Harvard University Math 231br Notes for Advanced Algebraic Topology Reuben Stern Spring Semester 2017 Contents 0.1 Preliminaries .................................. 4 0.2 Administrative Stuff .............................. 4 1 January 23, 2017 5 1.1 Overview of Course............................... 5 2 January 25, 2017 6 2.1 Some facts about the Category Spaces .................... 6 2.2 Example: Locality and Sheafiness....................... 6 2.3 Example: Products of Spaces ......................... 7 2.4 Example: Gluings (or Quotient Spaces).................... 7 2.5 Example: Space of Functions.......................... 8 2.6 Pairs of Spaces ................................. 9 3 January 27, 2017 11 3.1 The Punchline from Last Class ........................ 11 3.2 Perpsectives on the Fundamental Group ................... 11 4 January 30, 2017 15 4.1 Higher Homotopy Groups ........................... 15 4.2 Exact Sequences of Spaces........................... 16 5 February 1, 2017 18 5.1 Finishing Things from Last Time ....................... 18 5.2 Dual Results: Exact Sequences ........................ 18 5.3 Relative Homotopy Groups........................... 19 6 February 3, 2017 21 6.1 The Action of the Fundamental Group.................... 21 7 February 6, 2017 22 7.1 Fibrations.................................... 22 8 February 8, 2017 23 8.1 Fiber Bundles and Examples.......................... 23 9 February 10, 2017 24 9.1 CW Complexes................................. 24 10 February 13, 2017 25 10.1 The Homotopy Theory of CW Complexes .................. 25 10.2 Homotopy Excision and Corollaries...................... 25 1 Math 231br CONTENTS 10.3 Introduction to Stable Homotopy....................... 25 11 February 15, 2017 26 11.1 Why Model Categories? ............................ 26 11.2 Definitions, Examples, and Basic Properties ................. 28 11.3 Homotopy Relations .............................. 31 12 February 17, 2017 33 12.1 The Homotopy Category of a Model Category................ 33 12.2 Brief Introduction to Derived Functors.................... 38 13 February 22, 2017 40 13.1 The Homotopy Theory of CW Complexes, II................. 40 13.2 Eilenberg-MacLane Spaces........................... 40 14 February 24, 2017 41 14.1 Brown Representability............................. 41 15 February 27, 2017 44 15.1 Spectra ..................................... 44 15.2 The Category of Spectra............................ 45 16 March 1, 2017 47 16.1 Co/homology Theories from Spectra ..................... 47 17 March 3, 2017 51 17.1 The Hurewicz Theorem............................. 51 17.2 Spectral Sequences ............................... 51 18 March 6, 2017 54 18.1 Obstruction Theory............................... 54 19 March 8, 2017 55 19.1 A Complicated Example............................ 55 20 March 10, 2017 56 20.1 Smash Products................................. 56 21 March 20, 2017 57 21.1 The Serre Spectral Sequence.......................... 57 22 March 22, 2017 58 22.1 More About the Serre Spectral Sequence................... 58 22.2 The Transgressive Differential Lemma..................... 58 22.3 Multiplicative Structure ............................ 58 Reuben Stern 2 Spring 2017 Math 231br CONTENTS 23 March 24, 2017 59 23.1 More Spectral Sequences............................ 59 23.2 Multiplicative Extensions............................ 59 ∗ 23.3 Computing H (K(Z/2, 1); F2) ......................... 59 24 March 27, 2017 60 24.1 G-bundles and Fiber Bundles ......................... 60 25 Properties of Chern Classes 64 26 The Steenrod Algebra and the Bar Spectral Sequence 65 27 April 10, 2017 66 27.1 The Steenrod Algebra, part II......................... 66 28 April 12, 2017 67 28.1 A Serre Spectral Sequence Trick........................ 67 28.2 Serre Classes .................................. 67 29 April 14, 2017 68 29.1 Serre’s Method ................................. 68 30 April 17, 2017 69 30.1 The Adams Spectral Sequence......................... 69 30.2 A Fun Computation .............................. 69 A Collected Homework Problems 71 A.1 Problem Set 1.................................. 71 A.2 Problem Set 2.................................. 72 A.3 Problem Set 3.................................. 73 A.4 Problem Set 4.................................. 73 A.5 Problem Set 5.................................. 77 B Notes from Switzer 79 References 80 Reuben Stern 3 Spring 2017 Math 231br 0.1 Preliminaries 0.1 Preliminaries These notes were taken during the Spring semester of 2017, in Harvard’s Math 231br, Advanced Algebraic Topology. The course was taught by Eric Peterson, and met Mon- day/Wednesday/Friday from 2 to 3 pm. Allow me to elucidate the process for taking these notes: I take notes by hand during lecture, which I transfer to LATEX at night. It is an unfortunate consequence of this method that these notes do not capture the unique lecturing style of the professor. Indeed, I take full responsibility for any errors in exposition or mathematics, but all credit for genuinely clever remarks, proofs, or exposition will be due to the professor (and not to the scribe). In an appendix at the end of this document, you will find the collected homework problems (with solutions). I make no promises regarding the correctness of these solutions; consider yourself warned. Please send any and all corrections to [email protected]. They will be most appreciated. A rough syllabus for the course is, in Eric’s own words, to “cover some initial segment of the book.” To explain, the course plans to discuss general homology and cohomology theories, bordism homology and cohomology, stable homotopy groups of spheres, and spectral sequences as computational tools for homotopy groups. 0.2 Administrative Stuff The grading of the course is as follows: weekly-ish problem sets constitute about one third, a midterm paper is another third, and a final paper will be the final third. Reuben Stern 4 Spring 2017 Math 231br January 23, 2017 1 January 23, 2017 Didn’t attend lecture. Went over course logistics and outlined the focus of the class. 1.1 Overview of Course The goal of this class is to give an introduction to homotopy theory (of spaces). This is four-part: 1. Decomposition of spaces (co/fiber sequences). 2. Invariants constructed from decompositions (H∗, π∗) and their properties (theorems of Whitehead and Hurewicz, etc.). 3. Representability theorems (Brown, Adams) and the stable category (HZ, KU and KO, S, HG, etc.) 4. Computation (characteristic classes, Bott periodicity, the Steenrod operations, the Adams and Serre spectral sequences, etc.) As an example of the kind of analysis we can perform with these tools at our disposal, start with your favorite simply connected space, like Sn≥2. Its homotopy groups are notoriously difficult and important to compute. Theorem 1.1. (Hurewicz) If X is (n − 1)-connected (that is, πkX = 0 for all k < n), ∼ then Hn(X; Z) = πnX. We also have a decomposition X[n + 1, ∞] X K(πnX, n) by a fiber sequence such that πnX k = n πkX n + 1 ≤ k πkK(πnX, n) = , πkX[n + 1, ∞] = . 0 otherwise 0 otherwise Then we can also plug the data of H∗X and H∗K(πnX, n) into a gadget known as the Serre spectral sequence to get H∗X[n + 1, ∞]. Reuben Stern 5 Spring 2017 Math 231br January 25, 2017 2 January 25, 2017 This is the first lecture I attended. I noticed that the class was 75% graduate students, and got a little frightened. Need not fear though, as Eric is a warm and inviting lecturer! 2.1 Some facts about the Category Spaces Eric: “I can’t make it through a lecture without saying ‘category’ at least five times.” The goal of this lecture is to give a couple of technical lemmas and some basic con- structions, so that we don’t have to mention them explicitly ever again. There will be four examples in this class: a sheaf condition on a decomposition of a topological space, product spaces, gluings (or quotient spaces), and the space Y X of functions X → Y . 2.2 Example: Locality and Sheafiness S Let X be a topological space and X = j Aj a decomposition of X into a collection of locally finite closed subsets. Then a continuous function f : X → T (for an arbitrary space T ) is the same data as continuous functions fj : Aj → T agreeing on the overlap. We picture this as follows: Make pic- Equivalently, we demand fi|Ai∩Aj = fj|Ai∩Aj . This condition is called locality or a sheaf condition on the collection {fi}. For intution on where the term “sheaf condition” ture of comes from, here is an aside on sheaves! overlap Aside 2.1. (Sheaves.) A sheaf is an assignment from subsets (usually taken to be open) of a space X to arbitrary sets, together with restriction maps: if B ⊆ A ⊆ X, we get a map resA,B : F (A) → F (B).(Example: take F (A) = {continuous functions A → T }.) We ask that these data satisfy: • The restriction maps commute nicely: if C ⊆ B ⊆ A ⊆ X, then resB,C ◦ resA,B = resA,C . • For a cover {Aj} of X, we think about the following diagram: res res Aj ,Ak∩Aj X,Aj Q Q F (X) j F (Aj) k,` F (Ak ∩ A`) res Aj ,Aj ∩A` The sheaf condition is then equivalent to the leftmost map equalizing the right hand side. And before we move on, an aside on equalizers! As Eric keeps pointing out, this class will be very category-heavy. Reuben Stern 6 Spring 2017 Math 231br 2.3 Example: Products of Spaces f Aside 2.2. (Equalizers.) An equalizer (of sets) E ⊆ S T is the maximal subset g of S on which f and g restrict to the same function. Lemma 2.3. Suppose that F −−→h S is some function such that f ◦ h = g ◦ h. Then there is a unique factorization of h as F h S i ∃! E To conclude our previous example, the assignment F (A) = {continuous functions f : A → T } forms a sheaf. 2.3 Example: Products of Spaces For any two spaces X and Y , there is a space X × Y with the following property: any map T → X × Y is the same data as pairs of maps f : T → X and g; T → Y . If we write Spaces for the category of spaces1, then we write Spaces(T,X × Y ) to mean the set of continuous functions T → X × Y .