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Math 528: Algebraic Topology Class Notes

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Math 528: Algebraic Topology Class Notes Math 528: Algebraic Topology Class Notes Lectures by Denis Sjerve, notes by Benjamin Young Term 2, Spring 2005 2 Contents 1January4 9 1.1ARoughDefinitionofAlgebraicTopology........... 9 2January6 15 2.1TheMayer-VietorisSequenceinHomology........... 15 2.2Example:TwoSpaceswithIdenticalHomology........ 20 3 January 11 23 3.1Hatcher’sWebPage....................... 23 3.2CWComplexes.......................... 23 3.3CellularHomology........................ 26 3.4APreviewoftheCohomologyRing............... 28 3.5 Boundary Operators in Cellular Homology ........... 28 4 January 13 31 3 4 CONTENTS 4.1 Homology of RP n ......................... 31 4.2APairofAdjointFunctors.................... 33 5 January 18 35 5.1Assignment1 ........................... 35 5.2HomologywithCoefficients................... 36 5.3Application:LefschetzFixedPointTheorem.......... 38 6 January 20 43 6.1TensorProducts.......................... 43 6.2LefschetzFixedPointTheorem................. 45 6.3Cohomology............................ 47 7 January 25 49 7.1Examples:LefschetzFixedPointFormula........... 49 7.2ApplyingCohomology...................... 53 7.3History:TheHopfInvariant1Problem............. 53 7.4AxiomaticDescriptionofCohomology............. 55 7.5MyProject............................ 55 8 January 27 57 CONTENTS 5 8.1ADifferenceBetweenHomologyandCohomology....... 57 8.2AxiomsforUnreducedCohomology............... 57 8.3Eilenberg-SteenrodAxioms.................... 58 8.4ConstructionofaCohomologyTheory............. 59 8.5UniversalCoefficientTheoreminCohomology......... 62 9February1 65 9.1CommentsontheAssignment.................. 65 9.2 Proof of the UCT inCohomology................ 67 9.3 Properties of Ext(A, G) ..................... 69 10 February 3 71 10.1NaturalityintheUCT...................... 71 10.2ProofoftheUCT......................... 72 10.3SomeHomologicalAlgebra.................... 73 10.4GroupHomologyandCohomology............... 75 10.5TheMilnorConstruction..................... 77 11 February 8 81 11.1ASeminarbyJosephMaher................... 81 11.2Products.............................. 82 6 CONTENTS 12 February 23 87 12.1Assignment2 ........................... 87 12.2Examplesofcomputingcohomologyrings............ 88 12.3CohomologyOperations..................... 91 12.4Axiomsforthemod2SteenrodAlgebra............ 92 13 February 25 95 13.1AxiomaticdevelopmentofSteenrodAlgebra.......... 95 13.2Application:Hopfinvariant1problem............. 97 13.3 Poincar´eDuality......................... 98 13.4 Orientability . ........................... 99 14 March 1 101 14.1Lasttime.............................101 14.2Orientablecoverings.......................102 14.3 Other ways of defining orientability ...............103 14.4 Poincar´eDuality.........................106 14.5Reading:Chapter4,Higherhomotopygroups.........106 15 March 3 109 15.1Thepath-loopfibration......................109 CONTENTS 7 15.2 The James Reduced Product Construction JX .........111 15.3 The Infinite Symmetric Product SP∞(X) ...........112 15.4HigherHomotopyGroups....................113 16 March 8 115 16.1MoreontheMilnorSimplicialPathLoopSpaces........115 16.2RelativeHomotopyGroups....................117 8 CONTENTS Chapter 1 January 4 1.1 A Rough Definition of Algebraic Topol- ogy Algebraic topology is a formal procedure for encompassing all functorial re- lationships between the worlds of topology and algebra: world of topological problems −→F world of algebraic problems Examples: 1. The retraction problem: Suppose X is a topological space and A ⊆ X is a subspace. Does there exist a continuous map r : X → A such that r(a)=a for all a ∈ A? r is called a retraction and A is called a retract of X. Ifaretraction∃ then we have a factorization of the identity map i r on A : A → X → A, where r ◦ i = idA. F (i) F (r) Functoriality of F means that the composite F (A) → F (X) → F (A) F (r) F (i) (respectively F (A) → F (X) → F (A)) is the identity on F (A)ifF is a 9 10 CHAPTER 1. JANUARY 4 covariant (respectively contravariant) functor. As an example consider the retraction problem for X the n-disk and A its boundary, n>1: Sn−1 = ∂(Dn) →i Dn →r ∂(Dn)=Sn−1. Suppose that the functor F is the nth homology group: n i∗ n r∗ n Hn−1(∂(D )) → Hn−1(D ) → Hn−1(∂D ) || || || Z −→i∗ 0 −→r∗ Z Such a factorization is clearly not possible, so ∂Dn is not a retract of Dn 2. When does a self map f : X → X have a fixed point? That is, when does ∃ x ∈ X such that f(x)=x? For example suppose f : X → X, where X = Dn. Assume that f(x) =6 x for all x ∈ Dn.Thenwecan project f(x) through x onto a point r(x) ∈ ∂Dn, as follows: r(x) x f(x) Dn Then r : Dn → ∂Dn is continuous and r(x)=x if x ∈ ∂Dn.Thusr is a retraction of Dn onto its boundary, a contradiction. Thus f must have a fixed point. 3. What finite groups G admit fixed point free actions on some sphere Sn? That is, when does ∃ amapG × Sn → Sn,(g,x) 7→ g · x,such that h · (g · x)=(hg) · x, id · x = x, and for any g =6 id, g · x =6 x for all x ∈ Sn. 1.1. A ROUGH DEFINITION OF ALGEBRAIC TOPOLOGY 11 This is “still” unsolved (although some of the ideas involved in the supposed proof of the Poincar´e conjecture would do it for dimension 3). However, lots is known about this problem. For example, any cyclic group G = Zn admits a fixed-point free action on any odd-dimensional sphere: X 2k−1 k S = {(z1,...,zk) ⊆ C | ziz¯i =1}. A generator for G is T : S1 → S1, T (x)=ξx,whereξ = e2πi/n.Then a fixed point free action of G on S2k−1 is given by T (z1,...,zk)=(ξz1,...,ξzk). There are other actions as well. Exercise: Construct some other fixed point free actions of G on S2k−1. 4. Suppose M n is a smooth manifold of dimension n. What is the span of M, that is what is the largest integer k such that there exists a k-plane varing continuously with respect to x? This means that at each point x ∈ M we have k linearly independent tangent vectors v1(x),...,vk(x) in TxM, varying continuously with respect to x. $x$ Tx(M) Definition: if k = n then we say that M is parallelizable. In all cases k ≤ n. In the case of the 2-sphere we can’t find a non-zero tangent vector which varies continuously over the sphere, so k = 0. This is the famous “fuzzy ball” theorem. On the other hand S1 is parallelizable. 12 CHAPTER 1. JANUARY 4 S3 is also parallizable. To see this consider R4 with basis the unit quaternions 1,i,j,k.Thus a typical quaternion is q = q0 +q1i+q2j+q3k, 4 where the qi are real. R becomes a division algebra, where we multiply quaternions using the rules i2 = j2 = k2 = −1,ij = k, ji = −k, jk = i, kj = −i, ki = j, ik = −j and the distributive law. That is 0 0 0 0 0 qq =(q0 + q1i + q2j + q3k)(q0 + q1i + q2j + q3k)=r0 + r1i + r2j + r3k where 0 − 0 − 0 − 0 r0 = q0q0 q1q1 q2q2 q3q3 0 0 0 − 0 r1 = q0q1 + q1q0 + q2q3 q3q2 0 − 0 0 0 r2 = q0q2 q1q3 + q2q0 + q3q1 0 0 − 0 0 r3 = q0q3 + q1q2 q2q1 + q3q0 The conjugate of a quaternion q = q0 + q1i + q2j + Pq3k is defined by − − − 2 q¯ = q0 q1i q2j q3. It is routine√ to show that qq¯ = n qn. We define the norm of a quaternion by |q| = qq.¯ Then |qq0| = |q|||q0| The space of unit quaternions X { | 2 } q0 + q1i + q2j + q3k qn =1 n is just the 3-sphere, and it is a group. Pick three linearly independent vectors at some fixed point in S3. Then use the group structure to translate this frame to all of S3. 1.1. A ROUGH DEFINITION OF ALGEBRAIC TOPOLOGY 13 5. The homeomorphism problem. When is X homeomorphic to Y ? −−−f → X Y F y F y F (f) F (x) −−−→ F (Y ) 6. The homotopy equivalence problem. When is X homotopically equiv- alent to Y ? f p 7. The lifting problem. Given X → B and E → b, can we find a map f˜ : X → E such that pf˜ ' f? 8. The embedding problem for manifolds. What is the smallest k such that the n-dimensional manifold M can be embedded into Rn+k? Let Sn be the unit sphere in Rn+1 and RP n = real projective space of dimension n: RP n def= Sn/x ∼−x. Alternatively, RP n is the space of lines through the origin in Rn+1. Unsolved problem: what is the smallest k such that RP n ⊆ Rn+k? 9. Immersion problem: What is the least k such that RP n immerses into Rn+k? embedding $S^1$ R^2 immersion 14 CHAPTER 1. JANUARY 4 10. The computation of homotopy groups of spheres. def k πk(X) = the set of homotopy classes of maps f : S → X. It is known that πk(X) is a group ∀ k ≥ 1andthatπk(X) is abelian n ∀k ≥ 2. What is πk(S )? The Freudenthal suspension theorem states n n+1 that πk(S ) ≈ πk+1(S )ifk<2n − 1. For example, 3 4 5 π4(S ) ≈ π5(S ) ≈ π6(S ) ≈··· . 2 We know that these groups are all ≈ Z2 and π3(S )=Z. Chapter 2 January 6 2.1 The Mayer-Vietoris Sequence in Homol- ogy Recall the van Kampen Theorem: Suppose X isaspacewithabasepoint x0,andX1 and X2 are path connected subspaces such that x0 ∈ X1 ∩ X2, X = X1 ∪ X2 and X1 ∩ X2 is path connected. Consider the diagram X ∩ X −−−i1→ X 1 2 1 i2y j1y j2 X2 −−−→ X Apply the fundamental group ‘functor’ π1 to this diagram: 15 16 CHAPTER 2. JANUARY 6 i π (X ∩ X ) −−−1#→ π (X ) 1 1 2 1 1 i2#y j1#y j2# π1(X2) −−−→ π1(X) Question: How do we compute π1(X) from this data? There exists a group homomorphism from the free product π1(X1) ∗ π1(X2) into π1(X), given by c1 · c2 7→ j1#(c1) · j2#(c2). Fact: This map is onto π1(X). However, there exists a kernel coming from −1 π1(X1 ∩ X2). In fact, i1#(α) · i2#(α ), for every α ∈ π1(X1 ∩ X2), is in the kernel because j1#i1# = j2#i2#. Theorem: (van Kampen): Suppose all the spaces X1,X2,X1 ∩ X2 contain thebasepointx0 ∈ X = X1 ∪ X2, and every space is path connected. Then π1(X) ≈ π1(X1) ∗ π2(X2)/K where K is the normal subgroup generated by −1 all elements of the form i1#(α) · i2#(α ), where α ∈ Π2(X1 ∩ X2).
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