. EG ≃ ∗ and EG Ð → BF a fibration, since ωG is locally trivial G-bundle over a CW complex. Therefore p−1(∗) ≃ ΩBG. The constructions we have for En G, Bn G, EG and BG are clearly functorial on group homomorphisms. So B is a functor Top group Ð → Top . (probably we need some topological restrictions on the category of Topological groups we are considering.) Recall that Ω can be considered as a functor Top Ð → Top group . This is due to Milnor. The first thing one has to show is an equivalence of categories between simplicial sets and topological spaces. Then he works on simplicial sets to create a fibration P X Ð → X. P X turns out to be contractible so the fiber is the loop space ΩX. The way the fibration is constructed ΩX has a natural group structure on it as well. From now on we work with CW complexes. So all coverings are numerable since CW complexes are paracompact. Theorem 3.1. For a principal G-bundle, ξ, over B, there is f ∶ B Ð → BG such that ξ ≅ f ∗ (ωG ). Proof. Since ξ is numerable there is {un } over B and Un = u−1 n (0, 1], trivializing ξ for all n ≥ 0. (Quesiton: Why can we assume {Un } is a countable collection?) There is a trivialization hn ∶ Un × G ≅ E(ξ∣Un ) ⊆ Eξ ). Let gn ∶ Un × G Ð → G be the projection and p ∶ E(ξ) Ð → B. Define g ∶ E(ξ) Ð → EG −1 z ↦ (u0 (p(z)g0 h−1 0 (z), ⋯, un (p(z))gn hn (z), ⋯). If hn is not defined, then un p(z) = 0. We conclude that g is well-defined and that g(zs) = g(z)s for all s ∈ G since hn is G-equivariant. So g induces f ∶ B Ð → BG and (g, f ) ∶ ξ Ð → ωG is a bundle map, i.e. E(ξ) g B ∗ commutes. Hence ξ ≅ f (ωG ). f / EG / BG 3. MILNOR’S CONSTRUCTION Theorem 3.2. For f and g are maps B Ð → BG, and f ∗ (ωG ) ≅ g ∗ (ωG ) then f ≃ g. This completes the proof of [B, BG] Ð → kG (B) being a bijection. 67

Algebraic Topology Math 528, Spring 2011 Professor Donald Stanley Notes by: Pooya Ronagh

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