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Algebraic Topology: Math 231Br Notes ALGEBRAIC TOPOLOGY: MATH 231BR NOTES AARON LANDESMAN CONTENTS 1. Introduction 4 2. 1/25/16 5 2.1. Overview 5 2.2. Vector Bundles 5 2.3. Tautological bundles on projective spaces and Grassmannians 7 2.4. Operations on vector bundles 8 3. 1/27/16 8 3.1. Logistics 8 3.2. Constructions with Vector Bundles 8 3.3. Grassmannians and the universal bundle 10 4. 1/29/16 12 5. 2/1/16 15 5.1. Characteristic Classes 15 5.2. Leray-Hirsch Theorem 17 6. 2/3/16 18 6.1. Review 18 7. 2/5/16 21 8. 2/10/16 24 8.1. Reviewing Leray-Hirsch 24 8.2. Review of Chern and Stiefel-Whitney Classes 25 8.3. Examples and Calculations 26 9. 2/12/15 28 9.1. Logistics 28 9.2. Applications of Stiefel-Whitney classes 28 9.3. Pontryagin Classes 31 10. 2/17/16 31 10.1. Theory on Pontryagin classes 31 10.2. Calculations and Examples with Pontryagin Classes 33 10.3. Euler Classes 34 11. 2/19/16 35 11.1. Review and Thom Classes 35 11.2. Euler Classes 37 11.3. Examples of Thom and Euler Classes 38 12. 2/22/16 39 12.1. More on Euler Classes 39 12.2. K Theory 41 13. 2/24/16 44 1 2 AARON LANDESMAN 13.1. Examples for K theory 44 13.2. Reduced K theory 45 13.3. Products 45 14. 2/26/16 48 14.1. Equivalent definitions of relative K theory 48 14.2. Back to Smash Products 49 15. 2/29/16 52 15.1. Review 52 15.2. Fredholm operators and index 53 16. 3/2/16 55 16.1. More on Fredholm operators 55 16.2. The index of a family 56 16.3. More examples of Fredholm operators 58 17. 3/4/2016 58 17.1. Toeplitz Operators 60 18. 3/7/2016 61 19. 3/9/16 65 19.1. Review of infinite dimensional groups 65 19.2. Real K theory 67 19.3. Symplectic K theory 68 20. 3/11/2016 69 20.1. Chern character 70 20.2. 73 21. 3/21/2016 75 21.1. Clifford Algebras and Clifford Modules 76 22. 3/23/2016 78 22.1. How to obtain the Z/2-grading 79 22.2. Bundles of Clifford modules 80 23. 3/25/16 81 23.1. Clifford Modules 81 24. 3/28/16 84 24.1. 86 25. 3/30/16 88 25.1. Review 88 25.2. Calculating A^ 89 25.3. Relating our computations to A^ 91 26. 4/1/16 93 27. 4/4/16 97 27.1. Formalities 97 27.2. Notations 98 27.3. The exact sequence for K-theory in negative degrees 99 28. 4/6/16 101 28.1. Review 101 28.2. Bordism and cobordism 105 29. 4/8/2016 106 29.1. Oriented cobordism 107 ALGEBRAIC TOPOLOGY: MATH 231BR NOTES 3 30. 4/11/16 108 30.1. Review 108 31. 4/13/16 112 31.1. Oriented bordism as a homology theory 112 32. 4/15/16 115 32.1. Review 115 33. 4/18/16 119 33.1. Review 119 33.2. The relations between framed cobordism and homotopy groups 119 34. 4/20/16 122 34.1. Stabilization 122 34.2. J-homomorphism 123 35. 4/22/16 126 35.1. Low stable homotopy groups 126 35.2. More about Π3 127 35.3. Another integrality property for A^ 128 36. 4/25/16 130 36.1. Spectra 130 36.2. More involved examples of spectra 131 36.3. Fundamental groups of spectra 133 37. 4/27/16 134 37.1. Review 134 37.2. Constructing the long exact sequence for cohomology 135 37.3. Examples of cohomology theories for spectra 136 37.4. Stable homotopy and stable homotopy 137 4 AARON LANDESMAN 1. INTRODUCTION Peter Kronheimer taught a course (Math 231br) on algebraic topology and algebraic K theory at Harvard in Spring 2016. These are my “live-TEXed“ notes from the course. Conventions are as follows: Each lecture gets its own “chapter,” and appears in the table of contents with the date. Of course, these notes are not a faithful representation of the course, either in the math- ematics itself or in the quotes, jokes, and philosophical musings; in particular, the errors are my fault. By the same token, any virtues in the notes are to be credited to the lecturer and not the scribe. 1 Thanks to James Tao for taking notes on the days I missed class. Please email suggestions to [email protected]. 1This introduction has been adapted from Akhil Matthew’s introduction to his notes, with his permission. ALGEBRAIC TOPOLOGY: MATH 231BR NOTES 5 2. 1/25/16 2.1. Overview. This course will begin with (1) Vector bundles (2) characteristic classes (3) topological K-theory (4) Bott’s periodicity theorem (about the homotopy groups of the orthogonal and uni- tary groups, or equivalently about classifying vector bundles of large rank on spheres) Remark 2.1. There are many approaches to Bott periodicity. We give a proof in class following an argument of Atiyah. This introduces index theory for Fredholm operators and related things. Remark 2.2. K-theory is like ordinary homology (originally called an extraordinary ho- mology theory, but with the homology of a point not equal to Z). The other archetype of a general homology theory like this is cobordism theory. If there’s time, we’ll also talk about generalized homology theories, using stable homo- topy and spectra. Some books, useful for this class include (1) Milnor and Stashelf’s Characteristic Classes (2) Atiyah’s K-theory (3) Atiyah’s collected works (4) Hatcher’s Vector bundles and K-theory (online and incomplete) Remark 2.3. Atiyah’s K-theory addresses his book for someone who hasn’t taken 231a. For example, he proves Brouwer’s fixed point theorem. logistical information: (1) Information for the CA: Name: Adrian Zahariuc email: [email protected] (2) There will be slightly less than weekly homework. Homework 1 is up already. 2.2. Vector Bundles. Definition 2.4. Let X be a topological space. A real (or complex) vector bundle on X (or over X) is a topological space E with a continuous map φ : E X and a real (or complex) -1 vector space structure on each fiber Ex := φ (x). This must satisfy the additional condi- tion of being locally trivial, meaning that there is an open cover! U of X so that for each -1 U 2 U with U ⊂ X, the restriction of E to U, notated EU := φ (U) ⊂ E is trivial. Here, trivial means there exists a homeomorphism φ φ n EU U × R (2.1) φU π U id U or to U × Cn for the complex case, where n φjEx : Ex fxg × R is linear. ! 6 AARON LANDESMAN Example 2.5. (1) The trivial vector bundle E = X × Rn with φ : X × Rn X the projection. 1 (2) The Mobius¨ vector bundle on S . Take E˜ = I × R and I = [0, 1]. Then, take E!= E˜ / ∼ with ∼ the equivalence relation with (0, t) ∼ (1, -t). (3) The tangent bundle of Sn. Recall Sn = x 2 Rn+1 : jxj = 1 . Define TSn := (x, v) : x 2 Sn, v 2 Rn+1, x · v = 0 . We certainly have a projection map TSn Sn where the fibers are vector spaces n ∼ ? n+1 (TS )x = x ⊂ R . We shall now check local triviality, which we shall often not do in the future. Consider ! n U := fx : x 2 S , xn+1 > 0g . We have an inclusion TU TSn Over U this is trivial with ! TU U × Rn (x, v) 7 (x, π(v)) n+1 n n+1 ! with π : R R = x 2 R : xn+1 = 0 . (4) If X ⊂ Rn is a smooth manifold, then! TX = f(!x, v) ⊂ X × Rn : x 2 X, v 2 Rn, v is tangent to X at xg . (5) The normal bundle to X, ν(X) = f(x, v) : v is orthogonal to all vectors tangent to X at x g . Definition 2.6. Recall, a section of E X is a map s : X E where φ ◦ s = idX. Lemma 2.7. The M¨obiusbundle is not isomorphic to the trivial vector bundle on the circle. ! ! We give two proofs. Proof 1. The Mobius¨ bundle is not orientable, but the trivial bundle is, as can be seen by determining whether the bundle remains connected or is disconnected, after removing the image of the 0 section. Proof 2. The Mobius¨ bundle is not trivial because it has no nonvanishing section. as can be seen by the intermediate value theorem. Definition 2.8. Two vector bundles p : E X, q : F X are isomorphic over X if φ E F ! ! (2.2) p q X ALGEBRAIC TOPOLOGY: MATH 231BR NOTES 7 commutes, where φ is a homeomorphism and φjEx : Ex Fx is a linear isomorphism. ! 2.3. Tautological bundles on projective spaces and Grassmannians. So far we’ve only looked at real vector bundles, but we will now consider complex ones. n n Definition 2.9. Let CP , or PC denote x : x ⊂ Cn+1 : x is a 1 dimensional linear subspace Definition 2.10. The tautological bundle over CPn be the bundle L := f(x, v) : x 2 CPn, v 2 xg where x is thought of simultaneously as a 1-dimensional space in Cn+1 and a point of CPn.
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